Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.4% → 83.7%

Time: 10.3s

Alternatives: 17

Speedup: 0.7×

Specification

Math

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

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: 66.4% accurate, 1.0× speedup

Math

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

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: 83.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ t_3 := \mathsf{fma}\left(\frac{h \cdot \left(D \cdot M\right)}{\left(-\ell\right) \cdot \left(d + d\right)}, 0.25 \cdot \frac{D \cdot M}{d}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-158}:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(t\_0 \cdot 0.25, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+187}:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.125 \cdot \frac{t\_0 \cdot \left(M \cdot D\right)}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) d))
        (t_1
         (*
          (* (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)))))
        (t_2 (* (sqrt (/ d l)) (sqrt (/ d h))))
        (t_3
         (*
          (fma (/ (* h (* D M)) (* (- l) (+ d d))) (* 0.25 (/ (* D M) d)) 1.0)
          (/ (fabs d) (sqrt (* h l))))))
   (if (<= t_1 -2e-158)
     (* t_2 (fma (* t_0 0.25) (* (* (/ D (+ d d)) M) (/ (- h) l)) 1.0))
     (if (<= t_1 0.0)
       t_3
       (if (<= t_1 5e+187)
         (* t_2 (fma (/ h l) (* -0.125 (/ (* t_0 (* M D)) d)) 1.0))
         t_3)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (M * D) / d;
	double t_1 = (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)));
	double t_2 = sqrt((d / l)) * sqrt((d / h));
	double t_3 = fma(((h * (D * M)) / (-l * (d + d))), (0.25 * ((D * M) / d)), 1.0) * (fabs(d) / sqrt((h * l)));
	double tmp;
	if (t_1 <= -2e-158) {
		tmp = t_2 * fma((t_0 * 0.25), (((D / (d + d)) * M) * (-h / l)), 1.0);
	} else if (t_1 <= 0.0) {
		tmp = t_3;
	} else if (t_1 <= 5e+187) {
		tmp = t_2 * fma((h / l), (-0.125 * ((t_0 * (M * D)) / d)), 1.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(M * D) / d)
	t_1 = 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))))
	t_2 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	t_3 = Float64(fma(Float64(Float64(h * Float64(D * M)) / Float64(Float64(-l) * Float64(d + d))), Float64(0.25 * Float64(Float64(D * M) / d)), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))))
	tmp = 0.0
	if (t_1 <= -2e-158)
		tmp = Float64(t_2 * fma(Float64(t_0 * 0.25), Float64(Float64(Float64(D / Float64(d + d)) * M) * Float64(Float64(-h) / l)), 1.0));
	elseif (t_1 <= 0.0)
		tmp = t_3;
	elseif (t_1 <= 5e+187)
		tmp = Float64(t_2 * fma(Float64(h / l), Float64(-0.125 * Float64(Float64(t_0 * Float64(M * D)) / d)), 1.0));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(h * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[((-l) * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-158], N[(t$95$2 * N[(N[(t$95$0 * 0.25), $MachinePrecision] * N[(N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * N[((-h) / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$3, If[LessEqual[t$95$1, 5e+187], N[(t$95$2 * N[(N[(h / l), $MachinePrecision] * N[(-0.125 * N[(N[(t$95$0 * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

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)))) < -2.00000000000000013e-158

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 66.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \color{blue}{0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   7. Applied rewrites 66.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{d} \cdot 0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   if -2.00000000000000013e-158 < (*.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)))) < -0.0 or 5.0000000000000001e187 < (*.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 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 66.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \color{blue}{0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   7. Applied rewrites 66.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{d} \cdot 0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   9. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(D \cdot M\right)}{\left(-\ell\right) \cdot \left(d + d\right)}, 0.25 \cdot \frac{D \cdot M}{d}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if -0.0 < (*.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)))) < 5.0000000000000001e187

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 65.9%

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

Alternative 2: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_0 \leq 5 \cdot 10^{+187}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h \cdot \left(D \cdot M\right)}{\left(-\ell\right) \cdot \left(d + d\right)}, 0.25 \cdot \frac{D \cdot M}{d}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
   (if (<=
        (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) t_0)
        5e+187)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) t_0)
     (*
      (fma (/ (* h (* D M)) (* (- l) (+ d d))) (* 0.25 (/ (* D M) d)) 1.0)
      (/ (fabs d) (sqrt (* h l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * t_0) <= 5e+187) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_0;
	} else {
		tmp = fma(((h * (D * M)) / (-l * (d + d))), (0.25 * ((D * M) / d)), 1.0) * (fabs(d) / sqrt((h * l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * t_0) <= 5e+187)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * t_0);
	else
		tmp = Float64(fma(Float64(Float64(h * Float64(D * M)) / Float64(Float64(-l) * Float64(d + d))), Float64(0.25 * Float64(Float64(D * M) / d)), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, If[LessEqual[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] * t$95$0), $MachinePrecision], 5e+187], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(h * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[((-l) * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000001e187

