Henrywood and Agarwal, Equation (12)

Percentage Accurate: 35.8% → 78.3%

Time: 8.0s

Alternatives: 16

Speedup: 7.7×

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: 35.8% 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: 78.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d\_m + d\_m}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+221}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\_m\right)}{h}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-307}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot d\_m\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\right) \cdot \left(1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ D (+ d_m d_m)))))
   (if (<= l -1e+221)
     (/ (* (sqrt (/ h l)) (- d_m)) h)
     (if (<= l -2e-307)
       (*
        (* (/ 1.0 (sqrt (* l h))) d_m)
        (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))
       (*
        (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m)
        (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l)))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double tmp;
	if (l <= -1e+221) {
		tmp = (sqrt((h / l)) * -d_m) / h;
	} else if (l <= -2e-307) {
		tmp = ((1.0 / sqrt((l * h))) * d_m) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	} else {
		tmp = ((1.0 / (sqrt(l) * sqrt(h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d / (d_m + d_m))
    if (l <= (-1d+221)) then
        tmp = (sqrt((h / l)) * -d_m) / h
    else if (l <= (-2d-307)) then
        tmp = ((1.0d0 / sqrt((l * h))) * d_m) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))
    else
        tmp = ((1.0d0 / (sqrt(l) * sqrt(h))) * d_m) * (1.0d0 - ((((t_0 * t_0) * 0.5d0) * h) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double tmp;
	if (l <= -1e+221) {
		tmp = (Math.sqrt((h / l)) * -d_m) / h;
	} else if (l <= -2e-307) {
		tmp = ((1.0 / Math.sqrt((l * h))) * d_m) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	} else {
		tmp = ((1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = M * (D / (d_m + d_m))
	tmp = 0
	if l <= -1e+221:
		tmp = (math.sqrt((h / l)) * -d_m) / h
	elif l <= -2e-307:
		tmp = ((1.0 / math.sqrt((l * h))) * d_m) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))
	else:
		tmp = ((1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(M * Float64(D / Float64(d_m + d_m)))
	tmp = 0.0
	if (l <= -1e+221)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d_m)) / h);
	elseif (l <= -2e-307)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(l * h))) * d_m) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = M * (D / (d_m + d_m));
	tmp = 0.0;
	if (l <= -1e+221)
		tmp = (sqrt((h / l)) * -d_m) / h;
	elseif (l <= -2e-307)
		tmp = ((1.0 / sqrt((l * h))) * d_m) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	else
		tmp = ((1.0 / (sqrt(l) * sqrt(h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+221], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[l, -2e-307], N[(N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1e221

   1. Initial program 35.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot -1\right)}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}{h} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)}{h} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      10. lift-/.f64 25.0

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   7. Applied rewrites 25.0%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   if -1e221 < l < -1.99999999999999982e-307

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 70.7

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

   6. Applied rewrites 70.7%

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

   if -1.99999999999999982e-307 < l

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 74.7

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

   8. Applied rewrites 74.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 41.5

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

   10. Applied rewrites 41.5%

       \[\leadsto \left(\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 77.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d\_m + d\_m}\\ t_1 := 1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\_m\right)}{h}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-307}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ D (+ d_m d_m))))
        (t_1 (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l))))
   (if (<= l -1e+216)
     (/ (* (sqrt (/ h l)) (- d_m)) h)
     (if (<= l -2e-307)
       (* (* (sqrt (/ 1.0 (* l h))) d_m) t_1)
       (* (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m) t_1)))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double t_1 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	double tmp;
	if (l <= -1e+216) {
		tmp = (sqrt((h / l)) * -d_m) / h;
	} else if (l <= -2e-307) {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * t_1;
	} else {
		tmp = ((1.0 / (sqrt(l) * sqrt(h))) * d_m) * t_1;
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = m * (d / (d_m + d_m))
    t_1 = 1.0d0 - ((((t_0 * t_0) * 0.5d0) * h) / l)
    if (l <= (-1d+216)) then
        tmp = (sqrt((h / l)) * -d_m) / h
    else if (l <= (-2d-307)) then
        tmp = (sqrt((1.0d0 / (l * h))) * d_m) * t_1
    else
        tmp = ((1.0d0 / (sqrt(l) * sqrt(h))) * d_m) * t_1
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double t_1 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	double tmp;
	if (l <= -1e+216) {
		tmp = (Math.sqrt((h / l)) * -d_m) / h;
	} else if (l <= -2e-307) {
		tmp = (Math.sqrt((1.0 / (l * h))) * d_m) * t_1;
	} else {
		tmp = ((1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m) * t_1;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = M * (D / (d_m + d_m))
	t_1 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l)
	tmp = 0
	if l <= -1e+216:
		tmp = (math.sqrt((h / l)) * -d_m) / h
	elif l <= -2e-307:
		tmp = (math.sqrt((1.0 / (l * h))) * d_m) * t_1
	else:
		tmp = ((1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m) * t_1
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(M * Float64(D / Float64(d_m + d_m)))
	t_1 = Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l))
	tmp = 0.0
	if (l <= -1e+216)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d_m)) / h);
	elseif (l <= -2e-307)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * t_1);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m) * t_1);
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = M * (D / (d_m + d_m));
	t_1 = 1.0 - ((((t_0 * t_0) * 0.5) * h) / l);
	tmp = 0.0;
	if (l <= -1e+216)
		tmp = (sqrt((h / l)) * -d_m) / h;
	elseif (l <= -2e-307)
		tmp = (sqrt((1.0 / (l * h))) * d_m) * t_1;
	else
		tmp = ((1.0 / (sqrt(l) * sqrt(h))) * d_m) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+216], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[l, -2e-307], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1e216

