Henrywood and Agarwal, Equation (12)

Percentage Accurate: 36.1% → 73.4%

Time: 12.6s

Alternatives: 23

Speedup: 10.2×

Specification

Math

\[\begin{array}{l} \\ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 36.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 73.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left(\ell \cdot d\_m\right) \cdot t\_0\\ t_2 := \frac{D\_m \cdot M\_m}{d\_m + d\_m}\\ \mathbf{if}\;\ell \leq -2:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot D\_m}{t\_1}, -0.125, \frac{d\_m}{t\_0}\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+67}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(t\_2 \cdot t\_2\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)}{t\_1} \cdot -0.125, h, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))
        (t_1 (* (* l d_m) t_0))
        (t_2 (/ (* D_m M_m) (+ d_m d_m))))
   (if (<= l -2.0)
     (fma (/ (* (* (* (* M_m M_m) h) D_m) D_m) t_1) -0.125 (/ d_m t_0))
     (if (<= l 6e+67)
       (*
        (/ (fabs d_m) (sqrt (* h l)))
        (- 1.0 (* (* (* t_2 t_2) 0.5) (/ h l))))
       (fma
        (* (/ (* (* D_m M_m) (* D_m M_m)) t_1) -0.125)
        h
        (* (/ d_m (* (sqrt l) (sqrt h))) 1.0))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double t_1 = (l * d_m) * t_0;
	double t_2 = (D_m * M_m) / (d_m + d_m);
	double tmp;
	if (l <= -2.0) {
		tmp = fma((((((M_m * M_m) * h) * D_m) * D_m) / t_1), -0.125, (d_m / t_0));
	} else if (l <= 6e+67) {
		tmp = (fabs(d_m) / sqrt((h * l))) * (1.0 - (((t_2 * t_2) * 0.5) * (h / l)));
	} else {
		tmp = fma(((((D_m * M_m) * (D_m * M_m)) / t_1) * -0.125), h, ((d_m / (sqrt(l) * sqrt(h))) * 1.0));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(Float64(l * d_m) * t_0)
	t_2 = Float64(Float64(D_m * M_m) / Float64(d_m + d_m))
	tmp = 0.0
	if (l <= -2.0)
		tmp = fma(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * D_m) / t_1), -0.125, Float64(d_m / t_0));
	elseif (l <= 6e+67)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(t_2 * t_2) * 0.5) * Float64(h / l))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(D_m * M_m) * Float64(D_m * M_m)) / t_1) * -0.125), h, Float64(Float64(d_m / Float64(sqrt(l) * sqrt(h))) * 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(l * d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.0], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / t$95$1), $MachinePrecision] * -0.125 + N[(d$95$m / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6e+67], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * 0.5), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * -0.125), $MachinePrecision] * h + N[(N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in h around 0

       \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} + \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\color{blue}{\sqrt{h}} \cdot \sqrt{\ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\sqrt{h \cdot \ell}} \]

       3. mult-flip N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       4. metadata-eval N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)}, \frac{-1}{8}, d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   12. Applied rewrites 60.0%

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}}, \color{blue}{-0.125}, \frac{d}{\sqrt{\ell \cdot h}}\right) \]

   if -2 < l < 6.0000000000000002e67

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(M \cdot \frac{D}{\color{blue}{d + d}}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(M \cdot \color{blue}{\frac{D}{d + d}}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\color{blue}{\left(M \cdot \frac{D}{d + d}\right)} \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. associate-/l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\color{blue}{\frac{M \cdot D}{d + d}} \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{\color{blue}{D \cdot M}}{d + d} \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\color{blue}{\frac{D \cdot M}{d + d}} \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{\color{blue}{D \cdot M}}{d + d} \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-+.f64 68.5

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{\color{blue}{d + d}} \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

   7. Applied rewrites 68.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\color{blue}{\frac{D \cdot M}{d + d}} \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d + d} \cdot \left(M \cdot \frac{D}{\color{blue}{d + d}}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d + d} \cdot \left(M \cdot \color{blue}{\frac{D}{d + d}}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d + d} \cdot \color{blue}{\left(M \cdot \frac{D}{d + d}\right)}\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. associate-/l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d + d} \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d + d} \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d + d} \cdot \color{blue}{\frac{D \cdot M}{d + d}}\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d + d} \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-+.f64 69.8

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d + d} \cdot \frac{D \cdot M}{\color{blue}{d + d}}\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

   9. Applied rewrites 69.8%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d + d} \cdot \color{blue}{\frac{D \cdot M}{d + d}}\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \]

   if 6.0000000000000002e67 < l

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot \ell\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot \ell\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. lower-*.f64 36.7

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   9. Applied rewrites 36.7%

      \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 73.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left(\ell \cdot d\_m\right) \cdot t\_0\\ t_2 := M\_m \cdot \frac{D\_m}{d\_m + d\_m}\\ \mathbf{if}\;\ell \leq -2:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot D\_m}{t\_1}, -0.125, \frac{d\_m}{t\_0}\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+67}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(t\_2 \cdot t\_2\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)}{t\_1} \cdot -0.125, h, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))
        (t_1 (* (* l d_m) t_0))
        (t_2 (* M_m (/ D_m (+ d_m d_m)))))
   (if (<= l -2.0)
     (fma (/ (* (* (* (* M_m M_m) h) D_m) D_m) t_1) -0.125 (/ d_m t_0))
     (if (<= l 6e+67)
       (*
        (/ (fabs d_m) (sqrt (* h l)))
        (- 1.0 (* (* (* t_2 t_2) 0.5) (/ h l))))
       (fma
        (* (/ (* (* D_m M_m) (* D_m M_m)) t_1) -0.125)
        h
        (* (/ d_m (* (sqrt l) (sqrt h))) 1.0))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double t_1 = (l * d_m) * t_0;
	double t_2 = M_m * (D_m / (d_m + d_m));
	double tmp;
	if (l <= -2.0) {
		tmp = fma((((((M_m * M_m) * h) * D_m) * D_m) / t_1), -0.125, (d_m / t_0));
	} else if (l <= 6e+67) {
		tmp = (fabs(d_m) / sqrt((h * l))) * (1.0 - (((t_2 * t_2) * 0.5) * (h / l)));
	} else {
		tmp = fma(((((D_m * M_m) * (D_m * M_m)) / t_1) * -0.125), h, ((d_m / (sqrt(l) * sqrt(h))) * 1.0));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(Float64(l * d_m) * t_0)
	t_2 = Float64(M_m * Float64(D_m / Float64(d_m + d_m)))
	tmp = 0.0
	if (l <= -2.0)
		tmp = fma(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * D_m) / t_1), -0.125, Float64(d_m / t_0));
	elseif (l <= 6e+67)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(t_2 * t_2) * 0.5) * Float64(h / l))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(D_m * M_m) * Float64(D_m * M_m)) / t_1) * -0.125), h, Float64(Float64(d_m / Float64(sqrt(l) * sqrt(h))) * 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(l * d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(M$95$m * N[(D$95$m / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.0], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / t$95$1), $MachinePrecision] * -0.125 + N[(d$95$m / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6e+67], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * 0.5), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * -0.125), $MachinePrecision] * h + N[(N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in h around 0

       \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} + \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\color{blue}{\sqrt{h}} \cdot \sqrt{\ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\sqrt{h \cdot \ell}} \]

       3. mult-flip N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       4. metadata-eval N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)}, \frac{-1}{8}, d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   12. Applied rewrites 60.0%

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}}, \color{blue}{-0.125}, \frac{d}{\sqrt{\ell \cdot h}}\right) \]

   if -2 < l < 6.0000000000000002e67

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)} \]

   if 6.0000000000000002e67 < l

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot \ell\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot \ell\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. lower-*.f64 36.7

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   9. Applied rewrites 36.7%

      \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left(\ell \cdot d\_m\right) \cdot t\_0\\ t_2 := \left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\\ \mathbf{if}\;\ell \leq -2:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot D\_m}{t\_1}, -0.125, \frac{d\_m}{t\_0}\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-166}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(\frac{t\_2}{\left(d\_m + d\_m\right) \cdot \left(d\_m + d\_m\right)} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_2}{t\_1} \cdot -0.125, h, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))
        (t_1 (* (* l d_m) t_0))
        (t_2 (* (* D_m M_m) (* D_m M_m))))
   (if (<= l -2.0)
     (fma (/ (* (* (* (* M_m M_m) h) D_m) D_m) t_1) -0.125 (/ d_m t_0))
     (if (<= l 9.5e-166)
       (*
        (/ (fabs d_m) (sqrt (* h l)))
        (- 1.0 (/ (* (* (/ t_2 (* (+ d_m d_m) (+ d_m d_m))) 0.5) h) l)))
       (fma (* (/ t_2 t_1) -0.125) h (* (/ d_m (* (sqrt l) (sqrt h))) 1.0))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double t_1 = (l * d_m) * t_0;
	double t_2 = (D_m * M_m) * (D_m * M_m);
	double tmp;
	if (l <= -2.0) {
		tmp = fma((((((M_m * M_m) * h) * D_m) * D_m) / t_1), -0.125, (d_m / t_0));
	} else if (l <= 9.5e-166) {
		tmp = (fabs(d_m) / sqrt((h * l))) * (1.0 - ((((t_2 / ((d_m + d_m) * (d_m + d_m))) * 0.5) * h) / l));
	} else {
		tmp = fma(((t_2 / t_1) * -0.125), h, ((d_m / (sqrt(l) * sqrt(h))) * 1.0));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(Float64(l * d_m) * t_0)
	t_2 = Float64(Float64(D_m * M_m) * Float64(D_m * M_m))
	tmp = 0.0
	if (l <= -2.0)
		tmp = fma(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * D_m) / t_1), -0.125, Float64(d_m / t_0));
	elseif (l <= 9.5e-166)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_2 / Float64(Float64(d_m + d_m) * Float64(d_m + d_m))) * 0.5) * h) / l)));
	else
		tmp = fma(Float64(Float64(t_2 / t_1) * -0.125), h, Float64(Float64(d_m / Float64(sqrt(l) * sqrt(h))) * 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(l * d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.0], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / t$95$1), $MachinePrecision] * -0.125 + N[(d$95$m / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.5e-166], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$2 / N[(N[(d$95$m + d$95$m), $MachinePrecision] * N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 / t$95$1), $MachinePrecision] * -0.125), $MachinePrecision] * h + N[(N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in h around 0