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.4

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.4

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.4

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

   3. Applied rewrites 66.4%

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

   if 5.0000000000000001e187 < (*.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 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 66.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \color{blue}{0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   7. Applied rewrites 66.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{d} \cdot 0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   9. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(D \cdot M\right)}{\left(-\ell\right) \cdot \left(d + d\right)}, 0.25 \cdot \frac{D \cdot M}{d}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot 0.25, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+187}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h \cdot \left(D \cdot M\right)}{\left(-\ell\right) \cdot \left(d + d\right)}, 0.25 \cdot \frac{D \cdot M}{d}, 1\right) \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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)))))
        (t_1 (/ (fabs d) (sqrt (* h l)))))
   (if (<= t_0 0.0)
     (*
      t_1
      (fma (* (/ (* M D) d) 0.25) (* (* (/ D (+ d d)) M) (/ (- h) l)) 1.0))
     (if (<= t_0 5e+187)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (*
        (fma (/ (* h (* D M)) (* (- l) (+ d d))) (* 0.25 (/ (* D M) d)) 1.0)
        t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = fabs(d) / sqrt((h * l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1 * fma((((M * D) / d) * 0.25), (((D / (d + d)) * M) * (-h / l)), 1.0);
	} else if (t_0 <= 5e+187) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = fma(((h * (D * M)) / (-l * (d + d))), (0.25 * ((D * M) / d)), 1.0) * t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	t_1 = Float64(abs(d) / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(t_1 * fma(Float64(Float64(Float64(M * D) / d) * 0.25), Float64(Float64(Float64(D / Float64(d + d)) * M) * Float64(Float64(-h) / l)), 1.0));
	elseif (t_0 <= 5e+187)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(fma(Float64(Float64(h * Float64(D * M)) / Float64(Float64(-l) * Float64(d + d))), Float64(0.25 * Float64(Float64(D * M) / d)), 1.0) * t_1);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(t$95$1 * N[(N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(N[(D / N[(d + d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * N[((-h) / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+187], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(h * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[((-l) * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $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)))) < -0.0

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 66.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \color{blue}{0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   7. Applied rewrites 66.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{d} \cdot 0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell}} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell} \cdot \color{blue}{\frac{d}{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      7. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      12. lower-unsound-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      13. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      15. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      16. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      17. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      18. lower-unsound-sqrt.f64 69.8

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot 0.25, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   9. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot 0.25, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   if -0.0 < (*.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)))) < 5.0000000000000001e187

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 38.9%

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

   if 5.0000000000000001e187 < (*.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 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 66.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \color{blue}{0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   7. Applied rewrites 66.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{d} \cdot 0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   9. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(D \cdot M\right)}{\left(-\ell\right) \cdot \left(d + d\right)}, 0.25 \cdot \frac{D \cdot M}{d}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+187}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.125 \cdot \frac{\frac{M \cdot D}{d} \cdot \left(M \cdot D\right)}{d}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h \cdot \left(D \cdot M\right)}{\left(-\ell\right) \cdot \left(d + d\right)}, 0.25 \cdot \frac{D \cdot M}{d}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (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))))
      5e+187)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (fma (/ h l) (* -0.125 (/ (* (/ (* M D) d) (* M D)) d)) 1.0))
   (*
    (fma (/ (* h (* D M)) (* (- l) (+ d d))) (* 0.25 (/ (* D M) d)) 1.0)
    (/ (fabs d) (sqrt (* h l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((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)))) <= 5e+187) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * fma((h / l), (-0.125 * ((((M * D) / d) * (M * D)) / d)), 1.0);
	} else {
		tmp = fma(((h * (D * M)) / (-l * (d + d))), (0.25 * ((D * M) / d)), 1.0) * (fabs(d) / sqrt((h * l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (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)))) <= 5e+187)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * fma(Float64(h / l), Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) / d) * Float64(M * D)) / d)), 1.0));
	else
		tmp = Float64(fma(Float64(Float64(h * Float64(D * M)) / Float64(Float64(-l) * Float64(d + d))), Float64(0.25 * Float64(Float64(D * M) / d)), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[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], 5e+187], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(h * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[((-l) * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000001e187

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 65.9%

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

   if 5.0000000000000001e187 < (*.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 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 66.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \color{blue}{0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   7. Applied rewrites 66.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{d} \cdot 0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   9. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(D \cdot M\right)}{\left(-\ell\right) \cdot \left(d + d\right)}, 0.25 \cdot \frac{D \cdot M}{d}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \mathsf{fma}\left(\frac{h \cdot \left(D \cdot M\right)}{\left(-\ell\right) \cdot \left(d + d\right)}, 0.25 \cdot \frac{D \cdot M}{d}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+187}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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)))))
        (t_1
         (*
          (fma (/ (* h (* D M)) (* (- l) (+ d d))) (* 0.25 (/ (* D M) d)) 1.0)
          (/ (fabs d) (sqrt (* h l))))))
   (if (<= t_0 0.0)
     t_1
     (if (<= t_0 5e+187) (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = fma(((h * (D * M)) / (-l * (d + d))), (0.25 * ((D * M) / d)), 1.0) * (fabs(d) / sqrt((h * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+187) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	t_1 = Float64(fma(Float64(Float64(h * Float64(D * M)) / Float64(Float64(-l) * Float64(d + d))), Float64(0.25 * Float64(Float64(D * M) / d)), 1.0) * Float64(abs(d) / sqrt(Float64(h * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+187)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(N[(N[(h * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[((-l) * N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+187], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or 5.0000000000000001e187 < (*.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 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 66.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\frac{M \cdot D}{d} \cdot \color{blue}{0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   7. Applied rewrites 66.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{M \cdot D}{d} \cdot 0.25}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   9. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{h \cdot \left(D \cdot M\right)}{\left(-\ell\right) \cdot \left(d + d\right)}, 0.25 \cdot \frac{D \cdot M}{d}, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if -0.0 < (*.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)))) < 5.0000000000000001e187