   1. Initial program 35.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot -1\right)}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}{h} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)}{h} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      10. lift-/.f64 25.0

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   7. Applied rewrites 25.0%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   if -1e216 < l < -1.99999999999999982e-307

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

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

   if -1.99999999999999982e-307 < l

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 74.7

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

   8. Applied rewrites 74.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 41.5

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

   10. Applied rewrites 41.5%

       \[\leadsto \left(\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{D}{d\_m + d\_m}\\ t_1 := {\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)}\\ t_2 := \left(t\_1 \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := M \cdot t\_0\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-264}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\_m\right) \cdot \left(1 - \left(\left(M \cdot \left(t\_0 \cdot \left(t\_0 \cdot M\right)\right)\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\left(t\_1 \cdot \frac{\sqrt{d\_m}}{\sqrt{\ell}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d\_m}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\left(t\_3 \cdot t\_3\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (/ D (+ d_m d_m)))
        (t_1 (pow (/ d_m h) (/ 1.0 2.0)))
        (t_2
         (*
          (* t_1 (pow (/ d_m l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l)))))
        (t_3 (* M t_0)))
   (if (<= t_2 2e-264)
     (*
      (* (sqrt (/ (/ 1.0 l) h)) d_m)
      (- 1.0 (* (* (* M (* t_0 (* t_0 M))) 0.5) (/ h l))))
     (if (<= t_2 2e+238)
       (* (* t_1 (/ (sqrt d_m) (sqrt l))) 1.0)
       (*
        (/ (* 1.0 d_m) (sqrt (* l h)))
        (- 1.0 (/ (* (* (* t_3 t_3) 0.5) h) l)))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = D / (d_m + d_m);
	double t_1 = pow((d_m / h), (1.0 / 2.0));
	double t_2 = (t_1 * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double t_3 = M * t_0;
	double tmp;
	if (t_2 <= 2e-264) {
		tmp = (sqrt(((1.0 / l) / h)) * d_m) * (1.0 - (((M * (t_0 * (t_0 * M))) * 0.5) * (h / l)));
	} else if (t_2 <= 2e+238) {
		tmp = (t_1 * (sqrt(d_m) / sqrt(l))) * 1.0;
	} else {
		tmp = ((1.0 * d_m) / sqrt((l * h))) * (1.0 - ((((t_3 * t_3) * 0.5) * h) / l));
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = d / (d_m + d_m)
    t_1 = (d_m / h) ** (1.0d0 / 2.0d0)
    t_2 = (t_1 * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))
    t_3 = m * t_0
    if (t_2 <= 2d-264) then
        tmp = (sqrt(((1.0d0 / l) / h)) * d_m) * (1.0d0 - (((m * (t_0 * (t_0 * m))) * 0.5d0) * (h / l)))
    else if (t_2 <= 2d+238) then
        tmp = (t_1 * (sqrt(d_m) / sqrt(l))) * 1.0d0
    else
        tmp = ((1.0d0 * d_m) / sqrt((l * h))) * (1.0d0 - ((((t_3 * t_3) * 0.5d0) * h) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = D / (d_m + d_m);
	double t_1 = Math.pow((d_m / h), (1.0 / 2.0));
	double t_2 = (t_1 * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double t_3 = M * t_0;
	double tmp;
	if (t_2 <= 2e-264) {
		tmp = (Math.sqrt(((1.0 / l) / h)) * d_m) * (1.0 - (((M * (t_0 * (t_0 * M))) * 0.5) * (h / l)));
	} else if (t_2 <= 2e+238) {
		tmp = (t_1 * (Math.sqrt(d_m) / Math.sqrt(l))) * 1.0;
	} else {
		tmp = ((1.0 * d_m) / Math.sqrt((l * h))) * (1.0 - ((((t_3 * t_3) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = D / (d_m + d_m)
	t_1 = math.pow((d_m / h), (1.0 / 2.0))
	t_2 = (t_1 * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))
	t_3 = M * t_0
	tmp = 0
	if t_2 <= 2e-264:
		tmp = (math.sqrt(((1.0 / l) / h)) * d_m) * (1.0 - (((M * (t_0 * (t_0 * M))) * 0.5) * (h / l)))
	elif t_2 <= 2e+238:
		tmp = (t_1 * (math.sqrt(d_m) / math.sqrt(l))) * 1.0
	else:
		tmp = ((1.0 * d_m) / math.sqrt((l * h))) * (1.0 - ((((t_3 * t_3) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(D / Float64(d_m + d_m))
	t_1 = Float64(d_m / h) ^ Float64(1.0 / 2.0)
	t_2 = Float64(Float64(t_1 * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(M * t_0)
	tmp = 0.0
	if (t_2 <= 2e-264)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d_m) * Float64(1.0 - Float64(Float64(Float64(M * Float64(t_0 * Float64(t_0 * M))) * 0.5) * Float64(h / l))));
	elseif (t_2 <= 2e+238)
		tmp = Float64(Float64(t_1 * Float64(sqrt(d_m) / sqrt(l))) * 1.0);
	else
		tmp = Float64(Float64(Float64(1.0 * d_m) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_3 * t_3) * 0.5) * h) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = D / (d_m + d_m);
	t_1 = (d_m / h) ^ (1.0 / 2.0);
	t_2 = (t_1 * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	t_3 = M * t_0;
	tmp = 0.0;
	if (t_2 <= 2e-264)
		tmp = (sqrt(((1.0 / l) / h)) * d_m) * (1.0 - (((M * (t_0 * (t_0 * M))) * 0.5) * (h / l)));
	elseif (t_2 <= 2e+238)
		tmp = (t_1 * (sqrt(d_m) / sqrt(l))) * 1.0;
	else
		tmp = ((1.0 * d_m) / sqrt((l * h))) * (1.0 - ((((t_3 * t_3) * 0.5) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * 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 * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(M * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-264], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(1.0 - N[(N[(N[(M * N[(t$95$0 * N[(t$95$0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+238], N[(N[(t$95$1 * N[(N[Sqrt[d$95$m], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(1.0 * d$95$m), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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)))) < 2e-264