       \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} + \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\color{blue}{\sqrt{h}} \cdot \sqrt{\ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\sqrt{h \cdot \ell}} \]

       3. mult-flip N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       4. metadata-eval N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)}, \frac{-1}{8}, d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   12. Applied rewrites 60.0%

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}}, \color{blue}{-0.125}, \frac{d}{\sqrt{\ell \cdot h}}\right) \]

   if -2 < l < 9.50000000000000046e-166

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\ell}}\right) \]

   7. Applied rewrites 61.3%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d + d\right) \cdot \left(d + d\right)} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   if 9.50000000000000046e-166 < l

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot \ell\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot \ell\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. lower-*.f64 36.7

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   9. Applied rewrites 36.7%

      \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 70.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\\ t_1 := \sqrt{\ell \cdot h}\\ t_2 := \left(\ell \cdot d\_m\right) \cdot t\_1\\ \mathbf{if}\;\ell \leq -2:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot D\_m}{t\_2}, -0.125, \frac{d\_m}{t\_1}\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{t\_0}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot -0.125, h, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{t\_2} \cdot -0.125, h, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (* D_m M_m) (* D_m M_m)))
        (t_1 (sqrt (* l h)))
        (t_2 (* (* l d_m) t_1)))
   (if (<= l -2.0)
     (fma (/ (* (* (* (* M_m M_m) h) D_m) D_m) t_2) -0.125 (/ d_m t_1))
     (if (<= l 1.15e-170)
       (*
        (/ (fabs d_m) (sqrt (* h l)))
        (fma (* (/ t_0 (* (* d_m d_m) l)) -0.125) h 1.0))
       (fma (* (/ t_0 t_2) -0.125) h (* (/ d_m (* (sqrt l) (sqrt h))) 1.0))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m * M_m) * (D_m * M_m);
	double t_1 = sqrt((l * h));
	double t_2 = (l * d_m) * t_1;
	double tmp;
	if (l <= -2.0) {
		tmp = fma((((((M_m * M_m) * h) * D_m) * D_m) / t_2), -0.125, (d_m / t_1));
	} else if (l <= 1.15e-170) {
		tmp = (fabs(d_m) / sqrt((h * l))) * fma(((t_0 / ((d_m * d_m) * l)) * -0.125), h, 1.0);
	} else {
		tmp = fma(((t_0 / t_2) * -0.125), h, ((d_m / (sqrt(l) * sqrt(h))) * 1.0));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m * M_m) * Float64(D_m * M_m))
	t_1 = sqrt(Float64(l * h))
	t_2 = Float64(Float64(l * d_m) * t_1)
	tmp = 0.0
	if (l <= -2.0)
		tmp = fma(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * D_m) / t_2), -0.125, Float64(d_m / t_1));
	elseif (l <= 1.15e-170)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(h * l))) * fma(Float64(Float64(t_0 / Float64(Float64(d_m * d_m) * l)) * -0.125), h, 1.0));
	else
		tmp = fma(Float64(Float64(t_0 / t_2) * -0.125), h, Float64(Float64(d_m / Float64(sqrt(l) * sqrt(h))) * 1.0));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * d$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[l, -2.0], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / t$95$2), $MachinePrecision] * -0.125 + N[(d$95$m / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.15e-170], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 / t$95$2), $MachinePrecision] * -0.125), $MachinePrecision] * h + N[(N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in h around 0

       \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} + \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\color{blue}{\sqrt{h}} \cdot \sqrt{\ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\sqrt{h \cdot \ell}} \]

       3. mult-flip N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       4. metadata-eval N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)}, \frac{-1}{8}, d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   12. Applied rewrites 60.0%

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}}, \color{blue}{-0.125}, \frac{d}{\sqrt{\ell \cdot h}}\right) \]

   if -2 < l < 1.14999999999999993e-170

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)} \]

   6. Taylor expanded in h around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)\right)} \]

   8. Applied rewrites 60.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, 1\right)} \]

   if 1.14999999999999993e-170 < l

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in d around 0

      \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot \ell\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot \ell\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. lower-*.f64 36.7

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   9. Applied rewrites 36.7%

      \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 68.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \frac{d\_m}{t\_0}\\ \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{-178}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 1.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot -0.125, h, 1\right)\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot D\_m}{\left(\ell \cdot d\_m\right) \cdot t\_0}, -0.125, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot \frac{M\_m \cdot M\_m}{\frac{\ell \cdot d\_m}{t\_0}}\right) \cdot -0.125}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))) (t_1 (/ d_m t_0)))
   (if (<= (* M_m D_m) 5e-178)
     t_1
     (if (<= (* M_m D_m) 1.5e+106)
       (*
        (/ (fabs d_m) (sqrt (* h l)))
        (fma
         (* (/ (* (* D_m M_m) (* D_m M_m)) (* (* d_m d_m) l)) -0.125)
         h
         1.0))
       (if (<= (* M_m D_m) 1e+202)
         (fma
          (/ (* (* (* (* M_m M_m) h) D_m) D_m) (* (* l d_m) t_0))
          -0.125
          t_1)
         (/
          (* (* (* D_m D_m) (/ (* M_m M_m) (/ (* l d_m) t_0))) -0.125)
          l))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double t_1 = d_m / t_0;
	double tmp;
	if ((M_m * D_m) <= 5e-178) {
		tmp = t_1;
	} else if ((M_m * D_m) <= 1.5e+106) {
		tmp = (fabs(d_m) / sqrt((h * l))) * fma(((((D_m * M_m) * (D_m * M_m)) / ((d_m * d_m) * l)) * -0.125), h, 1.0);
	} else if ((M_m * D_m) <= 1e+202) {
		tmp = fma((((((M_m * M_m) * h) * D_m) * D_m) / ((l * d_m) * t_0)), -0.125, t_1);
	} else {
		tmp = (((D_m * D_m) * ((M_m * M_m) / ((l * d_m) / t_0))) * -0.125) / l;
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(d_m / t_0)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e-178)
		tmp = t_1;
	elseif (Float64(M_m * D_m) <= 1.5e+106)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(h * l))) * fma(Float64(Float64(Float64(Float64(D_m * M_m) * Float64(D_m * M_m)) / Float64(Float64(d_m * d_m) * l)) * -0.125), h, 1.0));
	elseif (Float64(M_m * D_m) <= 1e+202)
		tmp = fma(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * D_m) / Float64(Float64(l * d_m) * t_0)), -0.125, t_1);
	else
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * Float64(Float64(M_m * M_m) / Float64(Float64(l * d_m) / t_0))) * -0.125) / l);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(d$95$m / t$95$0), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e-178], t$95$1, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1.5e+106], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+202], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * -0.125 + t$95$1), $MachinePrecision], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 M D) < 4.99999999999999976e-178

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 4.99999999999999976e-178 < (*.f64 M D) < 1.5e106

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)} \]

   6. Taylor expanded in h around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)\right)} \]

   8. Applied rewrites 60.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, 1\right)} \]

   if 1.5e106 < (*.f64 M D) < 9.999999999999999e201

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in h around 0

       \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} + \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\color{blue}{\sqrt{h}} \cdot \sqrt{\ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\sqrt{h \cdot \ell}} \]

       3. mult-flip N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       4. metadata-eval N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)}, \frac{-1}{8}, d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   12. Applied rewrites 60.0%

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}}, \color{blue}{-0.125}, \frac{d}{\sqrt{\ell \cdot h}}\right) \]

   if 9.999999999999999e201 < (*.f64 M D)

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in h around inf

       \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}}{\ell} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \frac{-1}{8}}{\ell} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \frac{-1}{8}}{\ell} \]