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 38.9%

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

Alternative 6: 73.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \frac{\frac{M \cdot D}{d} \cdot \left(M \cdot D\right)}{d}, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h \cdot \ell} \cdot d}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{\ell}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+213}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \frac{\ell - \left(\left(0.125 \cdot \frac{\left(D \cdot M\right) \cdot D}{d \cdot d}\right) \cdot M\right) \cdot h}{\ell}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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))))))
   (if (<= t_0 -5e-151)
     (*
      (fma (* -0.125 (/ (* (/ (* M D) d) (* M D)) d)) (/ h l) 1.0)
      (sqrt (* (/ d (* h l)) d)))
     (if (<= t_0 0.0)
       (/ (* -1.0 (* d (* l (sqrt (/ 1.0 (* h l)))))) l)
       (if (<= t_0 5e+213)
         (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
         (*
          (/ (fabs d) (sqrt (* h l)))
          (/ (- l (* (* (* 0.125 (/ (* (* D M) D) (* d d))) M) h)) l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double tmp;
	if (t_0 <= -5e-151) {
		tmp = fma((-0.125 * ((((M * D) / d) * (M * D)) / d)), (h / l), 1.0) * sqrt(((d / (h * l)) * d));
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 * (d * (l * sqrt((1.0 / (h * l)))))) / l;
	} else if (t_0 <= 5e+213) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * ((l - (((0.125 * (((D * M) * D) / (d * d))) * M) * h)) / l);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	tmp = 0.0
	if (t_0 <= -5e-151)
		tmp = Float64(fma(Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) / d) * Float64(M * D)) / d)), Float64(h / l), 1.0) * sqrt(Float64(Float64(d / Float64(h * l)) * d)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 * Float64(d * Float64(l * sqrt(Float64(1.0 / Float64(h * l)))))) / l);
	elseif (t_0 <= 5e+213)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(Float64(l - Float64(Float64(Float64(0.125 * Float64(Float64(Float64(D * M) * D) / Float64(d * d))) * M) * h)) / l));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -5e-151], N[(N[(N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 * N[(d * N[(l * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t$95$0, 5e+213], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(l - N[(N[(N[(0.125 * N[(N[(N[(D * M), $MachinePrecision] * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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)))) < -5.00000000000000003e-151

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 50.6%

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

   if -5.00000000000000003e-151 < (*.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)))) < -0.0

   1. Initial program 66.4%

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

   4. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 23.0

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

   6. Applied rewrites 23.0%

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

   if -0.0 < (*.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)))) < 4.9999999999999998e213

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 38.9%

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

   if 4.9999999999999998e213 < (*.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 66.4%

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

   5. Applied rewrites 60.4%

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

Alternative 7: 70.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_1 := \ell - \left(\left(0.125 \cdot \frac{\left(D \cdot M\right) \cdot D}{d \cdot d}\right) \cdot M\right) \cdot h\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{t\_0 \cdot t\_1}{\ell}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{\ell}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+213}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{t\_1}{\ell}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (fabs d) (sqrt (* h l))))
        (t_1 (- l (* (* (* 0.125 (/ (* (* D M) D) (* d d))) M) h)))
        (t_2
         (*
          (* (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))))))
   (if (<= t_2 -5e-151)
     (/ (* t_0 t_1) l)
     (if (<= t_2 0.0)
       (/ (* -1.0 (* d (* l (sqrt (/ 1.0 (* h l)))))) l)
       (if (<= t_2 5e+213)
         (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
         (* t_0 (/ t_1 l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(d) / sqrt((h * l));
	double t_1 = l - (((0.125 * (((D * M) * D) / (d * d))) * M) * h);
	double t_2 = (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)));
	double tmp;
	if (t_2 <= -5e-151) {
		tmp = (t_0 * t_1) / l;
	} else if (t_2 <= 0.0) {
		tmp = (-1.0 * (d * (l * sqrt((1.0 / (h * l)))))) / l;
	} else if (t_2 <= 5e+213) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_0 * (t_1 / l);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = abs(d) / sqrt((h * l))
    t_1 = l - (((0.125d0 * (((d_1 * m) * d_1) / (d * d))) * m) * h)
    t_2 = (((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)))
    if (t_2 <= (-5d-151)) then
        tmp = (t_0 * t_1) / l
    else if (t_2 <= 0.0d0) then
        tmp = ((-1.0d0) * (d * (l * sqrt((1.0d0 / (h * l)))))) / l
    else if (t_2 <= 5d+213) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = t_0 * (t_1 / l)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.abs(d) / Math.sqrt((h * l));
	double t_1 = l - (((0.125 * (((D * M) * D) / (d * d))) * M) * h);
	double t_2 = (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)));
	double tmp;
	if (t_2 <= -5e-151) {
		tmp = (t_0 * t_1) / l;
	} else if (t_2 <= 0.0) {
		tmp = (-1.0 * (d * (l * Math.sqrt((1.0 / (h * l)))))) / l;
	} else if (t_2 <= 5e+213) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_0 * (t_1 / l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.fabs(d) / math.sqrt((h * l))
	t_1 = l - (((0.125 * (((D * M) * D) / (d * d))) * M) * h)
	t_2 = (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)))
	tmp = 0
	if t_2 <= -5e-151:
		tmp = (t_0 * t_1) / l
	elif t_2 <= 0.0:
		tmp = (-1.0 * (d * (l * math.sqrt((1.0 / (h * l)))))) / l
	elif t_2 <= 5e+213:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_0 * (t_1 / l)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(d) / sqrt(Float64(h * l)))
	t_1 = Float64(l - Float64(Float64(Float64(0.125 * Float64(Float64(Float64(D * M) * D) / Float64(d * d))) * M) * h))
	t_2 = 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))))
	tmp = 0.0
	if (t_2 <= -5e-151)
		tmp = Float64(Float64(t_0 * t_1) / l);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-1.0 * Float64(d * Float64(l * sqrt(Float64(1.0 / Float64(h * l)))))) / l);
	elseif (t_2 <= 5e+213)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(t_0 * Float64(t_1 / l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = abs(d) / sqrt((h * l));
	t_1 = l - (((0.125 * (((D * M) * D) / (d * d))) * M) * h);
	t_2 = (((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)));
	tmp = 0.0;
	if (t_2 <= -5e-151)
		tmp = (t_0 * t_1) / l;
	elseif (t_2 <= 0.0)
		tmp = (-1.0 * (d * (l * sqrt((1.0 / (h * l)))))) / l;
	elseif (t_2 <= 5e+213)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_0 * (t_1 / l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(l - N[(N[(N[(0.125 * N[(N[(N[(D * M), $MachinePrecision] * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -5e-151], N[(N[(t$95$0 * t$95$1), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(-1.0 * N[(d * N[(l * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t$95$2, 5e+213], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(t$95$0 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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)))) < -5.00000000000000003e-151