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.2

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

   8. Applied rewrites 75.2%

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

      1. metadata-eval N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 69.6%

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

   if 2e-264 < (*.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)))) < 2.0000000000000001e238

   1. Initial program 35.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 35.2

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

   3. Applied rewrites 35.2%

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

   4. Taylor expanded in d around inf

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

   6. Applied rewrites 22.5%

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

   if 2.0000000000000001e238 < (*.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 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.2

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

   8. Applied rewrites 75.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 74.8

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

   10. Applied rewrites 74.8%

       \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d\_m + d\_m}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\_m\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\_m\right) \cdot \left(1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ D (+ d_m d_m)))))
   (if (<= l -1e+216)
     (/ (* (sqrt (/ h l)) (- d_m)) h)
     (*
      (* (sqrt (/ (/ 1.0 l) h)) d_m)
      (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double tmp;
	if (l <= -1e+216) {
		tmp = (sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = (sqrt(((1.0 / l) / h)) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d / (d_m + d_m))
    if (l <= (-1d+216)) then
        tmp = (sqrt((h / l)) * -d_m) / h
    else
        tmp = (sqrt(((1.0d0 / l) / h)) * d_m) * (1.0d0 - ((((t_0 * t_0) * 0.5d0) * h) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double tmp;
	if (l <= -1e+216) {
		tmp = (Math.sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = (Math.sqrt(((1.0 / l) / h)) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = M * (D / (d_m + d_m))
	tmp = 0
	if l <= -1e+216:
		tmp = (math.sqrt((h / l)) * -d_m) / h
	else:
		tmp = (math.sqrt(((1.0 / l) / h)) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(M * Float64(D / Float64(d_m + d_m)))
	tmp = 0.0
	if (l <= -1e+216)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d_m)) / h);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d_m) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = M * (D / (d_m + d_m));
	tmp = 0.0;
	if (l <= -1e+216)
		tmp = (sqrt((h / l)) * -d_m) / h;
	else
		tmp = (sqrt(((1.0 / l) / h)) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+216], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1e216

   1. Initial program 35.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot -1\right)}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}{h} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)}{h} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      10. lift-/.f64 25.0

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   7. Applied rewrites 25.0%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   if -1e216 < l

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.2

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

   8. Applied rewrites 75.2%

      \[\leadsto \left(\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d\_m + d\_m}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\_m\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d\_m}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ D (+ d_m d_m)))))
   (if (<= l -1e+216)
     (/ (* (sqrt (/ h l)) (- d_m)) h)
     (*
      (/ (* 1.0 d_m) (sqrt (* l h)))
      (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double tmp;
	if (l <= -1e+216) {
		tmp = (sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = ((1.0 * d_m) / sqrt((l * h))) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d / (d_m + d_m))
    if (l <= (-1d+216)) then
        tmp = (sqrt((h / l)) * -d_m) / h
    else
        tmp = ((1.0d0 * d_m) / sqrt((l * h))) * (1.0d0 - ((((t_0 * t_0) * 0.5d0) * h) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double tmp;
	if (l <= -1e+216) {
		tmp = (Math.sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = ((1.0 * d_m) / Math.sqrt((l * h))) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = M * (D / (d_m + d_m))
	tmp = 0
	if l <= -1e+216:
		tmp = (math.sqrt((h / l)) * -d_m) / h
	else:
		tmp = ((1.0 * d_m) / math.sqrt((l * h))) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(M * Float64(D / Float64(d_m + d_m)))
	tmp = 0.0
	if (l <= -1e+216)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d_m)) / h);
	else
		tmp = Float64(Float64(Float64(1.0 * d_m) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = M * (D / (d_m + d_m));
	tmp = 0.0;
	if (l <= -1e+216)
		tmp = (sqrt((h / l)) * -d_m) / h;
	else
		tmp = ((1.0 * d_m) / sqrt((l * h))) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+216], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(1.0 * d$95$m), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1e216

   1. Initial program 35.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot -1\right)}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}{h} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)}{h} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      10. lift-/.f64 25.0

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   7. Applied rewrites 25.0%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   if -1e216 < l

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.2

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

   8. Applied rewrites 75.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-sqrt.f64 74.8

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

   10. Applied rewrites 74.8%

       \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d\_m + d\_m}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\_m\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\ell \cdot h}} \cdot d\_m\right) \cdot \left(1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ D (+ d_m d_m)))))
   (if (<= l -1e+216)
     (/ (* (sqrt (/ h l)) (- d_m)) h)
     (*
      (* (/ 1.0 (sqrt (* l h))) d_m)
      (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double tmp;
	if (l <= -1e+216) {
		tmp = (sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = ((1.0 / sqrt((l * h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d / (d_m + d_m))
    if (l <= (-1d+216)) then
        tmp = (sqrt((h / l)) * -d_m) / h
    else
        tmp = ((1.0d0 / sqrt((l * h))) * d_m) * (1.0d0 - ((((t_0 * t_0) * 0.5d0) * h) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double tmp;
	if (l <= -1e+216) {
		tmp = (Math.sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = ((1.0 / Math.sqrt((l * h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = M * (D / (d_m + d_m))
	tmp = 0
	if l <= -1e+216:
		tmp = (math.sqrt((h / l)) * -d_m) / h
	else:
		tmp = ((1.0 / math.sqrt((l * h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(M * Float64(D / Float64(d_m + d_m)))
	tmp = 0.0
	if (l <= -1e+216)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d_m)) / h);
	else
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(l * h))) * d_m) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = M * (D / (d_m + d_m));
	tmp = 0.0;
	if (l <= -1e+216)
		tmp = (sqrt((h / l)) * -d_m) / h;
	else
		tmp = ((1.0 / sqrt((l * h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+216], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1e216