   12. Applied rewrites 29.3%

       \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{\frac{\ell \cdot d}{\sqrt{\ell \cdot h}}}\right) \cdot -0.125}{\ell} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)\\ t_1 := \sqrt{\ell \cdot h}\\ t_2 := \sqrt{\frac{\ell}{h}}\\ \mathbf{if}\;\ell \leq -2:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot D\_m}{\left(\ell \cdot d\_m\right) \cdot t\_1}, -0.125, \frac{d\_m}{t\_1}\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{-208}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{t\_0}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot -0.125, h, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{\left(t\_2 \cdot \ell\right) \cdot d\_m}, -0.125, \frac{d\_m}{t\_2 \cdot h}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (* D_m M_m) (* D_m M_m)))
        (t_1 (sqrt (* l h)))
        (t_2 (sqrt (/ l h))))
   (if (<= l -2.0)
     (fma
      (/ (* (* (* (* M_m M_m) h) D_m) D_m) (* (* l d_m) t_1))
      -0.125
      (/ d_m t_1))
     (if (<= l 6.2e-208)
       (*
        (/ (fabs d_m) (sqrt (* h l)))
        (fma (* (/ t_0 (* (* d_m d_m) l)) -0.125) h 1.0))
       (fma (/ t_0 (* (* t_2 l) d_m)) -0.125 (/ d_m (* t_2 h)))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m * M_m) * (D_m * M_m);
	double t_1 = sqrt((l * h));
	double t_2 = sqrt((l / h));
	double tmp;
	if (l <= -2.0) {
		tmp = fma((((((M_m * M_m) * h) * D_m) * D_m) / ((l * d_m) * t_1)), -0.125, (d_m / t_1));
	} else if (l <= 6.2e-208) {
		tmp = (fabs(d_m) / sqrt((h * l))) * fma(((t_0 / ((d_m * d_m) * l)) * -0.125), h, 1.0);
	} else {
		tmp = fma((t_0 / ((t_2 * l) * d_m)), -0.125, (d_m / (t_2 * h)));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m * M_m) * Float64(D_m * M_m))
	t_1 = sqrt(Float64(l * h))
	t_2 = sqrt(Float64(l / h))
	tmp = 0.0
	if (l <= -2.0)
		tmp = fma(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * D_m) / Float64(Float64(l * d_m) * t_1)), -0.125, Float64(d_m / t_1));
	elseif (l <= 6.2e-208)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(h * l))) * fma(Float64(Float64(t_0 / Float64(Float64(d_m * d_m) * l)) * -0.125), h, 1.0));
	else
		tmp = fma(Float64(t_0 / Float64(Float64(t_2 * l) * d_m)), -0.125, Float64(d_m / Float64(t_2 * h)));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.0], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(d$95$m / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.2e-208], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(N[(t$95$2 * l), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(d$95$m / N[(t$95$2 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in h around 0

       \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} + \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\color{blue}{\sqrt{h}} \cdot \sqrt{\ell}} \]

       2. sqrt-prod N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + \frac{d}{\sqrt{h \cdot \ell}} \]

       3. mult-flip N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       4. metadata-eval N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \]

       5. sqrt-div N/A

          \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} + d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)}, \frac{-1}{8}, d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   12. Applied rewrites 60.0%

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}}, \color{blue}{-0.125}, \frac{d}{\sqrt{\ell \cdot h}}\right) \]

   if -2 < l < 6.1999999999999996e-208

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)} \]

   6. Taylor expanded in h around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)\right)} \]

   8. Applied rewrites 60.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, 1\right)} \]

   if 6.1999999999999996e-208 < l

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in h around inf

      \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} + \color{blue}{\frac{d}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} + \frac{d}{\color{blue}{h} \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)}, \frac{-1}{8}, \frac{d}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)}\right) \]

   9. Applied rewrites 33.1%

      \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d}, \color{blue}{-0.125}, \frac{d}{\sqrt{\frac{\ell}{h}} \cdot h}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{-178}:\\ \;\;\;\;\frac{d\_m}{t\_0}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 10^{+143}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot -0.125, h, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot \frac{M\_m \cdot M\_m}{\frac{\ell \cdot d\_m}{t\_0}}\right) \cdot -0.125}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= (* M_m D_m) 5e-178)
     (/ d_m t_0)
     (if (<= (* M_m D_m) 1e+143)
       (*
        (/ (fabs d_m) (sqrt (* h l)))
        (fma
         (* (/ (* (* D_m M_m) (* D_m M_m)) (* (* d_m d_m) l)) -0.125)
         h
         1.0))
       (/ (* (* (* D_m D_m) (/ (* M_m M_m) (/ (* l d_m) t_0))) -0.125) l)))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if ((M_m * D_m) <= 5e-178) {
		tmp = d_m / t_0;
	} else if ((M_m * D_m) <= 1e+143) {
		tmp = (fabs(d_m) / sqrt((h * l))) * fma(((((D_m * M_m) * (D_m * M_m)) / ((d_m * d_m) * l)) * -0.125), h, 1.0);
	} else {
		tmp = (((D_m * D_m) * ((M_m * M_m) / ((l * d_m) / t_0))) * -0.125) / l;
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e-178)
		tmp = Float64(d_m / t_0);
	elseif (Float64(M_m * D_m) <= 1e+143)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(h * l))) * fma(Float64(Float64(Float64(Float64(D_m * M_m) * Float64(D_m * M_m)) / Float64(Float64(d_m * d_m) * l)) * -0.125), h, 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * Float64(Float64(M_m * M_m) / Float64(Float64(l * d_m) / t_0))) * -0.125) / l);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e-178], N[(d$95$m / t$95$0), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 1e+143], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 4.99999999999999976e-178

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 4.99999999999999976e-178 < (*.f64 M D) < 1e143

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)} \]

   6. Taylor expanded in h around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)\right)} \]

   8. Applied rewrites 60.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, 1\right)} \]

   if 1e143 < (*.f64 M D)

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in h around inf

       \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}}{\ell} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \frac{-1}{8}}{\ell} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \frac{-1}{8}}{\ell} \]

   12. Applied rewrites 29.3%

       \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{\frac{\ell \cdot d}{\sqrt{\ell \cdot h}}}\right) \cdot -0.125}{\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \leq 1.55 \cdot 10^{-84}:\\ \;\;\;\;\frac{d\_m}{t\_0}\\ \mathbf{elif}\;M\_m \leq 3.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\left(\frac{h}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot \left(M\_m \cdot M\_m\right)\right) \cdot -0.125, D\_m \cdot D\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot \frac{M\_m \cdot M\_m}{\frac{\ell \cdot d\_m}{t\_0}}\right) \cdot -0.125}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= M_m 1.55e-84)
     (/ d_m t_0)
     (if (<= M_m 3.6e+75)
       (*
        (/ (fabs d_m) (sqrt (* h l)))
        (fma
         (* (* (/ h (* (* d_m d_m) l)) (* M_m M_m)) -0.125)
         (* D_m D_m)
         1.0))
       (/ (* (* (* D_m D_m) (/ (* M_m M_m) (/ (* l d_m) t_0))) -0.125) l)))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (M_m <= 1.55e-84) {
		tmp = d_m / t_0;
	} else if (M_m <= 3.6e+75) {
		tmp = (fabs(d_m) / sqrt((h * l))) * fma((((h / ((d_m * d_m) * l)) * (M_m * M_m)) * -0.125), (D_m * D_m), 1.0);
	} else {
		tmp = (((D_m * D_m) * ((M_m * M_m) / ((l * d_m) / t_0))) * -0.125) / l;
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (M_m <= 1.55e-84)
		tmp = Float64(d_m / t_0);
	elseif (M_m <= 3.6e+75)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(h * l))) * fma(Float64(Float64(Float64(h / Float64(Float64(d_m * d_m) * l)) * Float64(M_m * M_m)) * -0.125), Float64(D_m * D_m), 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * Float64(Float64(M_m * M_m) / Float64(Float64(l * d_m) / t_0))) * -0.125) / l);
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 1.55e-84], N[(d$95$m / t$95$0), $MachinePrecision], If[LessEqual[M$95$m, 3.6e+75], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(h / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.55000000000000001e-84

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 1.55000000000000001e-84 < M < 3.6e75

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in D around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left({D}^{2} \cdot \left(\frac{1}{{D}^{2}} - \frac{1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\frac{1}{{D}^{2}} - \frac{1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \color{blue}{{D}^{2}}\right) \]

      2. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\frac{1}{{D}^{2}} - \frac{1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \left(D \cdot \color{blue}{D}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{1}{{D}^{2}} - \frac{1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{1}{{D}^{2}} - \frac{1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D}\right) \]

   4. Applied rewrites 22.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\left(\frac{1}{D \cdot D} - \left(\left(M \cdot M\right) \cdot \frac{h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot 0.125\right) \cdot D\right) \cdot D\right)} \]

   6. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\left(\frac{h}{\left(d \cdot d\right) \cdot \ell} \cdot \left(M \cdot M\right)\right) \cdot -0.125, D \cdot D, 1\right)} \]

   if 3.6e75 < M

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in h around inf

       \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}}{\ell} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \frac{-1}{8}}{\ell} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \frac{-1}{8}}{\ell} \]

   12. Applied rewrites 29.3%

       \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{\frac{\ell \cdot d}{\sqrt{\ell \cdot h}}}\right) \cdot -0.125}{\ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 58.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \leq 2.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{d\_m}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{1}{t\_0} \cdot h\right) \cdot \left(M\_m \cdot M\_m\right)\right) \cdot \left(D\_m \cdot D\_m\right)}{\ell \cdot d\_m} \cdot -0.125\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= M_m 2.5e-34)
     (/ d_m t_0)
     (*
      (/ (* (* (* (/ 1.0 t_0) h) (* M_m M_m)) (* D_m D_m)) (* l d_m))
      -0.125))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (M_m <= 2.5e-34) {
		tmp = d_m / t_0;
	} else {
		tmp = (((((1.0 / t_0) * h) * (M_m * M_m)) * (D_m * D_m)) / (l * d_m)) * -0.125;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (m_m <= 2.5d-34) then
        tmp = d_m / t_0
    else
        tmp = (((((1.0d0 / t_0) * h) * (m_m * m_m)) * (d_m_1 * d_m_1)) / (l * d_m)) * (-0.125d0)
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (M_m <= 2.5e-34) {
		tmp = d_m / t_0;
	} else {
		tmp = (((((1.0 / t_0) * h) * (M_m * M_m)) * (D_m * D_m)) / (l * d_m)) * -0.125;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if M_m <= 2.5e-34:
		tmp = d_m / t_0
	else:
		tmp = (((((1.0 / t_0) * h) * (M_m * M_m)) * (D_m * D_m)) / (l * d_m)) * -0.125
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (M_m <= 2.5e-34)
		tmp = Float64(d_m / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(1.0 / t_0) * h) * Float64(M_m * M_m)) * Float64(D_m * D_m)) / Float64(l * d_m)) * -0.125);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (M_m <= 2.5e-34)
		tmp = d_m / t_0;
	else
		tmp = (((((1.0 / t_0) * h) * (M_m * M_m)) * (D_m * D_m)) / (l * d_m)) * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 2.5e-34], N[(d$95$m / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * h), $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.5000000000000001e-34