   1. Initial program 66.4%

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

   5. Applied rewrites 56.3%

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

   if -5.00000000000000003e-151 < (*.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)))) < -0.0

   1. Initial program 66.4%

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

   4. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 23.0

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

   6. Applied rewrites 23.0%

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

   if -0.0 < (*.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)))) < 4.9999999999999998e213

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 38.9%

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

   if 4.9999999999999998e213 < (*.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 66.4%

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

   5. Applied rewrites 60.4%

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

Alternative 8: 70.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \frac{\ell - \left(\left(0.125 \cdot \frac{\left(D \cdot M\right) \cdot D}{d \cdot d}\right) \cdot M\right) \cdot h}{\ell}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{\ell}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+213}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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)))))
        (t_1
         (*
          (/ (fabs d) (sqrt (* h l)))
          (/ (- l (* (* (* 0.125 (/ (* (* D M) D) (* d d))) M) h)) l))))
   (if (<= t_0 -5e-151)
     t_1
     (if (<= t_0 0.0)
       (/ (* -1.0 (* d (* l (sqrt (/ 1.0 (* h l)))))) l)
       (if (<= t_0 5e+213) (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = (fabs(d) / sqrt((h * l))) * ((l - (((0.125 * (((D * M) * D) / (d * d))) * M) * h)) / l);
	double tmp;
	if (t_0 <= -5e-151) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 * (d * (l * sqrt((1.0 / (h * l)))))) / l;
	} else if (t_0 <= 5e+213) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((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)))
    t_1 = (abs(d) / sqrt((h * l))) * ((l - (((0.125d0 * (((d_1 * m) * d_1) / (d * d))) * m) * h)) / l)
    if (t_0 <= (-5d-151)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = ((-1.0d0) * (d * (l * sqrt((1.0d0 / (h * l)))))) / l
    else if (t_0 <= 5d+213) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = (Math.abs(d) / Math.sqrt((h * l))) * ((l - (((0.125 * (((D * M) * D) / (d * d))) * M) * h)) / l);
	double tmp;
	if (t_0 <= -5e-151) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 * (d * (l * Math.sqrt((1.0 / (h * l)))))) / l;
	} else if (t_0 <= 5e+213) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (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)))
	t_1 = (math.fabs(d) / math.sqrt((h * l))) * ((l - (((0.125 * (((D * M) * D) / (d * d))) * M) * h)) / l)
	tmp = 0
	if t_0 <= -5e-151:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (-1.0 * (d * (l * math.sqrt((1.0 / (h * l)))))) / l
	elif t_0 <= 5e+213:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(Float64(l - Float64(Float64(Float64(0.125 * Float64(Float64(Float64(D * M) * D) / Float64(d * d))) * M) * h)) / l))
	tmp = 0.0
	if (t_0 <= -5e-151)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 * Float64(d * Float64(l * sqrt(Float64(1.0 / Float64(h * l)))))) / l);
	elseif (t_0 <= 5e+213)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((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)));
	t_1 = (abs(d) / sqrt((h * l))) * ((l - (((0.125 * (((D * M) * D) / (d * d))) * M) * h)) / l);
	tmp = 0.0;
	if (t_0 <= -5e-151)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = (-1.0 * (d * (l * sqrt((1.0 / (h * l)))))) / l;
	elseif (t_0 <= 5e+213)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(l - N[(N[(N[(0.125 * N[(N[(N[(D * M), $MachinePrecision] * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-151], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 * N[(d * N[(l * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t$95$0, 5e+213], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]]

