Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.9% → 45.1%

Time: 9.7s

Alternatives: 14

Speedup: 1.5×

• could not determine a ground truth

  1. c0 = -1.207067377958195e+290
  2. w = -1.8980383851575216e-257
  3. h = -1.109729496371574e-233
  4. D = 1.6977917324979095e-277
  5. d = -2.1398271748761704e+294
  6. M = -2.0500512898291504e+240

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 24.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 45.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\\ \mathbf{if}\;M\_m \leq 3.9 \cdot 10^{-229}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{-M\_m \cdot M\_m}}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, t\_0, M\_m\right) \cdot \left(c0 \cdot t\_0 - M\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* (/ d (* (* w h) D)) (/ d D))))
   (if (<= M_m 3.9e-229)
     (* 0.5 (/ (* c0 (sqrt (- (* M_m M_m)))) w))
     (*
      (/ c0 (* 2.0 w))
      (fma
       (/ c0 (* w h))
       (* (/ d D) (/ d D))
       (sqrt (* (fma c0 t_0 M_m) (- (* c0 t_0) M_m))))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (d / ((w * h) * D)) * (d / D);
	double tmp;
	if (M_m <= 3.9e-229) {
		tmp = 0.5 * ((c0 * sqrt(-(M_m * M_m))) / w);
	} else {
		tmp = (c0 / (2.0 * w)) * fma((c0 / (w * h)), ((d / D) * (d / D)), sqrt((fma(c0, t_0, M_m) * ((c0 * t_0) - M_m))));
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(d / Float64(Float64(w * h) * D)) * Float64(d / D))
	tmp = 0.0
	if (M_m <= 3.9e-229)
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(Float64(-Float64(M_m * M_m)))) / w));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * fma(Float64(c0 / Float64(w * h)), Float64(Float64(d / D) * Float64(d / D)), sqrt(Float64(fma(c0, t_0, M_m) * Float64(Float64(c0 * t_0) - M_m)))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(d / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 3.9e-229], N[(0.5 * N[(N[(c0 * N[Sqrt[(-N[(M$95$m * M$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(c0 * t$95$0 + M$95$m), $MachinePrecision] * N[(N[(c0 * t$95$0), $MachinePrecision] - M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 3.89999999999999985e-229

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]

   if 3.89999999999999985e-229 < M

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \color{blue}{M \cdot M}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      4. difference-of-squares N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Applied rewrites 30.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}}\right) \]

   5. Applied rewrites 29.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{\color{blue}{D \cdot D}}, \sqrt{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{d \cdot d}{D \cdot D}}, \sqrt{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{\color{blue}{d \cdot d}}{D \cdot D}, \sqrt{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      4. times-frac N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{d}{D} \cdot \frac{d}{D}}, \sqrt{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{d}{D} \cdot \frac{d}{D}}, \sqrt{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{d}{D}} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      7. lift-/.f64 31.4

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \color{blue}{\frac{d}{D}}, \sqrt{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

   7. Applied rewrites 31.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \color{blue}{\frac{d}{D} \cdot \frac{d}{D}}, \sqrt{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{\color{blue}{d \cdot d}}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \color{blue}{\frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{d \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      4. frac-times N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \color{blue}{\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \color{blue}{\frac{d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{d}{\left(w \cdot h\right) \cdot D} \cdot \color{blue}{\frac{d}{D}}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

      7. lift-*.f64 31.3

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \color{blue}{\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

   9. Applied rewrites 31.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \color{blue}{\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}, M\right) \cdot \left(c0 \cdot \frac{\color{blue}{d \cdot d}}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}, M\right) \cdot \left(c0 \cdot \color{blue}{\frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M\right)}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\color{blue}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}} - M\right)}\right) \]

       4. frac-times N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}, M\right) \cdot \left(c0 \cdot \color{blue}{\left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M\right)}\right) \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}, M\right) \cdot \left(c0 \cdot \left(\color{blue}{\frac{d}{\left(w \cdot h\right) \cdot D}} \cdot \frac{d}{D}\right) - M\right)}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}, M\right) \cdot \left(c0 \cdot \left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \color{blue}{\frac{d}{D}}\right) - M\right)}\right) \]

       7. lift-*.f64 36.5

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}, M\right) \cdot \left(c0 \cdot \color{blue}{\left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M\right)}\right) \]