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.5000000000000001e-34 < M

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 32.9%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \color{blue}{1} \]

   7. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \]

      3. pow-prod-down N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \]

      6. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \cdot \color{blue}{\frac{-1}{8}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \cdot \color{blue}{\frac{-1}{8}} \]

   9. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot h\right) \cdot \left(M \cdot M\right)\right) \cdot \left(D \cdot D\right)}{\ell \cdot d} \cdot -0.125} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(D\_m \cdot D\_m\right)}{\ell \cdot d\_m} \cdot -0.125\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= M_m 2.5e-34)
   (/ d_m (sqrt (* l h)))
   (*
    (/ (* (* (* (* M_m M_m) h) (sqrt (/ 1.0 (* h l)))) (* D_m D_m)) (* l d_m))
    -0.125)))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 2.5e-34) {
		tmp = d_m / sqrt((l * h));
	} else {
		tmp = (((((M_m * M_m) * h) * sqrt((1.0 / (h * l)))) * (D_m * D_m)) / (l * d_m)) * -0.125;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 2.5d-34) then
        tmp = d_m / sqrt((l * h))
    else
        tmp = (((((m_m * m_m) * h) * sqrt((1.0d0 / (h * l)))) * (d_m_1 * d_m_1)) / (l * d_m)) * (-0.125d0)
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 2.5e-34) {
		tmp = d_m / Math.sqrt((l * h));
	} else {
		tmp = (((((M_m * M_m) * h) * Math.sqrt((1.0 / (h * l)))) * (D_m * D_m)) / (l * d_m)) * -0.125;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	tmp = 0
	if M_m <= 2.5e-34:
		tmp = d_m / math.sqrt((l * h))
	else:
		tmp = (((((M_m * M_m) * h) * math.sqrt((1.0 / (h * l)))) * (D_m * D_m)) / (l * d_m)) * -0.125
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (M_m <= 2.5e-34)
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(D_m * D_m)) / Float64(l * d_m)) * -0.125);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	tmp = 0.0;
	if (M_m <= 2.5e-34)
		tmp = d_m / sqrt((l * h));
	else
		tmp = (((((M_m * M_m) * h) * sqrt((1.0 / (h * l)))) * (D_m * D_m)) / (l * d_m)) * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[M$95$m, 2.5e-34], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.5000000000000001e-34

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.5000000000000001e-34 < M

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell}} \]
   5. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \]

      2. pow1/2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \]

      4. pow1/2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \]

      6. *-commutative N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \]

      7. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \cdot \color{blue}{\frac{-1}{8}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell} \cdot \color{blue}{\frac{-1}{8}} \]

   6. Applied rewrites 31.2%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(D \cdot D\right)}{\ell \cdot d} \cdot -0.125} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \leq 2.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{d\_m}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D\_m \cdot D\_m\right) \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot h}{\left(\left|d\_m\right| \cdot t\_0\right) \cdot \ell}\right) \cdot -0.125\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= M_m 2.6e-34)
     (/ d_m t_0)
     (*
      (* (* D_m D_m) (/ (* (* M_m M_m) h) (* (* (fabs d_m) t_0) l)))
      -0.125))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (M_m <= 2.6e-34) {
		tmp = d_m / t_0;
	} else {
		tmp = ((D_m * D_m) * (((M_m * M_m) * h) / ((fabs(d_m) * t_0) * l))) * -0.125;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (m_m <= 2.6d-34) then
        tmp = d_m / t_0
    else
        tmp = ((d_m_1 * d_m_1) * (((m_m * m_m) * h) / ((abs(d_m) * t_0) * l))) * (-0.125d0)
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (M_m <= 2.6e-34) {
		tmp = d_m / t_0;
	} else {
		tmp = ((D_m * D_m) * (((M_m * M_m) * h) / ((Math.abs(d_m) * t_0) * l))) * -0.125;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if M_m <= 2.6e-34:
		tmp = d_m / t_0
	else:
		tmp = ((D_m * D_m) * (((M_m * M_m) * h) / ((math.fabs(d_m) * t_0) * l))) * -0.125
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (M_m <= 2.6e-34)
		tmp = Float64(d_m / t_0);
	else
		tmp = Float64(Float64(Float64(D_m * D_m) * Float64(Float64(Float64(M_m * M_m) * h) / Float64(Float64(abs(d_m) * t_0) * l))) * -0.125);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (M_m <= 2.6e-34)
		tmp = d_m / t_0;
	else
		tmp = ((D_m * D_m) * (((M_m * M_m) * h) / ((abs(d_m) * t_0) * l))) * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 2.6e-34], N[(d$95$m / t$95$0), $MachinePrecision], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(N[Abs[d$95$m], $MachinePrecision] * t$95$0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.5999999999999999e-34

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.5999999999999999e-34 < M

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{1}{d \cdot h}} \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{1}{d \cdot h}} \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)} \]

   4. Applied rewrites 18.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\left(\left(\left(\frac{\frac{h}{\sqrt{h \cdot d}}}{\sqrt{\ell \cdot d}} \cdot M\right) \cdot M\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell}, -0.125, \sqrt{\frac{1}{\left(h \cdot d\right) \cdot \left(\ell \cdot d\right)}}\right) \cdot d\right) \cdot d} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \left(\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \left(\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \left(\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\right)} \cdot \frac{-1}{8} \]

   7. Applied rewrites 24.1%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\sqrt{h \cdot \left(\left(d \cdot d\right) \cdot \ell\right)} \cdot \ell} \cdot \color{blue}{-0.125} \]

   9. Applied rewrites 30.4%

      \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(\left|d\right| \cdot \sqrt{\ell \cdot h}\right) \cdot \ell}\right) \cdot -0.125 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 58.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \leq 3.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{d\_m}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(D\_m \cdot D\_m\right) \cdot \frac{M\_m \cdot M\_m}{\frac{\ell \cdot d\_m}{t\_0}}\right) \cdot -0.125}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= M_m 3.8e-33)
     (/ d_m t_0)
     (/ (* (* (* D_m D_m) (/ (* M_m M_m) (/ (* l d_m) t_0))) -0.125) l))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (M_m <= 3.8e-33) {
		tmp = d_m / t_0;
	} else {
		tmp = (((D_m * D_m) * ((M_m * M_m) / ((l * d_m) / t_0))) * -0.125) / l;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (m_m <= 3.8d-33) then
        tmp = d_m / t_0
    else
        tmp = (((d_m_1 * d_m_1) * ((m_m * m_m) / ((l * d_m) / t_0))) * (-0.125d0)) / l
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (M_m <= 3.8e-33) {
		tmp = d_m / t_0;
	} else {
		tmp = (((D_m * D_m) * ((M_m * M_m) / ((l * d_m) / t_0))) * -0.125) / l;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if M_m <= 3.8e-33:
		tmp = d_m / t_0
	else:
		tmp = (((D_m * D_m) * ((M_m * M_m) / ((l * d_m) / t_0))) * -0.125) / l
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (M_m <= 3.8e-33)
		tmp = Float64(d_m / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(D_m * D_m) * Float64(Float64(M_m * M_m) / Float64(Float64(l * d_m) / t_0))) * -0.125) / l);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (M_m <= 3.8e-33)
		tmp = d_m / t_0;
	else
		tmp = (((D_m * D_m) * ((M_m * M_m) / ((l * d_m) / t_0))) * -0.125) / l;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 3.8e-33], N[(d$95$m / t$95$0), $MachinePrecision], N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 3.79999999999999994e-33

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 3.79999999999999994e-33 < M

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in h around inf

       \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}}{\ell} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \frac{-1}{8}}{\ell} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \frac{-1}{8}}{\ell} \]

   12. Applied rewrites 29.3%

       \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{\frac{\ell \cdot d}{\sqrt{\ell \cdot h}}}\right) \cdot -0.125}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \leq 2.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{d\_m}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot D\_m}{t\_0 \cdot d\_m} \cdot -0.125}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= M_m 2.5e-34)
     (/ d_m t_0)
     (/ (* (/ (* (* (* (* M_m M_m) h) D_m) D_m) (* t_0 d_m)) -0.125) l))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (M_m <= 2.5e-34) {
		tmp = d_m / t_0;
	} else {
		tmp = ((((((M_m * M_m) * h) * D_m) * D_m) / (t_0 * d_m)) * -0.125) / l;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (m_m <= 2.5d-34) then
        tmp = d_m / t_0
    else
        tmp = ((((((m_m * m_m) * h) * d_m_1) * d_m_1) / (t_0 * d_m)) * (-0.125d0)) / l
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (M_m <= 2.5e-34) {
		tmp = d_m / t_0;
	} else {
		tmp = ((((((M_m * M_m) * h) * D_m) * D_m) / (t_0 * d_m)) * -0.125) / l;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if M_m <= 2.5e-34:
		tmp = d_m / t_0
	else:
		tmp = ((((((M_m * M_m) * h) * D_m) * D_m) / (t_0 * d_m)) * -0.125) / l
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (M_m <= 2.5e-34)
		tmp = Float64(d_m / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * D_m) / Float64(t_0 * d_m)) * -0.125) / l);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (M_m <= 2.5e-34)
		tmp = d_m / t_0;
	else
		tmp = ((((((M_m * M_m) * h) * D_m) * D_m) / (t_0 * d_m)) * -0.125) / l;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 2.5e-34], N[(d$95$m / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(t$95$0 * d$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.5000000000000001e-34