   1. Initial program 35.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot -1\right)}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}{h} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)}{h} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      10. lift-/.f64 25.0

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   7. Applied rewrites 25.0%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   if -1e216 < l

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 74.7

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

   8. Applied rewrites 74.7%

      \[\leadsto \left(\frac{1}{\sqrt{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d\_m + d\_m}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\_m\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot \left(1 - \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ D (+ d_m d_m)))))
   (if (<= l -1e+216)
     (/ (* (sqrt (/ h l)) (- d_m)) h)
     (*
      (* (sqrt (/ 1.0 (* l h))) d_m)
      (- 1.0 (/ (* (* (* t_0 t_0) 0.5) h) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double tmp;
	if (l <= -1e+216) {
		tmp = (sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d / (d_m + d_m))
    if (l <= (-1d+216)) then
        tmp = (sqrt((h / l)) * -d_m) / h
    else
        tmp = (sqrt((1.0d0 / (l * h))) * d_m) * (1.0d0 - ((((t_0 * t_0) * 0.5d0) * h) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = M * (D / (d_m + d_m));
	double tmp;
	if (l <= -1e+216) {
		tmp = (Math.sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = M * (D / (d_m + d_m))
	tmp = 0
	if l <= -1e+216:
		tmp = (math.sqrt((h / l)) * -d_m) / h
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(M * Float64(D / Float64(d_m + d_m)))
	tmp = 0.0
	if (l <= -1e+216)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d_m)) / h);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * Float64(1.0 - Float64(Float64(Float64(Float64(t_0 * t_0) * 0.5) * h) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = M * (D / (d_m + d_m));
	tmp = 0.0;
	if (l <= -1e+216)
		tmp = (sqrt((h / l)) * -d_m) / h;
	else
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((t_0 * t_0) * 0.5) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+216], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1e216

   1. Initial program 35.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot -1\right)}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}{h} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)}{h} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      10. lift-/.f64 25.0

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   7. Applied rewrites 25.0%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   if -1e216 < l

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{D}{d\_m + d\_m}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\_m\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot \left(1 - \frac{\left(\left(M \cdot \left(t\_0 \cdot \left(t\_0 \cdot M\right)\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (/ D (+ d_m d_m))))
   (if (<= l -1e+216)
     (/ (* (sqrt (/ h l)) (- d_m)) h)
     (*
      (* (sqrt (/ 1.0 (* l h))) d_m)
      (- 1.0 (/ (* (* (* M (* t_0 (* t_0 M))) 0.5) h) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = D / (d_m + d_m);
	double tmp;
	if (l <= -1e+216) {
		tmp = (sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / l));
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / (d_m + d_m)
    if (l <= (-1d+216)) then
        tmp = (sqrt((h / l)) * -d_m) / h
    else
        tmp = (sqrt((1.0d0 / (l * h))) * d_m) * (1.0d0 - ((((m * (t_0 * (t_0 * m))) * 0.5d0) * h) / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = D / (d_m + d_m);
	double tmp;
	if (l <= -1e+216) {
		tmp = (Math.sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = D / (d_m + d_m)
	tmp = 0
	if l <= -1e+216:
		tmp = (math.sqrt((h / l)) * -d_m) / h
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(D / Float64(d_m + d_m))
	tmp = 0.0
	if (l <= -1e+216)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d_m)) / h);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * Float64(1.0 - Float64(Float64(Float64(Float64(M * Float64(t_0 * Float64(t_0 * M))) * 0.5) * h) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = D / (d_m + d_m);
	tmp = 0.0;
	if (l <= -1e+216)
		tmp = (sqrt((h / l)) * -d_m) / h;
	else
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+216], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M * N[(t$95$0 * N[(t$95$0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1e216

   1. Initial program 35.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot -1\right)}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}{h} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)}{h} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      10. lift-/.f64 25.0

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   7. Applied rewrites 25.0%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   if -1e216 < l