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)))) < -5.00000000000000003e-151 or 4.9999999999999998e213 < (*.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 66.4%

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

   5. Applied rewrites 60.4%

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

   if -5.00000000000000003e-151 < (*.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)))) < -0.0

   1. Initial program 66.4%

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

   4. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 23.0

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

   6. Applied rewrites 23.0%

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

   if -0.0 < (*.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)))) < 4.9999999999999998e213

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 38.9%

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

Alternative 9: 52.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{-1 \cdot \left(h \cdot \left(t\_0 \cdot \sqrt{\sqrt{\frac{d}{\ell} \cdot \frac{d}{\ell}}}\right)\right)}{h}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{\ell}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left(t\_1 \cdot t\_0\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot t\_1}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2
         (*
          (* (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))))))
   (if (<= t_2 -5e-151)
     (/ (* -1.0 (* h (* t_0 (sqrt (sqrt (* (/ d l) (/ d l))))))) h)
     (if (<= t_2 0.0)
       (/ (* -1.0 (* d (* l (sqrt (/ 1.0 (* h l)))))) l)
       (if (<= t_2 INFINITY)
         (* (* t_1 t_0) 1.0)
         (/ (* (* -1.0 (* d (sqrt (/ h d)))) t_1) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = (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)));
	double tmp;
	if (t_2 <= -5e-151) {
		tmp = (-1.0 * (h * (t_0 * sqrt(sqrt(((d / l) * (d / l))))))) / h;
	} else if (t_2 <= 0.0) {
		tmp = (-1.0 * (d * (l * sqrt((1.0 / (h * l)))))) / l;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (t_1 * t_0) * 1.0;
	} else {
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_1) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / h));
	double t_1 = Math.sqrt((d / l));
	double t_2 = (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)));
	double tmp;
	if (t_2 <= -5e-151) {
		tmp = (-1.0 * (h * (t_0 * Math.sqrt(Math.sqrt(((d / l) * (d / l))))))) / h;
	} else if (t_2 <= 0.0) {
		tmp = (-1.0 * (d * (l * Math.sqrt((1.0 / (h * l)))))) / l;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (t_1 * t_0) * 1.0;
	} else {
		tmp = ((-1.0 * (d * Math.sqrt((h / d)))) * t_1) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = math.sqrt((d / l))
	t_2 = (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)))
	tmp = 0
	if t_2 <= -5e-151:
		tmp = (-1.0 * (h * (t_0 * math.sqrt(math.sqrt(((d / l) * (d / l))))))) / h
	elif t_2 <= 0.0:
		tmp = (-1.0 * (d * (l * math.sqrt((1.0 / (h * l)))))) / l
	elif t_2 <= math.inf:
		tmp = (t_1 * t_0) * 1.0
	else:
		tmp = ((-1.0 * (d * math.sqrt((h / d)))) * t_1) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = 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))))
	tmp = 0.0
	if (t_2 <= -5e-151)
		tmp = Float64(Float64(-1.0 * Float64(h * Float64(t_0 * sqrt(sqrt(Float64(Float64(d / l) * Float64(d / l))))))) / h);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-1.0 * Float64(d * Float64(l * sqrt(Float64(1.0 / Float64(h * l)))))) / l);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(t_1 * t_0) * 1.0);
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(d * sqrt(Float64(h / d)))) * t_1) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = sqrt((d / l));
	t_2 = (((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)));
	tmp = 0.0;
	if (t_2 <= -5e-151)
		tmp = (-1.0 * (h * (t_0 * sqrt(sqrt(((d / l) * (d / l))))))) / h;
	elseif (t_2 <= 0.0)
		tmp = (-1.0 * (d * (l * sqrt((1.0 / (h * l)))))) / l;
	elseif (t_2 <= Inf)
		tmp = (t_1 * t_0) * 1.0;
	else
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_1) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -5e-151], N[(N[(-1.0 * N[(h * N[(t$95$0 * N[Sqrt[N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(-1.0 * N[(d * N[(l * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(t$95$1 * t$95$0), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(-1.0 * N[(d * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / h), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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)))) < -5.00000000000000003e-151

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.6

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 9.6

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

   7. Applied rewrites 9.6%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 13.7

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

   9. Applied rewrites 13.7%

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

   if -5.00000000000000003e-151 < (*.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)))) < -0.0

   1. Initial program 66.4%

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

   4. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 23.0

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

   6. Applied rewrites 23.0%

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

   if -0.0 < (*.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)))) < +inf.0

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 38.9%

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

   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 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.6