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.5000000000000001e-34 < M

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in l around 0

       \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \sqrt{h \cdot \ell}}}{\ell} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}}{\ell} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8}}{\ell} \]

   12. Applied rewrites 32.0%

       \[\leadsto \frac{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{\sqrt{\ell \cdot h} \cdot d} \cdot -0.125}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \leq 2.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{d\_m}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot D\_m}{\left(\ell \cdot d\_m\right) \cdot t\_0} \cdot -0.125\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= M_m 2.1e-34)
     (/ d_m t_0)
     (* (/ (* (* (* (* M_m M_m) h) D_m) D_m) (* (* l d_m) t_0)) -0.125))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (M_m <= 2.1e-34) {
		tmp = d_m / t_0;
	} else {
		tmp = (((((M_m * M_m) * h) * D_m) * D_m) / ((l * d_m) * t_0)) * -0.125;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (m_m <= 2.1d-34) then
        tmp = d_m / t_0
    else
        tmp = (((((m_m * m_m) * h) * d_m_1) * d_m_1) / ((l * d_m) * t_0)) * (-0.125d0)
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (M_m <= 2.1e-34) {
		tmp = d_m / t_0;
	} else {
		tmp = (((((M_m * M_m) * h) * D_m) * D_m) / ((l * d_m) * t_0)) * -0.125;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if M_m <= 2.1e-34:
		tmp = d_m / t_0
	else:
		tmp = (((((M_m * M_m) * h) * D_m) * D_m) / ((l * d_m) * t_0)) * -0.125
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (M_m <= 2.1e-34)
		tmp = Float64(d_m / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * D_m) / Float64(Float64(l * d_m) * t_0)) * -0.125);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (M_m <= 2.1e-34)
		tmp = d_m / t_0;
	else
		tmp = (((((M_m * M_m) * h) * D_m) * D_m) / ((l * d_m) * t_0)) * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 2.1e-34], N[(d$95$m / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.1000000000000001e-34

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.1000000000000001e-34 < M

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{1}{d \cdot h}} \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{1}{d \cdot h}} \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)} \]

   4. Applied rewrites 18.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\left(\left(\left(\frac{\frac{h}{\sqrt{h \cdot d}}}{\sqrt{\ell \cdot d}} \cdot M\right) \cdot M\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell}, -0.125, \sqrt{\frac{1}{\left(h \cdot d\right) \cdot \left(\ell \cdot d\right)}}\right) \cdot d\right) \cdot d} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \left(\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \left(\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot \left(\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\right)} \cdot \frac{-1}{8} \]

   7. Applied rewrites 24.1%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\sqrt{h \cdot \left(\left(d \cdot d\right) \cdot \ell\right)} \cdot \ell} \cdot \color{blue}{-0.125} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      3. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      4. pow2 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      11. pow2 N/A

          \[\leadsto \frac{\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      13. pow2 N/A

          \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8} \]

      16. associate-*r* N/A

          \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{\left(d \cdot \ell\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{\left(d \cdot \ell\right) \cdot \sqrt{h \cdot \ell}} \cdot \frac{-1}{8} \]

   10. Applied rewrites 31.6%

       \[\leadsto \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}} \cdot -0.125 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-214}:\\ \;\;\;\;\frac{\left(D\_m \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d\_m} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
       (-
        1.0
        (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0)) (/ h l))))
      -2e-214)
   (* (/ (* (* D_m M_m) (* D_m M_m)) (* (* (sqrt (/ l h)) l) d_m)) -0.125)
   (/ d_m (sqrt (* l h)))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -2e-214) {
		tmp = (((D_m * M_m) * (D_m * M_m)) / ((sqrt((l / h)) * l) * d_m)) * -0.125;
	} else {
		tmp = d_m / sqrt((l * h));
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m_1) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))) <= (-2d-214)) then
        tmp = (((d_m_1 * m_m) * (d_m_1 * m_m)) / ((sqrt((l / h)) * l) * d_m)) * (-0.125d0)
    else
        tmp = d_m / sqrt((l * h))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -2e-214) {
		tmp = (((D_m * M_m) * (D_m * M_m)) / ((Math.sqrt((l / h)) * l) * d_m)) * -0.125;
	} else {
		tmp = d_m / Math.sqrt((l * h));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -2e-214:
		tmp = (((D_m * M_m) * (D_m * M_m)) / ((math.sqrt((l / h)) * l) * d_m)) * -0.125
	else:
		tmp = d_m / math.sqrt((l * h))
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l)))) <= -2e-214)
		tmp = Float64(Float64(Float64(Float64(D_m * M_m) * Float64(D_m * M_m)) / Float64(Float64(sqrt(Float64(l / h)) * l) * d_m)) * -0.125);
	else
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0)) * (h / l)))) <= -2e-214)
		tmp = (((D_m * M_m) * (D_m * M_m)) / ((sqrt((l / h)) * l) * d_m)) * -0.125;
	else
		tmp = d_m / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-214], N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999983e-214

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in h around inf

      \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{{\left(D \cdot M\right)}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      5. unpow2 N/A

         \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot \sqrt{\frac{\ell}{h}}\right) \cdot d} \cdot \frac{-1}{8} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\ell \cdot \sqrt{\frac{\ell}{h}}\right) \cdot d} \cdot \frac{-1}{8} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot \frac{-1}{8} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot \frac{-1}{8} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot \frac{-1}{8} \]

      14. lower-/.f64 22.6

          \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot -0.125 \]

   9. Applied rewrites 22.6%

      \[\leadsto \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot \color{blue}{-0.125} \]

   if -1.99999999999999983e-214 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 46.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -9 \cdot 10^{+222}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{h}{\left(d\_m \cdot d\_m\right) \cdot \ell}}}{h} \cdot d\_m\right) \cdot d\_m\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\frac{-\frac{\sqrt{-h}}{\sqrt{-\ell}} \cdot d\_m}{h}\\ \mathbf{elif}\;h \leq 1.08 \cdot 10^{+46}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\_m\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= h -9e+222)
   (* (* (/ (sqrt (/ h (* (* d_m d_m) l))) h) d_m) d_m)
   (if (<= h -5e-292)
     (/ (- (* (/ (sqrt (- h)) (sqrt (- l))) d_m)) h)
     (if (<= h 1.08e+46)
       (/ d_m (sqrt (* l h)))
       (* (/ (* (fabs d_m) (sqrt (/ l h))) l) 1.0)))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -9e+222) {
		tmp = ((sqrt((h / ((d_m * d_m) * l))) / h) * d_m) * d_m;
	} else if (h <= -5e-292) {
		tmp = -((sqrt(-h) / sqrt(-l)) * d_m) / h;
	} else if (h <= 1.08e+46) {
		tmp = d_m / sqrt((l * h));
	} else {
		tmp = ((fabs(d_m) * sqrt((l / h))) / l) * 1.0;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (h <= (-9d+222)) then
        tmp = ((sqrt((h / ((d_m * d_m) * l))) / h) * d_m) * d_m
    else if (h <= (-5d-292)) then
        tmp = -((sqrt(-h) / sqrt(-l)) * d_m) / h
    else if (h <= 1.08d+46) then
        tmp = d_m / sqrt((l * h))
    else
        tmp = ((abs(d_m) * sqrt((l / h))) / l) * 1.0d0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -9e+222) {
		tmp = ((Math.sqrt((h / ((d_m * d_m) * l))) / h) * d_m) * d_m;
	} else if (h <= -5e-292) {
		tmp = -((Math.sqrt(-h) / Math.sqrt(-l)) * d_m) / h;
	} else if (h <= 1.08e+46) {
		tmp = d_m / Math.sqrt((l * h));
	} else {
		tmp = ((Math.abs(d_m) * Math.sqrt((l / h))) / l) * 1.0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	tmp = 0
	if h <= -9e+222:
		tmp = ((math.sqrt((h / ((d_m * d_m) * l))) / h) * d_m) * d_m
	elif h <= -5e-292:
		tmp = -((math.sqrt(-h) / math.sqrt(-l)) * d_m) / h
	elif h <= 1.08e+46:
		tmp = d_m / math.sqrt((l * h))
	else:
		tmp = ((math.fabs(d_m) * math.sqrt((l / h))) / l) * 1.0
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -9e+222)
		tmp = Float64(Float64(Float64(sqrt(Float64(h / Float64(Float64(d_m * d_m) * l))) / h) * d_m) * d_m);
	elseif (h <= -5e-292)
		tmp = Float64(Float64(-Float64(Float64(sqrt(Float64(-h)) / sqrt(Float64(-l))) * d_m)) / h);
	elseif (h <= 1.08e+46)
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(Float64(abs(d_m) * sqrt(Float64(l / h))) / l) * 1.0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -9e+222)
		tmp = ((sqrt((h / ((d_m * d_m) * l))) / h) * d_m) * d_m;
	elseif (h <= -5e-292)
		tmp = -((sqrt(-h) / sqrt(-l)) * d_m) / h;
	elseif (h <= 1.08e+46)
		tmp = d_m / sqrt((l * h));
	else
		tmp = ((abs(d_m) * sqrt((l / h))) / l) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -9e+222], N[(N[(N[(N[Sqrt[N[(h / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision] * d$95$m), $MachinePrecision] * d$95$m), $MachinePrecision], If[LessEqual[h, -5e-292], N[((-N[(N[(N[Sqrt[(-h)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision]) / h), $MachinePrecision], If[LessEqual[h, 1.08e+46], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[d$95$m], $MachinePrecision] * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if h < -8.99999999999999978e222