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

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

      1. metadata-eval N/A

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

   8. Applied rewrites 73.6%

      \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \left(1 - \color{blue}{\frac{\left(\left(M \cdot \left(\frac{D}{d + d} \cdot \left(\frac{D}{d + d} \cdot M\right)\right)\right) \cdot 0.5\right) \cdot h}{\ell}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\left(-\left(-d\_m\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{d\_m} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<= (* M D) 2e+64)
   (* (- (- d_m)) (sqrt (/ (/ 1.0 h) l)))
   (* (/ (* -0.125 (* (* D M) (* D M))) d_m) (sqrt (/ h (* (* l l) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if ((M * D) <= 2e+64) {
		tmp = -(-d_m) * sqrt(((1.0 / h) / l));
	} else {
		tmp = ((-0.125 * ((D * M) * (D * M))) / d_m) * sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m * d) <= 2d+64) then
        tmp = -(-d_m) * sqrt(((1.0d0 / h) / l))
    else
        tmp = (((-0.125d0) * ((d * m) * (d * m))) / d_m) * sqrt((h / ((l * l) * l)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if ((M * D) <= 2e+64) {
		tmp = -(-d_m) * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = ((-0.125 * ((D * M) * (D * M))) / d_m) * Math.sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if (M * D) <= 2e+64:
		tmp = -(-d_m) * math.sqrt(((1.0 / h) / l))
	else:
		tmp = ((-0.125 * ((D * M) * (D * M))) / d_m) * math.sqrt((h / ((l * l) * l)))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (Float64(M * D) <= 2e+64)
		tmp = Float64(Float64(-Float64(-d_m)) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(Float64(Float64(-0.125 * Float64(Float64(D * M) * Float64(D * M))) / d_m) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if ((M * D) <= 2e+64)
		tmp = -(-d_m) * sqrt(((1.0 / h) / l));
	else
		tmp = ((-0.125 * ((D * M) * (D * M))) / d_m) * sqrt((h / ((l * l) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[N[(M * D), $MachinePrecision], 2e+64], N[((-(-d$95$m)) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.00000000000000004e64

   1. Initial program 35.8%

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

   2. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto -\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      12. lower-*.f64 43.1

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   4. Applied rewrites 43.1%

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

      1. lift-neg.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{\ell \cdot h}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      7. metadata-eval N/A

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

      8. sqrt-pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-pow2 N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      16. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      18. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      19. lower-neg.f64 N/A

          \[\leadsto \left(-\left(\mathsf{neg}\left(d\right)\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      20. lower-neg.f64 N/A

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

   6. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. *-commutative N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. associate-/r* N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

      6. lower-/.f64 43.4

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

   8. Applied rewrites 43.4%

      \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

   if 2.00000000000000004e64 < (*.f64 M D)

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      14. associate-/l* N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{\color{blue}{3}}}} \]

   7. Applied rewrites 28.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-/l* N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 27.9

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

   9. Applied rewrites 27.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. unswap-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 32.9

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

   11. Applied rewrites 32.9%

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

Alternative 10: 50.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\left(-\left(-d\_m\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \frac{M}{d\_m}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<= (* M D) 2e+64)
   (* (- (- d_m)) (sqrt (/ (/ 1.0 h) l)))
   (* (* (* -0.125 (* D D)) (* M (/ M d_m))) (sqrt (/ h (* (* l l) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if ((M * D) <= 2e+64) {
		tmp = -(-d_m) * sqrt(((1.0 / h) / l));
	} else {
		tmp = ((-0.125 * (D * D)) * (M * (M / d_m))) * sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m * d) <= 2d+64) then
        tmp = -(-d_m) * sqrt(((1.0d0 / h) / l))
    else
        tmp = (((-0.125d0) * (d * d)) * (m * (m / d_m))) * sqrt((h / ((l * l) * l)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if ((M * D) <= 2e+64) {
		tmp = -(-d_m) * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = ((-0.125 * (D * D)) * (M * (M / d_m))) * Math.sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if (M * D) <= 2e+64:
		tmp = -(-d_m) * math.sqrt(((1.0 / h) / l))
	else:
		tmp = ((-0.125 * (D * D)) * (M * (M / d_m))) * math.sqrt((h / ((l * l) * l)))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (Float64(M * D) <= 2e+64)
		tmp = Float64(Float64(-Float64(-d_m)) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(Float64(Float64(-0.125 * Float64(D * D)) * Float64(M * Float64(M / d_m))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if ((M * D) <= 2e+64)
		tmp = -(-d_m) * sqrt(((1.0 / h) / l));
	else
		tmp = ((-0.125 * (D * D)) * (M * (M / d_m))) * sqrt((h / ((l * l) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[N[(M * D), $MachinePrecision], 2e+64], N[((-(-d$95$m)) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.00000000000000004e64

   1. Initial program 35.8%

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

   2. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto -\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      12. lower-*.f64 43.1

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   4. Applied rewrites 43.1%

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

      1. lift-neg.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{\ell \cdot h}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      7. metadata-eval N/A

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

      8. sqrt-pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-pow2 N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      16. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      18. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      19. lower-neg.f64 N/A

          \[\leadsto \left(-\left(\mathsf{neg}\left(d\right)\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      20. lower-neg.f64 N/A

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

   6. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. *-commutative N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. associate-/r* N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

      6. lower-/.f64 43.4

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

   8. Applied rewrites 43.4%

      \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

   if 2.00000000000000004e64 < (*.f64 M D)

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.5

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      14. associate-/l* N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \frac{M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{\color{blue}{3}}}} \]

   7. Applied rewrites 28.6%

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

Alternative 11: 50.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\left(-\left(-d\_m\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d\_m}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<= (* M D) 2e+64)
   (* (- (- d_m)) (sqrt (/ (/ 1.0 h) l)))
   (* (* -0.125 (* (* D D) (/ (* M M) d_m))) (sqrt (/ h (* (* l l) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if ((M * D) <= 2e+64) {
		tmp = -(-d_m) * sqrt(((1.0 / h) / l));
	} else {
		tmp = (-0.125 * ((D * D) * ((M * M) / d_m))) * sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m * d) <= 2d+64) then
        tmp = -(-d_m) * sqrt(((1.0d0 / h) / l))
    else
        tmp = ((-0.125d0) * ((d * d) * ((m * m) / d_m))) * sqrt((h / ((l * l) * l)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if ((M * D) <= 2e+64) {
		tmp = -(-d_m) * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = (-0.125 * ((D * D) * ((M * M) / d_m))) * Math.sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if (M * D) <= 2e+64:
		tmp = -(-d_m) * math.sqrt(((1.0 / h) / l))
	else:
		tmp = (-0.125 * ((D * D) * ((M * M) / d_m))) * math.sqrt((h / ((l * l) * l)))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (Float64(M * D) <= 2e+64)
		tmp = Float64(Float64(-Float64(-d_m)) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(M * M) / d_m))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if ((M * D) <= 2e+64)
		tmp = -(-d_m) * sqrt(((1.0 / h) / l));
	else
		tmp = (-0.125 * ((D * D) * ((M * M) / d_m))) * sqrt((h / ((l * l) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[N[(M * D), $MachinePrecision], 2e+64], N[((-(-d$95$m)) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.00000000000000004e64