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 11.5

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

   7. Applied rewrites 11.5%

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

Alternative 10: 52.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-h\right)}{h}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{\ell}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot t\_0}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (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))))))
   (if (<= t_1 -5e-151)
     (/ (* (sqrt (* (/ d (* h l)) d)) (- h)) h)
     (if (<= t_1 0.0)
       (/ (* -1.0 (* d (* l (sqrt (/ 1.0 (* h l)))))) l)
       (if (<= t_1 INFINITY)
         (* (* t_0 (sqrt (/ d h))) 1.0)
         (/ (* (* -1.0 (* d (sqrt (/ h d)))) t_0) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (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)));
	double tmp;
	if (t_1 <= -5e-151) {
		tmp = (sqrt(((d / (h * l)) * d)) * -h) / h;
	} else if (t_1 <= 0.0) {
		tmp = (-1.0 * (d * (l * sqrt((1.0 / (h * l)))))) / l;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * sqrt((d / h))) * 1.0;
	} else {
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_0) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (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)));
	double tmp;
	if (t_1 <= -5e-151) {
		tmp = (Math.sqrt(((d / (h * l)) * d)) * -h) / h;
	} else if (t_1 <= 0.0) {
		tmp = (-1.0 * (d * (l * Math.sqrt((1.0 / (h * l)))))) / l;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = ((-1.0 * (d * Math.sqrt((h / d)))) * t_0) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = (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)))
	tmp = 0
	if t_1 <= -5e-151:
		tmp = (math.sqrt(((d / (h * l)) * d)) * -h) / h
	elif t_1 <= 0.0:
		tmp = (-1.0 * (d * (l * math.sqrt((1.0 / (h * l)))))) / l
	elif t_1 <= math.inf:
		tmp = (t_0 * math.sqrt((d / h))) * 1.0
	else:
		tmp = ((-1.0 * (d * math.sqrt((h / d)))) * t_0) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = 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))))
	tmp = 0.0
	if (t_1 <= -5e-151)
		tmp = Float64(Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(-h)) / h);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-1.0 * Float64(d * Float64(l * sqrt(Float64(1.0 / Float64(h * l)))))) / l);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(d * sqrt(Float64(h / d)))) * t_0) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = (((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)));
	tmp = 0.0;
	if (t_1 <= -5e-151)
		tmp = (sqrt(((d / (h * l)) * d)) * -h) / h;
	elseif (t_1 <= 0.0)
		tmp = (-1.0 * (d * (l * sqrt((1.0 / (h * l)))))) / l;
	elseif (t_1 <= Inf)
		tmp = (t_0 * sqrt((d / h))) * 1.0;
	else
		tmp = ((-1.0 * (d * sqrt((h / d)))) * t_0) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e-151], N[(N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * (-h)), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(-1.0 * N[(d * N[(l * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(-1.0 * N[(d * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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)))) < -5.00000000000000003e-151

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.6

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 9.6

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

   7. Applied rewrites 9.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-*.f64 9.6

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

   9. Applied rewrites 13.6%

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

   if -5.00000000000000003e-151 < (*.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)))) < -0.0

   1. Initial program 66.4%

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

   4. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 23.0

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

   6. Applied rewrites 23.0%

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

   if -0.0 < (*.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)))) < +inf.0

   1. Initial program 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 38.9%

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

   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 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.6

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 11.5

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

   7. Applied rewrites 11.5%

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

Alternative 11: 49.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-h\right)}{h}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \left(\ell \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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))))))
   (if (<= t_0 -5e-151)
     (/ (* (sqrt (* (/ d (* h l)) d)) (- h)) h)
     (if (<= t_0 0.0)
       (/ (* -1.0 (* d (* l (sqrt (/ 1.0 (* h l)))))) l)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double tmp;
	if (t_0 <= -5e-151) {
		tmp = (sqrt(((d / (h * l)) * d)) * -h) / h;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 * (d * (l * sqrt((1.0 / (h * l)))))) / l;
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((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)))
    if (t_0 <= (-5d-151)) then
        tmp = (sqrt(((d / (h * l)) * d)) * -h) / h
    else if (t_0 <= 0.0d0) then
        tmp = ((-1.0d0) * (d * (l * sqrt((1.0d0 / (h * l)))))) / l
    else
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double tmp;
	if (t_0 <= -5e-151) {
		tmp = (Math.sqrt(((d / (h * l)) * d)) * -h) / h;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 * (d * (l * Math.sqrt((1.0 / (h * l)))))) / l;
	} else {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (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)))
	tmp = 0
	if t_0 <= -5e-151:
		tmp = (math.sqrt(((d / (h * l)) * d)) * -h) / h
	elif t_0 <= 0.0:
		tmp = (-1.0 * (d * (l * math.sqrt((1.0 / (h * l)))))) / l
	else:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	tmp = 0.0
	if (t_0 <= -5e-151)
		tmp = Float64(Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(-h)) / h);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 * Float64(d * Float64(l * sqrt(Float64(1.0 / Float64(h * l)))))) / l);
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((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)));
	tmp = 0.0;
	if (t_0 <= -5e-151)
		tmp = (sqrt(((d / (h * l)) * d)) * -h) / h;
	elseif (t_0 <= 0.0)
		tmp = (-1.0 * (d * (l * sqrt((1.0 / (h * l)))))) / l;
	else
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -5e-151], N[(N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * (-h)), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 * N[(d * N[(l * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $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)))) < -5.00000000000000003e-151

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.6

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 9.6

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

   7. Applied rewrites 9.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-*.f64 9.6

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

   9. Applied rewrites 13.6%

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

   if -5.00000000000000003e-151 < (*.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)))) < -0.0