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{{d}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{1}{d \cdot h}} \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{1}{d \cdot h}} \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)} \]

   4. Applied rewrites 18.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\left(\left(\left(\frac{\frac{h}{\sqrt{h \cdot d}}}{\sqrt{\ell \cdot d}} \cdot M\right) \cdot M\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell}, -0.125, \sqrt{\frac{1}{\left(h \cdot d\right) \cdot \left(\ell \cdot d\right)}}\right) \cdot d\right) \cdot d} \]

   5. Taylor expanded in h around 0

      \[\leadsto \left(\frac{\sqrt{\frac{h}{{d}^{2} \cdot \ell}}}{h} \cdot d\right) \cdot d \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\frac{h}{{d}^{2} \cdot \ell}}}{h} \cdot d\right) \cdot d \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\frac{h}{{d}^{2} \cdot \ell}}}{h} \cdot d\right) \cdot d \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\frac{h}{{d}^{2} \cdot \ell}}}{h} \cdot d\right) \cdot d \]

      4. pow2 N/A

         \[\leadsto \left(\frac{\sqrt{\frac{h}{\left(d \cdot d\right) \cdot \ell}}}{h} \cdot d\right) \cdot d \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{\sqrt{\frac{h}{\left(d \cdot d\right) \cdot \ell}}}{h} \cdot d\right) \cdot d \]

      6. lift-*.f64 17.9

         \[\leadsto \left(\frac{\sqrt{\frac{h}{\left(d \cdot d\right) \cdot \ell}}}{h} \cdot d\right) \cdot d \]

   7. Applied rewrites 17.9%

      \[\leadsto \left(\frac{\sqrt{\frac{h}{\left(d \cdot d\right) \cdot \ell}}}{h} \cdot d\right) \cdot d \]

   if -8.99999999999999978e222 < h < -4.99999999999999981e-292

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot \left(h \cdot d\right)}}{h}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{-d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. mult-flip N/A

         \[\leadsto \frac{-\sqrt{h \cdot \frac{1}{\ell}} \cdot d}{h} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{-\left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right) \cdot d}{h} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right) \cdot d}{h} \]

      7. sqrt-unprod N/A

         \[\leadsto \frac{-\sqrt{h \cdot \frac{1}{\ell}} \cdot d}{h} \]

      8. mult-flip N/A

         \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      10. lift-/.f64 25.6

          \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 25.6%

      \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. frac-2neg N/A

         \[\leadsto \frac{-\sqrt{\frac{\mathsf{neg}\left(h\right)}{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      4. sqrt-div N/A

         \[\leadsto \frac{-\frac{\sqrt{\mathsf{neg}\left(h\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-\frac{\sqrt{\mathsf{neg}\left(h\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-\frac{\sqrt{\mathsf{neg}\left(h\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      7. lower-neg.f64 N/A

         \[\leadsto \frac{-\frac{\sqrt{-h}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{-\frac{\sqrt{-h}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      9. lower-neg.f64 22.7

         \[\leadsto \frac{-\frac{\sqrt{-h}}{\sqrt{-\ell}} \cdot d}{h} \]

   9. Applied rewrites 22.7%

      \[\leadsto \frac{-\frac{\sqrt{-h}}{\sqrt{-\ell}} \cdot d}{h} \]

   if -4.99999999999999981e-292 < h < 1.07999999999999994e46

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 1.07999999999999994e46 < h

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 32.9%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \color{blue}{1} \]

   7. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{{d}^{2} \cdot \ell}{h}}}{\ell}} \cdot 1 \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{d}^{2} \cdot \ell}{h}}}{\color{blue}{\ell}} \cdot 1 \]

      2. associate-/l* N/A

         \[\leadsto \frac{\sqrt{{d}^{2} \cdot \frac{\ell}{h}}}{\ell} \cdot 1 \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{{d}^{2}} \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      4. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d} \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      5. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      9. lower-/.f64 21.9

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

   9. Applied rewrites 21.9%

      \[\leadsto \color{blue}{\frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell}} \cdot 1 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 44.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{+196}:\\ \;\;\;\;d\_m \cdot \frac{\sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\frac{-\frac{\sqrt{-h}}{\sqrt{-\ell}} \cdot d\_m}{h}\\ \mathbf{elif}\;h \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\_m\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= h -2e+196)
   (* d_m (/ (sqrt (/ h l)) h))
   (if (<= h -5e-292)
     (/ (- (* (/ (sqrt (- h)) (sqrt (- l))) d_m)) h)
     (if (<= h 5e+34)
       (/ d_m (sqrt (* l h)))
       (* (/ (* (fabs d_m) (sqrt (/ l h))) l) 1.0)))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -2e+196) {
		tmp = d_m * (sqrt((h / l)) / h);
	} else if (h <= -5e-292) {
		tmp = -((sqrt(-h) / sqrt(-l)) * d_m) / h;
	} else if (h <= 5e+34) {
		tmp = d_m / sqrt((l * h));
	} else {
		tmp = ((fabs(d_m) * sqrt((l / h))) / l) * 1.0;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (h <= (-2d+196)) then
        tmp = d_m * (sqrt((h / l)) / h)
    else if (h <= (-5d-292)) then
        tmp = -((sqrt(-h) / sqrt(-l)) * d_m) / h
    else if (h <= 5d+34) then
        tmp = d_m / sqrt((l * h))
    else
        tmp = ((abs(d_m) * sqrt((l / h))) / l) * 1.0d0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -2e+196) {
		tmp = d_m * (Math.sqrt((h / l)) / h);
	} else if (h <= -5e-292) {
		tmp = -((Math.sqrt(-h) / Math.sqrt(-l)) * d_m) / h;
	} else if (h <= 5e+34) {
		tmp = d_m / Math.sqrt((l * h));
	} else {
		tmp = ((Math.abs(d_m) * Math.sqrt((l / h))) / l) * 1.0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	tmp = 0
	if h <= -2e+196:
		tmp = d_m * (math.sqrt((h / l)) / h)
	elif h <= -5e-292:
		tmp = -((math.sqrt(-h) / math.sqrt(-l)) * d_m) / h
	elif h <= 5e+34:
		tmp = d_m / math.sqrt((l * h))
	else:
		tmp = ((math.fabs(d_m) * math.sqrt((l / h))) / l) * 1.0
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -2e+196)
		tmp = Float64(d_m * Float64(sqrt(Float64(h / l)) / h));
	elseif (h <= -5e-292)
		tmp = Float64(Float64(-Float64(Float64(sqrt(Float64(-h)) / sqrt(Float64(-l))) * d_m)) / h);
	elseif (h <= 5e+34)
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(Float64(abs(d_m) * sqrt(Float64(l / h))) / l) * 1.0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -2e+196)
		tmp = d_m * (sqrt((h / l)) / h);
	elseif (h <= -5e-292)
		tmp = -((sqrt(-h) / sqrt(-l)) * d_m) / h;
	elseif (h <= 5e+34)
		tmp = d_m / sqrt((l * h));
	else
		tmp = ((abs(d_m) * sqrt((l / h))) / l) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -2e+196], N[(d$95$m * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-292], N[((-N[(N[(N[Sqrt[(-h)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision]) / h), $MachinePrecision], If[LessEqual[h, 5e+34], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[d$95$m], $MachinePrecision] * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.9999999999999999e196

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot \left(h \cdot d\right)}}{h}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

      3. mult-flip N/A

         \[\leadsto d \cdot \frac{\sqrt{h \cdot \frac{1}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto d \cdot \frac{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}{h} \]

      6. sqrt-unprod N/A

         \[\leadsto d \cdot \frac{\sqrt{h \cdot \frac{1}{\ell}}}{h} \]

      7. mult-flip N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

      9. lift-/.f64 26.0

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

   7. Applied rewrites 26.0%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{h}} \]

   if -1.9999999999999999e196 < h < -4.99999999999999981e-292

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot \left(h \cdot d\right)}}{h}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{-d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. mult-flip N/A

         \[\leadsto \frac{-\sqrt{h \cdot \frac{1}{\ell}} \cdot d}{h} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{-\left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right) \cdot d}{h} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right) \cdot d}{h} \]

      7. sqrt-unprod N/A

         \[\leadsto \frac{-\sqrt{h \cdot \frac{1}{\ell}} \cdot d}{h} \]

      8. mult-flip N/A

         \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      10. lift-/.f64 25.6

          \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 25.6%

      \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. frac-2neg N/A

         \[\leadsto \frac{-\sqrt{\frac{\mathsf{neg}\left(h\right)}{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      4. sqrt-div N/A

         \[\leadsto \frac{-\frac{\sqrt{\mathsf{neg}\left(h\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-\frac{\sqrt{\mathsf{neg}\left(h\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{-\frac{\sqrt{\mathsf{neg}\left(h\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      7. lower-neg.f64 N/A

         \[\leadsto \frac{-\frac{\sqrt{-h}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{-\frac{\sqrt{-h}}{\sqrt{\mathsf{neg}\left(\ell\right)}} \cdot d}{h} \]