   1. Initial program 35.8%

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

   2. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto -\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      12. lower-*.f64 43.1

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   4. Applied rewrites 43.1%

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

      1. lift-neg.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{\ell \cdot h}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      7. metadata-eval N/A

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

      8. sqrt-pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-pow2 N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      16. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      18. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      19. lower-neg.f64 N/A

          \[\leadsto \left(-\left(\mathsf{neg}\left(d\right)\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      20. lower-neg.f64 N/A

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

   6. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. *-commutative N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. associate-/r* N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

      6. lower-/.f64 43.4

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

   8. Applied rewrites 43.4%

      \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

   if 2.00000000000000004e64 < (*.f64 M D)

   1. Initial program 35.8%

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

   2. Taylor expanded in d around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 27.5

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

   4. Applied rewrites 27.5%

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

Alternative 12: 44.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-242}:\\ \;\;\;\;\left(d\_m \cdot \left(-\sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right) \cdot h\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(-\left(-d\_m\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\_m\right)}{h}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))))
   (if (<= t_0 -2e-242)
     (* (* d_m (- (sqrt (/ 1.0 (* (* (* h h) h) l))))) h)
     (if (<= t_0 INFINITY)
       (* (- (- d_m)) (sqrt (/ (/ 1.0 h) l)))
       (/ (* (sqrt (/ h l)) (- d_m)) h)))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-242) {
		tmp = (d_m * -sqrt((1.0 / (((h * h) * h) * l)))) * h;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = -(-d_m) * sqrt(((1.0 / h) / l));
	} else {
		tmp = (sqrt((h / l)) * -d_m) / h;
	}
	return tmp;
}

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (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 * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-242) {
		tmp = (d_m * -Math.sqrt((1.0 / (((h * h) * h) * l)))) * h;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = -(-d_m) * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = (Math.sqrt((h / l)) * -d_m) / h;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = (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 * D) / (2.0 * d_m)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -2e-242:
		tmp = (d_m * -math.sqrt((1.0 / (((h * h) * h) * l)))) * h
	elif t_0 <= math.inf:
		tmp = -(-d_m) * math.sqrt(((1.0 / h) / l))
	else:
		tmp = (math.sqrt((h / l)) * -d_m) / h
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = 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 * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -2e-242)
		tmp = Float64(Float64(d_m * Float64(-sqrt(Float64(1.0 / Float64(Float64(Float64(h * h) * h) * l))))) * h);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(-Float64(-d_m)) * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d_m)) / h);
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = (((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -2e-242)
		tmp = (d_m * -sqrt((1.0 / (((h * h) * h) * l)))) * h;
	elseif (t_0 <= Inf)
		tmp = -(-d_m) * sqrt(((1.0 / h) / l));
	else
		tmp = (sqrt((h / l)) * -d_m) / h;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = 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 * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-242], N[(N[(d$95$m * (-N[Sqrt[N[(1.0 / N[(N[(N[(h * h), $MachinePrecision] * h), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * h), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[((-(-d$95$m)) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision] / h), $MachinePrecision]]]]

Derivation

1. Split input into 3 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)))) < -2e-242

   1. Initial program 35.8%

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

   2. Taylor expanded in h around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right) + d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \cdot \color{blue}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{1}{h \cdot {\ell}^{3}}}\right) + d \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \cdot \color{blue}{h} \]

   4. Applied rewrites 14.1%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \cdot h \]
   6. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \left(\left(d \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \cdot h \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(d \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \cdot h \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right) \cdot h \]

      4. associate-*l* N/A

         \[\leadsto \left(d \cdot \left(-1 \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)\right) \cdot h \]

      5. lower-*.f64 N/A

         \[\leadsto \left(d \cdot \left(-1 \cdot \sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)\right) \cdot h \]

      6. mul-1-neg N/A

         \[\leadsto \left(d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)\right)\right) \cdot h \]

      7. lower-neg.f64 N/A

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{{h}^{3} \cdot \ell}}\right)\right) \cdot h \]

      8. pow3 N/A

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right) \cdot h \]

      9. lift-*.f64 N/A

         \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right) \cdot h \]

      10. lift-*.f64 N/A

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right) \cdot h \]

      11. lift-*.f64 N/A

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right) \cdot h \]

      12. lift-/.f64 N/A

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right) \cdot h \]

      13. lift-sqrt.f64 17.9

          \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right) \cdot h \]

   7. Applied rewrites 17.9%

      \[\leadsto \left(d \cdot \left(-\sqrt{\frac{1}{\left(\left(h \cdot h\right) \cdot h\right) \cdot \ell}}\right)\right) \cdot h \]

   if -2e-242 < (*.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)))) < +inf.0

   1. Initial program 35.8%

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

   2. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto -\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      12. lower-*.f64 43.1

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   4. Applied rewrites 43.1%

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

      1. lift-neg.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{\ell \cdot h}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      7. metadata-eval N/A