   1. Initial program 66.4%

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

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

   4. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 23.0

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

   6. Applied rewrites 23.0%

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

   if -0.0 < (*.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 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 38.9%

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

Alternative 12: 48.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{h \cdot \ell} \cdot d} \cdot \left(-h\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (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))))
      -2e-158)
   (/ (* (sqrt (* (/ d (* h l)) d)) (- h)) h)
   (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((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)))) <= -2e-158) {
		tmp = (sqrt(((d / (h * l)) * d)) * -h) / h;
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((((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)))) <= (-2d-158)) then
        tmp = (sqrt(((d / (h * l)) * d)) * -h) / h
    else
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((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)))) <= -2e-158) {
		tmp = (Math.sqrt(((d / (h * l)) * d)) * -h) / h;
	} else {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((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)))) <= -2e-158:
		tmp = (math.sqrt(((d / (h * l)) * d)) * -h) / h
	else:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (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)))) <= -2e-158)
		tmp = Float64(Float64(sqrt(Float64(Float64(d / Float64(h * l)) * d)) * Float64(-h)) / h);
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((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)))) <= -2e-158)
		tmp = (sqrt(((d / (h * l)) * d)) * -h) / h;
	else
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[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], -2e-158], N[(N[(N[Sqrt[N[(N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * (-h)), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000013e-158

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.6

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

   4. Applied rewrites 23.6%

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

   5. Taylor expanded in h around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 9.6

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

   7. Applied rewrites 9.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-*.f64 9.6

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

   9. Applied rewrites 13.6%

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

   if -2.00000000000000013e-158 < (*.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 66.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

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

      7. pow1/2 N/A

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

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

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

      10. metadata-eval 66.9

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

      11. lift-pow.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.9

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

   5. Applied rewrites 66.9%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 38.9%

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

Alternative 13: 46.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (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))))
      0.0)
   (/ (* -1.0 (* d (sqrt (/ h l)))) h)
   (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((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)))) <= 0.0) {
		tmp = (-1.0 * (d * sqrt((h / l)))) / h;
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((((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)))) <= 0.0d0) then
        tmp = ((-1.0d0) * (d * sqrt((h / l)))) / h
    else
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((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)))) <= 0.0) {
		tmp = (-1.0 * (d * Math.sqrt((h / l)))) / h;
	} else {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((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)))) <= 0.0:
		tmp = (-1.0 * (d * math.sqrt((h / l)))) / h
	else:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (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)))) <= 0.0)
		tmp = Float64(Float64(-1.0 * Float64(d * sqrt(Float64(h / l)))) / h);
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((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)))) <= 0.0)
		tmp = (-1.0 * (d * sqrt((h / l)))) / h;
	else
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[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], 0.0], N[(N[(-1.0 * N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.6

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

   4. Applied rewrites 23.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 21.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 21.2

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

   6. Applied rewrites 21.2%

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

   7. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      4. lower-/.f64 13.3

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   9. Applied rewrites 13.3%

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   if -0.0 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}\right)} \]

      4. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell} + 1\right)} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} + 1\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{h}{\ell}\right)\right)} + 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \left(\mathsf{neg}\left(\frac{h}{\ell}\right)\right) + 1\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \left(\mathsf{neg}\left(\frac{h}{\ell}\right)\right) + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{h}{\ell}\right)\right) + 1\right) \]

      10. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \left(\mathsf{neg}\left(\frac{h}{\ell}\right)\right) + 1\right) \]

      11. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\mathsf{neg}\left(\frac{h}{\ell}\right)\right)\right)} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \left(\mathsf{neg}\left(\frac{h}{\ell}\right)\right), 1\right)} \]

   3. Applied rewrites 66.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      3. lower-*.f64 66.9

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      5. metadata-eval 66.9

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      8. lift-sqrt.f64 66.9

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      10. metadata-eval 66.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

      13. lower-sqrt.f64 66.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \mathsf{fma}\left(\frac{0.5 \cdot \left(D \cdot M\right)}{d + d}, \left(\frac{D}{d + d} \cdot M\right) \cdot \frac{-h}{\ell}, 1\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 46.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := d \cdot \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{-1 \cdot t\_0}{h}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-158}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* d (sqrt (/ h l))))
        (t_1
         (*
          (* (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)))))
        (t_2 (/ (* -1.0 t_0) h)))
   (if (<= t_1 -2e-158) t_2 (if (<= t_1 INFINITY) (/ t_0 h) t_2))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d * sqrt((h / l));
	double t_1 = (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)));
	double t_2 = (-1.0 * t_0) / h;
	double tmp;
	if (t_1 <= -2e-158) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0 / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d * Math.sqrt((h / l));
	double t_1 = (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)));
	double t_2 = (-1.0 * t_0) / h;
	double tmp;
	if (t_1 <= -2e-158) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d * math.sqrt((h / l))
	t_1 = (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)))
	t_2 = (-1.0 * t_0) / h
	tmp = 0
	if t_1 <= -2e-158:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = t_0 / h
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d * sqrt(Float64(h / l)))
	t_1 = 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))))
	t_2 = Float64(Float64(-1.0 * t_0) / h)
	tmp = 0.0
	if (t_1 <= -2e-158)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(t_0 / h);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d * sqrt((h / l));
	t_1 = (((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)));
	t_2 = (-1.0 * t_0) / h;
	tmp = 0.0;
	if (t_1 <= -2e-158)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = t_0 / h;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(N[(-1.0 * t$95$0), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-158], t$95$2, If[LessEqual[t$95$1, Infinity], N[(t$95$0 / h), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000013e-158 or +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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.2