      9. lower-neg.f64 22.7

         \[\leadsto \frac{-\frac{\sqrt{-h}}{\sqrt{-\ell}} \cdot d}{h} \]

   9. Applied rewrites 22.7%

      \[\leadsto \frac{-\frac{\sqrt{-h}}{\sqrt{-\ell}} \cdot d}{h} \]

   if -4.99999999999999981e-292 < h < 4.9999999999999998e34

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 4.9999999999999998e34 < h

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 32.9%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \color{blue}{1} \]

   7. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{{d}^{2} \cdot \ell}{h}}}{\ell}} \cdot 1 \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{d}^{2} \cdot \ell}{h}}}{\color{blue}{\ell}} \cdot 1 \]

      2. associate-/l* N/A

         \[\leadsto \frac{\sqrt{{d}^{2} \cdot \frac{\ell}{h}}}{\ell} \cdot 1 \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{{d}^{2}} \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      4. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d} \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      5. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      9. lower-/.f64 21.9

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

   9. Applied rewrites 21.9%

      \[\leadsto \color{blue}{\frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell}} \cdot 1 \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 44.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{+196}:\\ \;\;\;\;d\_m \cdot \frac{\sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;h \leq 1.08 \cdot 10^{+46}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\_m\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= h -2e+196)
   (* d_m (/ (sqrt (/ h l)) h))
   (if (<= h 1.08e+46)
     (/ d_m (sqrt (* l h)))
     (* (/ (* (fabs d_m) (sqrt (/ l h))) l) 1.0))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -2e+196) {
		tmp = d_m * (sqrt((h / l)) / h);
	} else if (h <= 1.08e+46) {
		tmp = d_m / sqrt((l * h));
	} else {
		tmp = ((fabs(d_m) * sqrt((l / h))) / l) * 1.0;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (h <= (-2d+196)) then
        tmp = d_m * (sqrt((h / l)) / h)
    else if (h <= 1.08d+46) then
        tmp = d_m / sqrt((l * h))
    else
        tmp = ((abs(d_m) * sqrt((l / h))) / l) * 1.0d0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -2e+196) {
		tmp = d_m * (Math.sqrt((h / l)) / h);
	} else if (h <= 1.08e+46) {
		tmp = d_m / Math.sqrt((l * h));
	} else {
		tmp = ((Math.abs(d_m) * Math.sqrt((l / h))) / l) * 1.0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	tmp = 0
	if h <= -2e+196:
		tmp = d_m * (math.sqrt((h / l)) / h)
	elif h <= 1.08e+46:
		tmp = d_m / math.sqrt((l * h))
	else:
		tmp = ((math.fabs(d_m) * math.sqrt((l / h))) / l) * 1.0
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -2e+196)
		tmp = Float64(d_m * Float64(sqrt(Float64(h / l)) / h));
	elseif (h <= 1.08e+46)
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(Float64(abs(d_m) * sqrt(Float64(l / h))) / l) * 1.0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -2e+196)
		tmp = d_m * (sqrt((h / l)) / h);
	elseif (h <= 1.08e+46)
		tmp = d_m / sqrt((l * h));
	else
		tmp = ((abs(d_m) * sqrt((l / h))) / l) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -2e+196], N[(d$95$m * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.08e+46], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[d$95$m], $MachinePrecision] * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.9999999999999999e196

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot \left(h \cdot d\right)}}{h}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

      3. mult-flip N/A

         \[\leadsto d \cdot \frac{\sqrt{h \cdot \frac{1}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto d \cdot \frac{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}{h} \]

      6. sqrt-unprod N/A

         \[\leadsto d \cdot \frac{\sqrt{h \cdot \frac{1}{\ell}}}{h} \]

      7. mult-flip N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

      9. lift-/.f64 26.0

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

   7. Applied rewrites 26.0%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{h}} \]

   if -1.9999999999999999e196 < h < 1.07999999999999994e46

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 1.07999999999999994e46 < h

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 32.9%

      \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \color{blue}{1} \]

   7. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{{d}^{2} \cdot \ell}{h}}}{\ell}} \cdot 1 \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{{d}^{2} \cdot \ell}{h}}}{\color{blue}{\ell}} \cdot 1 \]

      2. associate-/l* N/A

         \[\leadsto \frac{\sqrt{{d}^{2} \cdot \frac{\ell}{h}}}{\ell} \cdot 1 \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{{d}^{2}} \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      4. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d} \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      5. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      7. lift-fabs.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

      9. lower-/.f64 21.9

         \[\leadsto \frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell} \cdot 1 \]

   9. Applied rewrites 21.9%

      \[\leadsto \color{blue}{\frac{\left|d\right| \cdot \sqrt{\frac{\ell}{h}}}{\ell}} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 44.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{+196}:\\ \;\;\;\;d\_m \cdot \frac{\sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;h \leq 7 \cdot 10^{-9}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{d\_m}{-\sqrt{\frac{\ell}{h}} \cdot h}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= h -2e+196)
   (* d_m (/ (sqrt (/ h l)) h))
   (if (<= h 7e-9)
     (/ d_m (sqrt (* l h)))
     (- (/ d_m (- (* (sqrt (/ l h)) h)))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -2e+196) {
		tmp = d_m * (sqrt((h / l)) / h);
	} else if (h <= 7e-9) {
		tmp = d_m / sqrt((l * h));
	} else {
		tmp = -(d_m / -(sqrt((l / h)) * h));
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (h <= (-2d+196)) then
        tmp = d_m * (sqrt((h / l)) / h)
    else if (h <= 7d-9) then
        tmp = d_m / sqrt((l * h))
    else
        tmp = -(d_m / -(sqrt((l / h)) * h))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= -2e+196) {
		tmp = d_m * (Math.sqrt((h / l)) / h);
	} else if (h <= 7e-9) {
		tmp = d_m / Math.sqrt((l * h));
	} else {
		tmp = -(d_m / -(Math.sqrt((l / h)) * h));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	tmp = 0
	if h <= -2e+196:
		tmp = d_m * (math.sqrt((h / l)) / h)
	elif h <= 7e-9:
		tmp = d_m / math.sqrt((l * h))
	else:
		tmp = -(d_m / -(math.sqrt((l / h)) * h))
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= -2e+196)
		tmp = Float64(d_m * Float64(sqrt(Float64(h / l)) / h));
	elseif (h <= 7e-9)
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	else
		tmp = Float64(-Float64(d_m / Float64(-Float64(sqrt(Float64(l / h)) * h))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= -2e+196)
		tmp = d_m * (sqrt((h / l)) / h);
	elseif (h <= 7e-9)
		tmp = d_m / sqrt((l * h));
	else
		tmp = -(d_m / -(sqrt((l / h)) * h));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, -2e+196], N[(d$95$m * N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 7e-9], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(d$95$m / (-N[(N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision] * h), $MachinePrecision])), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if h < -1.9999999999999999e196

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot \left(h \cdot d\right)}}{h}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

      3. mult-flip N/A

         \[\leadsto d \cdot \frac{\sqrt{h \cdot \frac{1}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto d \cdot \frac{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}{h} \]

      6. sqrt-unprod N/A

         \[\leadsto d \cdot \frac{\sqrt{h \cdot \frac{1}{\ell}}}{h} \]

      7. mult-flip N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

      9. lift-/.f64 26.0

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

   7. Applied rewrites 26.0%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{h}} \]

   if -1.9999999999999999e196 < h < 6.9999999999999998e-9

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 6.9999999999999998e-9 < h

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. pow1/2 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. pow1/2 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. *-commutative N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \frac{\sqrt{1}}{\sqrt{h \cdot \ell}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right) \]

      10. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. mult-flip N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto -\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \]

      14. *-commutative N/A

          \[\leadsto -\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]

      15. lower-/.f64 N/A

          \[\leadsto -\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   6. Applied rewrites 10.1%

      \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

   7. Taylor expanded in h around -inf

      \[\leadsto -\frac{d}{-1 \cdot \left(h \cdot \sqrt{\frac{\ell}{h}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto -\frac{d}{\mathsf{neg}\left(h \cdot \sqrt{\frac{\ell}{h}}\right)} \]

      2. sqrt-prod N/A

         \[\leadsto -\frac{d}{\mathsf{neg}\left(h \cdot \sqrt{\frac{\ell}{h}}\right)} \]

      3. *-commutative N/A

         \[\leadsto -\frac{d}{\mathsf{neg}\left(h \cdot \sqrt{\frac{\ell}{h}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto -\frac{d}{-h \cdot \sqrt{\frac{\ell}{h}}} \]

      5. *-commutative N/A

         \[\leadsto -\frac{d}{-\sqrt{\frac{\ell}{h}} \cdot h} \]

      6. lower-*.f64 N/A

         \[\leadsto -\frac{d}{-\sqrt{\frac{\ell}{h}} \cdot h} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto -\frac{d}{-\sqrt{\frac{\ell}{h}} \cdot h} \]

      8. lower-/.f64 25.6

         \[\leadsto -\frac{d}{-\sqrt{\frac{\ell}{h}} \cdot h} \]