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

      8. sqrt-pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-pow2 N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      16. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      18. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      19. lower-neg.f64 N/A

          \[\leadsto \left(-\left(\mathsf{neg}\left(d\right)\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      20. lower-neg.f64 N/A

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

   6. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. *-commutative N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. associate-/r* N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

      6. lower-/.f64 43.4

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

   8. Applied rewrites 43.4%

      \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

   if +inf.0 < (*.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 35.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot -1\right)}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}{h} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)}{h} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      10. lift-/.f64 25.0

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   7. Applied rewrites 25.0%

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

Alternative 13: 43.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := -\left(-d\_m\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-242}:\\ \;\;\;\;t\_1 \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\_m\right)}{h}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l)))))
        (t_1 (- (- d_m))))
   (if (<= t_0 -2e-242)
     (* t_1 (- (sqrt (/ 1.0 (* l h)))))
     (if (<= t_0 INFINITY)
       (* t_1 (sqrt (/ (/ 1.0 h) l)))
       (/ (* (sqrt (/ h l)) (- d_m)) h)))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double t_1 = -(-d_m);
	double tmp;
	if (t_0 <= -2e-242) {
		tmp = t_1 * -sqrt((1.0 / (l * h)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1 * sqrt(((1.0 / h) / l));
	} else {
		tmp = (sqrt((h / l)) * -d_m) / h;
	}
	return tmp;
}

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (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 * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double t_1 = -(-d_m);
	double tmp;
	if (t_0 <= -2e-242) {
		tmp = t_1 * -Math.sqrt((1.0 / (l * h)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * Math.sqrt(((1.0 / h) / l));
	} else {
		tmp = (Math.sqrt((h / l)) * -d_m) / h;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = (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 * D) / (2.0 * d_m)), 2.0)) * (h / l)))
	t_1 = -(-d_m)
	tmp = 0
	if t_0 <= -2e-242:
		tmp = t_1 * -math.sqrt((1.0 / (l * h)))
	elif t_0 <= math.inf:
		tmp = t_1 * math.sqrt(((1.0 / h) / l))
	else:
		tmp = (math.sqrt((h / l)) * -d_m) / h
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = 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 * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(-Float64(-d_m))
	tmp = 0.0
	if (t_0 <= -2e-242)
		tmp = Float64(t_1 * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	elseif (t_0 <= Inf)
		tmp = Float64(t_1 * sqrt(Float64(Float64(1.0 / h) / l)));
	else
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d_m)) / h);
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = (((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	t_1 = -(-d_m);
	tmp = 0.0;
	if (t_0 <= -2e-242)
		tmp = t_1 * -sqrt((1.0 / (l * h)));
	elseif (t_0 <= Inf)
		tmp = t_1 * sqrt(((1.0 / h) / l));
	else
		tmp = (sqrt((h / l)) * -d_m) / h;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = 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 * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-(-d$95$m))}, If[LessEqual[t$95$0, -2e-242], N[(t$95$1 * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$1 * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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)))) < -2e-242

   1. Initial program 35.8%

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

   2. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto -\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      12. lower-*.f64 43.1

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   4. Applied rewrites 43.1%

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

      1. lift-neg.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{\ell \cdot h}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      7. metadata-eval N/A

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

      8. sqrt-pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-pow2 N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      16. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      18. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      19. lower-neg.f64 N/A

          \[\leadsto \left(-\left(\mathsf{neg}\left(d\right)\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      20. lower-neg.f64 N/A

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

   6. Applied rewrites 43.1%

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

   7. Taylor expanded in h around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right) \]

      4. *-commutative N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

      10. lift-*.f64 10.6

          \[\leadsto \left(-\left(-d\right)\right) \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

   9. Applied rewrites 10.6%

      \[\leadsto \left(-\left(-d\right)\right) \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

   if -2e-242 < (*.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)))) < +inf.0

   1. Initial program 35.8%

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

   2. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto -\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      12. lower-*.f64 43.1

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   4. Applied rewrites 43.1%

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

      1. lift-neg.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{\ell \cdot h}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      7. metadata-eval N/A

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

      8. sqrt-pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-pow2 N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      16. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      18. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      19. lower-neg.f64 N/A

          \[\leadsto \left(-\left(\mathsf{neg}\left(d\right)\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      20. lower-neg.f64 N/A

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

   6. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. *-commutative N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. associate-/r* N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

      6. lower-/.f64 43.4

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

   8. Applied rewrites 43.4%

      \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}} \]

   if +inf.0 < (*.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 35.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot -1\right)}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}{h} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)}{h} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      10. lift-/.f64 25.0

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   7. Applied rewrites 25.0%

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

Alternative 14: 43.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\_m\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(-\left(-d\_m\right)\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<= l 6.8e-208)
   (/ (* (sqrt (/ h l)) (- d_m)) h)
   (* (- (- d_m)) (/ 1.0 (sqrt (* l h))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (l <= 6.8e-208) {
		tmp = (sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = -(-d_m) * (1.0 / sqrt((l * h)));
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: tmp
    if (l <= 6.8d-208) then
        tmp = (sqrt((h / l)) * -d_m) / h
    else
        tmp = -(-d_m) * (1.0d0 / sqrt((l * h)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (l <= 6.8e-208) {
		tmp = (Math.sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = -(-d_m) * (1.0 / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if l <= 6.8e-208:
		tmp = (math.sqrt((h / l)) * -d_m) / h
	else:
		tmp = -(-d_m) * (1.0 / math.sqrt((l * h)))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (l <= 6.8e-208)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d_m)) / h);
	else
		tmp = Float64(Float64(-Float64(-d_m)) * Float64(1.0 / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if (l <= 6.8e-208)
		tmp = (sqrt((h / l)) * -d_m) / h;
	else
		tmp = -(-d_m) * (1.0 / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[l, 6.8e-208], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision] / h), $MachinePrecision], N[((-(-d$95$m)) * N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.8e-208