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.2

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.2%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

      4. lower-/.f64 13.3

         \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   9. Applied rewrites 13.3%

      \[\leadsto \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{\ell}}\right)}{h} \]

   if -2.00000000000000013e-158 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.2

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.2

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.2%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 36.9

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 36.9%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 42.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;h \leq 5 \cdot 10^{-295}:\\ \;\;\;\;-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 5e-295)
   (* -1.0 (* d (sqrt (/ 1.0 (* h l)))))
   (/ (* d (/ (sqrt h) (sqrt l))) h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 5e-295) {
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (h <= 5d-295) then
        tmp = (-1.0d0) * (d * sqrt((1.0d0 / (h * l))))
    else
        tmp = (d * (sqrt(h) / sqrt(l))) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 5e-295) {
		tmp = -1.0 * (d * Math.sqrt((1.0 / (h * l))));
	} else {
		tmp = (d * (Math.sqrt(h) / Math.sqrt(l))) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 5e-295:
		tmp = -1.0 * (d * math.sqrt((1.0 / (h * l))))
	else:
		tmp = (d * (math.sqrt(h) / math.sqrt(l))) / h
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 5e-295)
		tmp = Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(Float64(d * Float64(sqrt(h) / sqrt(l))) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 5e-295)
		tmp = -1.0 * (d * sqrt((1.0 / (h * l))));
	else
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 5e-295], N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 5.00000000000000008e-295

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\frac{\left(\ell - \left(M \cdot \left(\frac{D \cdot \left(D \cdot M\right)}{4 \cdot \left(d \cdot d\right)} \cdot 0.5\right)\right) \cdot h\right) \cdot \sqrt{d \cdot \frac{d}{h \cdot \ell}}}{\ell}} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 25.9

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 25.9%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   if 5.00000000000000008e-295 < h

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.2

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.2

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.2%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 36.9

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 36.9%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       3. sqrt-div N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       4. lower-unsound-/.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       5. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       6. lower-unsound-sqrt.f64 22.0

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

   11. Applied rewrites 22.0%

       \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 40.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;d \leq 2.3 \cdot 10^{-307}:\\ \;\;\;\;\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 2.3e-307)
   (/ (* d (sqrt (/ h l))) h)
   (/ (* d (/ (sqrt h) (sqrt l))) h)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.3e-307) {
		tmp = (d * sqrt((h / l))) / h;
	} else {
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= 2.3d-307) then
        tmp = (d * sqrt((h / l))) / h
    else
        tmp = (d * (sqrt(h) / sqrt(l))) / h
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 2.3e-307) {
		tmp = (d * Math.sqrt((h / l))) / h;
	} else {
		tmp = (d * (Math.sqrt(h) / Math.sqrt(l))) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 2.3e-307:
		tmp = (d * math.sqrt((h / l))) / h
	else:
		tmp = (d * (math.sqrt(h) / math.sqrt(l))) / h
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 2.3e-307)
		tmp = Float64(Float64(d * sqrt(Float64(h / l))) / h);
	else
		tmp = Float64(Float64(d * Float64(sqrt(h) / sqrt(l))) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 2.3e-307)
		tmp = (d * sqrt((h / l))) / h;
	else
		tmp = (d * (sqrt(h) / sqrt(l))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 2.3e-307], N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.2999999999999999e-307

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.2

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.2

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.2%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 36.9

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 36.9%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   if 2.2999999999999999e-307 < d

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 21.2

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      9. lower-*.f64 21.2

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   6. Applied rewrites 21.2%

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   7. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-/.f64 36.9

         \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 36.9%

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

       3. sqrt-div N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       4. lower-unsound-/.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       5. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

       6. lower-unsound-sqrt.f64 22.0

          \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]

   11. Applied rewrites 22.0%

       \[\leadsto \frac{d \cdot \frac{\sqrt{h}}{\sqrt{\ell}}}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 36.9% accurate, 7.4× speedup

Math

\[\frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ (* d (sqrt (/ h l))) h))

C

double code(double d, double h, double l, double M, double D) {
	return (d * sqrt((h / l))) / h;
}

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 * sqrt((h / l))) / h
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (d * Math.sqrt((h / l))) / h;
}

Python

def code(d, h, l, M, D):
	return (d * math.sqrt((h / l))) / h

Julia

function code(d, h, l, M, D)
	return Float64(Float64(d * sqrt(Float64(h / l))) / h)
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (d * sqrt((h / l))) / h;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]

Derivation

1. Initial program 66.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in h around 0

   \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   6. lower-/.f64 23.6

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

4. Applied rewrites 23.6%

   \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. sqrt-unprod N/A

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   6. lower-*.f64 21.2

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

   8. *-commutative N/A

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   9. lower-*.f64 21.2

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

6. Applied rewrites 21.2%

   \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

7. Taylor expanded in d around 0

   \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   3. lower-/.f64 36.9

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

9. Applied rewrites 36.9%

   \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025170 
(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)))))