   9. Applied rewrites 25.6%

      \[\leadsto -\frac{d}{-\sqrt{\frac{\ell}{h}} \cdot h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 44.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;h \leq -2 \cdot 10^{+196}:\\ \;\;\;\;d\_m \cdot \frac{t\_0}{h}\\ \mathbf{elif}\;h \leq 2.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d\_m}{t\_0 \cdot \ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))))
   (if (<= h -2e+196)
     (* d_m (/ t_0 h))
     (if (<= h 2.5e+62) (/ d_m (sqrt (* l h))) (/ d_m (* t_0 l))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h / l));
	double tmp;
	if (h <= -2e+196) {
		tmp = d_m * (t_0 / h);
	} else if (h <= 2.5e+62) {
		tmp = d_m / sqrt((l * h));
	} else {
		tmp = d_m / (t_0 * l);
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h / l))
    if (h <= (-2d+196)) then
        tmp = d_m * (t_0 / h)
    else if (h <= 2.5d+62) then
        tmp = d_m / sqrt((l * h))
    else
        tmp = d_m / (t_0 * l)
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((h / l));
	double tmp;
	if (h <= -2e+196) {
		tmp = d_m * (t_0 / h);
	} else if (h <= 2.5e+62) {
		tmp = d_m / Math.sqrt((l * h));
	} else {
		tmp = d_m / (t_0 * l);
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = math.sqrt((h / l))
	tmp = 0
	if h <= -2e+196:
		tmp = d_m * (t_0 / h)
	elif h <= 2.5e+62:
		tmp = d_m / math.sqrt((l * h))
	else:
		tmp = d_m / (t_0 * l)
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h / l))
	tmp = 0.0
	if (h <= -2e+196)
		tmp = Float64(d_m * Float64(t_0 / h));
	elseif (h <= 2.5e+62)
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	else
		tmp = Float64(d_m / Float64(t_0 * l));
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = sqrt((h / l));
	tmp = 0.0;
	if (h <= -2e+196)
		tmp = d_m * (t_0 / h);
	elseif (h <= 2.5e+62)
		tmp = d_m / sqrt((l * h));
	else
		tmp = d_m / (t_0 * l);
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -2e+196], N[(d$95$m * N[(t$95$0 / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.5e+62], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d$95$m / N[(t$95$0 * l), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.9999999999999999e196

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot \left(h \cdot d\right)}}{h}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

      3. mult-flip N/A

         \[\leadsto d \cdot \frac{\sqrt{h \cdot \frac{1}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto d \cdot \frac{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}{h} \]

      5. lower-/.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}{h} \]

      6. sqrt-unprod N/A

         \[\leadsto d \cdot \frac{\sqrt{h \cdot \frac{1}{\ell}}}{h} \]

      7. mult-flip N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

      9. lift-/.f64 26.0

         \[\leadsto d \cdot \frac{\sqrt{\frac{h}{\ell}}}{h} \]

   7. Applied rewrites 26.0%

      \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{h}{\ell}}}{h}} \]

   if -1.9999999999999999e196 < h < 2.50000000000000014e62

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.50000000000000014e62 < h

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]

   5. Taylor expanded in l around inf

      \[\leadsto \frac{d}{\color{blue}{\ell \cdot \left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{d}{\ell \cdot \color{blue}{\left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{d}{\left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right) \cdot \ell} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{d}{\left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right) \cdot \ell} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \frac{1}{\ell}} \cdot \ell} \]

      5. mult-flip N/A

         \[\leadsto \frac{d}{\sqrt{\frac{h}{\ell}} \cdot \ell} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\frac{h}{\ell}} \cdot \ell} \]

      7. lift-/.f64 23.0

         \[\leadsto \frac{d}{\sqrt{\frac{h}{\ell}} \cdot \ell} \]

   7. Applied rewrites 23.0%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\frac{h}{\ell}} \cdot \ell}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 44.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 2.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\frac{h}{\ell}} \cdot \ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= h 2.5e+62) (/ d_m (sqrt (* l h))) (/ d_m (* (sqrt (/ h l)) l))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 2.5e+62) {
		tmp = d_m / sqrt((l * h));
	} else {
		tmp = d_m / (sqrt((h / l)) * l);
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (h <= 2.5d+62) then
        tmp = d_m / sqrt((l * h))
    else
        tmp = d_m / (sqrt((h / l)) * l)
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 2.5e+62) {
		tmp = d_m / Math.sqrt((l * h));
	} else {
		tmp = d_m / (Math.sqrt((h / l)) * l);
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	tmp = 0
	if h <= 2.5e+62:
		tmp = d_m / math.sqrt((l * h))
	else:
		tmp = d_m / (math.sqrt((h / l)) * l)
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= 2.5e+62)
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	else
		tmp = Float64(d_m / Float64(sqrt(Float64(h / l)) * l));
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= 2.5e+62)
		tmp = d_m / sqrt((l * h));
	else
		tmp = d_m / (sqrt((h / l)) * l);
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, 2.5e+62], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d$95$m / N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 2.50000000000000014e62

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.50000000000000014e62 < h

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]

   5. Taylor expanded in l around inf

      \[\leadsto \frac{d}{\color{blue}{\ell \cdot \left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{d}{\ell \cdot \color{blue}{\left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{d}{\left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right) \cdot \ell} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{d}{\left(\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}\right) \cdot \ell} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \frac{1}{\ell}} \cdot \ell} \]

      5. mult-flip N/A

         \[\leadsto \frac{d}{\sqrt{\frac{h}{\ell}} \cdot \ell} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{d}{\sqrt{\frac{h}{\ell}} \cdot \ell} \]

      7. lift-/.f64 23.0

         \[\leadsto \frac{d}{\sqrt{\frac{h}{\ell}} \cdot \ell} \]

   7. Applied rewrites 23.0%

      \[\leadsto \frac{d}{\color{blue}{\sqrt{\frac{h}{\ell}} \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 43.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-214}:\\ \;\;\;\;-\frac{d\_m}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
       (-
        1.0
        (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0)) (/ h l))))
      -2e-214)
   (- (/ d_m (sqrt (* h l))))
   (/ d_m (sqrt (* l h)))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -2e-214) {
		tmp = -(d_m / sqrt((h * l)));
	} else {
		tmp = d_m / sqrt((l * h));
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m_1) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))) <= (-2d-214)) then
        tmp = -(d_m / sqrt((h * l)))
    else
        tmp = d_m / sqrt((l * h))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -2e-214) {
		tmp = -(d_m / Math.sqrt((h * l)));
	} else {
		tmp = d_m / Math.sqrt((l * h));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -2e-214:
		tmp = -(d_m / math.sqrt((h * l)))
	else:
		tmp = d_m / math.sqrt((l * h))
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l)))) <= -2e-214)
		tmp = Float64(-Float64(d_m / sqrt(Float64(h * l))));
	else
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0)) * (h / l)))) <= -2e-214)
		tmp = -(d_m / sqrt((h * l)));
	else
		tmp = d_m / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-214], (-N[(d$95$m / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999983e-214

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow1/2 N/A

          \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lift-/.f64 55.9

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. pow1/2 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. pow1/2 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. metadata-eval N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. *-commutative N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \frac{\sqrt{1}}{\sqrt{h \cdot \ell}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right) \]

      10. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \frac{1}{\sqrt{h} \cdot \sqrt{\ell}}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. mult-flip N/A

          \[\leadsto \mathsf{neg}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto -\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \]

      14. *-commutative N/A

          \[\leadsto -\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]

      15. lower-/.f64 N/A

          \[\leadsto -\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \]

   6. Applied rewrites 10.1%

      \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

   if -1.99999999999999983e-214 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 36.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around inf

      \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. Taylor expanded in l around inf

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

   9. Applied rewrites 26.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

   10. Taylor expanded in d around inf

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   11. Step-by-step derivation

       1. sqrt-prod N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

       5. lift-*.f64 42.7

          \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   12. Applied rewrites 42.7%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 42.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \frac{d\_m}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m) :precision binary64 (/ d_m (sqrt (* l h))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	return d_m / sqrt((l * h));
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    code = d_m / sqrt((l * h))
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	return d_m / Math.sqrt((l * h));
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	return d_m / math.sqrt((l * h))

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	return Float64(d_m / sqrt(Float64(l * h)))
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp = code(d_m, h, l, M_m, D_m)
	tmp = d_m / sqrt((l * h));
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.1%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in h around inf

   \[\leadsto \color{blue}{h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} + \frac{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}}{h}\right)} \]

4. Applied rewrites 27.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(M \cdot M\right)\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   9. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot {M}^{2}\right)\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{d \cdot {M}^{2}}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(D \cdot \frac{{M}^{2} \cdot d}{\sqrt{h} \cdot \sqrt{\ell}}\right) \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

   14. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{D \cdot \left({M}^{2} \cdot d\right)}{\sqrt{h} \cdot \sqrt{\ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot \frac{-1}{8}, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

6. Applied rewrites 27.2%

   \[\leadsto \mathsf{fma}\left(\frac{\frac{\left(\left(M \cdot M\right) \cdot d\right) \cdot D}{\sqrt{h \cdot \ell}} \cdot D}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot 1\right) \]

7. Taylor expanded in l around inf

   \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\color{blue}{\ell}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{h}{\ell}}\right)} + \frac{d}{\sqrt{h} \cdot \sqrt{\frac{1}{\ell}}}}{\ell} \]

9. Applied rewrites 26.7%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{h}{\ell}} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\frac{h}{\ell}}}\right)}{\color{blue}{\ell}} \]

10. Taylor expanded in d around inf

    \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
11. Step-by-step derivation

    1. sqrt-prod N/A

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

    2. lower-/.f64 N/A

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

    3. lift-sqrt.f64 N/A

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

    4. *-commutative N/A

       \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

    5. lift-*.f64 42.7

       \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

12. Applied rewrites 42.7%

    \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025135 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))