   1. Initial program 35.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot -1\right)}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}{h} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)}{h} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      10. lift-/.f64 25.0

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   7. Applied rewrites 25.0%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   if 6.8e-208 < l

   1. Initial program 35.8%

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

   2. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto -\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      12. lower-*.f64 43.1

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   4. Applied rewrites 43.1%

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

      1. lift-neg.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{\ell \cdot h}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      7. metadata-eval N/A

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

      8. sqrt-pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-pow2 N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      16. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      18. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      19. lower-neg.f64 N/A

          \[\leadsto \left(-\left(\mathsf{neg}\left(d\right)\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      20. lower-neg.f64 N/A

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

   6. Applied rewrites 43.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \frac{1}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(-\left(-d\right)\right) \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

      10. lift-*.f64 43.2

          \[\leadsto \left(-\left(-d\right)\right) \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

   8. Applied rewrites 43.2%

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

Alternative 15: 40.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\_m\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<= l 6.8e-208)
   (/ (* (sqrt (/ h l)) (- d_m)) h)
   (* (sqrt (/ 1.0 (* l h))) d_m)))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (l <= 6.8e-208) {
		tmp = (sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = sqrt((1.0 / (l * h))) * d_m;
	}
	return tmp;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    real(8) :: tmp
    if (l <= 6.8d-208) then
        tmp = (sqrt((h / l)) * -d_m) / h
    else
        tmp = sqrt((1.0d0 / (l * h))) * d_m
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (l <= 6.8e-208) {
		tmp = (Math.sqrt((h / l)) * -d_m) / h;
	} else {
		tmp = Math.sqrt((1.0 / (l * h))) * d_m;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if l <= 6.8e-208:
		tmp = (math.sqrt((h / l)) * -d_m) / h
	else:
		tmp = math.sqrt((1.0 / (l * h))) * d_m
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (l <= 6.8e-208)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * Float64(-d_m)) / h);
	else
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m);
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if (l <= 6.8e-208)
		tmp = (sqrt((h / l)) * -d_m) / h;
	else
		tmp = sqrt((1.0 / (l * h))) * d_m;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[l, 6.8e-208], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision] / h), $MachinePrecision], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.8e-208

   1. Initial program 35.8%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(d \cdot -1\right)}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-1 \cdot d\right)}{h} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\mathsf{neg}\left(d\right)\right)}{h} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

      10. lift-/.f64 25.0

          \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   7. Applied rewrites 25.0%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \left(-d\right)}{h} \]

   if 6.8e-208 < l

   1. Initial program 35.8%

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

   2. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

         \[\leadsto -\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. *-commutative N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      12. lower-*.f64 43.1

          \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   4. Applied rewrites 43.1%

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

      1. lift-neg.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{\ell \cdot h}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

      7. metadata-eval N/A

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

      8. sqrt-pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-pow2 N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      16. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      18. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

      19. lower-neg.f64 N/A

          \[\leadsto \left(-\left(\mathsf{neg}\left(d\right)\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      20. lower-neg.f64 N/A

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

   6. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(-d\right)\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{\ell \cdot h}} \]

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 43.1

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

   8. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 40.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D) :precision binary64 (* (sqrt (/ 1.0 (* l h))) d_m))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	return sqrt((1.0 / (l * h))) * d_m;
}

Fortran

d_m =     private
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, d)
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
    real(8), intent (in) :: d
    code = sqrt((1.0d0 / (l * h))) * d_m
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	return Math.sqrt((1.0 / (l * h))) * d_m;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	return math.sqrt((1.0 / (l * h))) * d_m

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	return Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m)
end

MATLAB

d_m = abs(d);
function tmp = code(d_m, h, l, M, D)
	tmp = sqrt((1.0 / (l * h))) * d_m;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]

Derivation

1. Initial program 35.8%

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

2. Taylor expanded in d around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. sqrt-pow2 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

      \[\leadsto -\left(d \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   7. *-commutative N/A

      \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   8. lower-*.f64 N/A

      \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   10. lower-/.f64 N/A

       \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   11. *-commutative N/A

       \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   12. lower-*.f64 43.1

       \[\leadsto -\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

4. Applied rewrites 43.1%

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

   1. lift-neg.f64 N/A

      \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{\ell \cdot h}}} \]

   5. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{\ell \cdot h}} \]

   6. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{\ell \cdot h}} \]

   7. metadata-eval N/A

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

   8. sqrt-pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. sqrt-pow2 N/A

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

   15. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left(d \cdot {-1}^{1}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   16. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left(d \cdot -1\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   17. *-commutative N/A

       \[\leadsto \left(\mathsf{neg}\left(-1 \cdot d\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

   18. mul-1-neg N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{h \cdot \ell}} \]

   19. lower-neg.f64 N/A

       \[\leadsto \left(-\left(\mathsf{neg}\left(d\right)\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   20. lower-neg.f64 N/A

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

6. Applied rewrites 43.1%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(-d\right)\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\ell \cdot h}}} \]

   3. lift-neg.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d\right)\right)\right)\right) \cdot \sqrt{\frac{\color{blue}{1}}{\ell \cdot h}} \]

   4. remove-double-neg N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 43.1

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

8. Applied rewrites 43.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(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)))))