   11. Applied rewrites 36.5%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{\mathsf{fma}\left(c0, \frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}, M\right) \cdot \left(c0 \cdot \color{blue}{\left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right)} - M\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 44.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_0 + \sqrt{{t\_0}^{2} - M\_m \cdot M\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(-M\_m \cdot M\_m\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* c0 (/ (* d d) (* (* (* w h) D) D))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M_m M_m))))) INFINITY)
     (* t_1 (+ t_0 (sqrt (- (pow t_0 2.0) (* M_m M_m)))))
     (* 0.5 (/ (* c0 (pow (- (* M_m M_m)) 0.5)) w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 * ((d * d) / (((w * h) * D) * D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_1 * (t_0 + sqrt((pow(t_0, 2.0) - (M_m * M_m))));
	} else {
		tmp = 0.5 * ((c0 * pow(-(M_m * M_m), 0.5)) / w);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 * ((d * d) / (((w * h) * D) * D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (t_0 + Math.sqrt((Math.pow(t_0, 2.0) - (M_m * M_m))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(-(M_m * M_m), 0.5)) / w);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 * ((d * d) / (((w * h) * D) * D))
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M_m * M_m))))) <= math.inf:
		tmp = t_1 * (t_0 + math.sqrt((math.pow(t_0, 2.0) - (M_m * M_m))))
	else:
		tmp = 0.5 * ((c0 * math.pow(-(M_m * M_m), 0.5)) / w)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 * Float64(Float64(d * d) / Float64(Float64(Float64(w * h) * D) * D)))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_1 * Float64(t_0 + sqrt(Float64((t_0 ^ 2.0) - Float64(M_m * M_m)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M_m * M_m)) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 * ((d * d) / (((w * h) * D) * D));
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))))) <= Inf)
		tmp = t_1 * (t_0 + sqrt(((t_0 ^ 2.0) - (M_m * M_m))));
	else
		tmp = 0.5 * ((c0 * (-(M_m * M_m) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(t$95$0 + N[Sqrt[N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

   3. Applied rewrites 27.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} + \sqrt{{\left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}\right)}^{2} - M \cdot M}\right)} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-*.f64 23.1

         \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 23.1%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 41.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{w + w} \cdot \mathsf{fma}\left(c0, t\_0, \sqrt{{\left(c0 \cdot t\_0\right)}^{2} - M\_m \cdot M\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(-M\_m \cdot M\_m\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* d d) (* (* (* w h) D) D)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m)))))
        INFINITY)
     (*
      (/ c0 (+ w w))
      (fma c0 t_0 (sqrt (- (pow (* c0 t_0) 2.0) (* M_m M_m)))))
     (* 0.5 (/ (* c0 (pow (- (* M_m M_m)) 0.5)) w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (d * d) / (((w * h) * D) * D);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = (c0 / (w + w)) * fma(c0, t_0, sqrt((pow((c0 * t_0), 2.0) - (M_m * M_m))));
	} else {
		tmp = 0.5 * ((c0 * pow(-(M_m * M_m), 0.5)) / w);
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(d * d) / Float64(Float64(Float64(w * h) * D) * D))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(Float64(c0 / Float64(w + w)) * fma(c0, t_0, sqrt(Float64((Float64(c0 * t_0) ^ 2.0) - Float64(M_m * M_m)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M_m * M_m)) ^ 0.5)) / w));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(d * d), $MachinePrecision] / N[(N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(c0 * t$95$0 + N[Sqrt[N[(N[Power[N[(c0 * t$95$0), $MachinePrecision], 2.0], $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   3. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\frac{c0}{w + w} \cdot \mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, \sqrt{{\left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}\right)}^{2} - M \cdot M}\right)} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-*.f64 23.1

         \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 23.1%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 41.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + \sqrt{M\_m \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(-M\_m \cdot M\_m\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m))))) INFINITY)
     (*
      t_0
      (+ t_1 (sqrt (* M_m (- (* c0 (/ (* d d) (* (* (* w h) D) D))) M_m)))))
     (* 0.5 (/ (* c0 (pow (- (* M_m M_m)) 0.5)) w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_1 + sqrt((M_m * ((c0 * ((d * d) / (((w * h) * D) * D))) - M_m))));
	} else {
		tmp = 0.5 * ((c0 * pow(-(M_m * M_m), 0.5)) / w);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_1 + Math.sqrt((M_m * ((c0 * ((d * d) / (((w * h) * D) * D))) - M_m))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(-(M_m * M_m), 0.5)) / w);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= math.inf:
		tmp = t_0 * (t_1 + math.sqrt((M_m * ((c0 * ((d * d) / (((w * h) * D) * D))) - M_m))))
	else:
		tmp = 0.5 * ((c0 * math.pow(-(M_m * M_m), 0.5)) / w)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_0 * Float64(t_1 + sqrt(Float64(M_m * Float64(Float64(c0 * Float64(Float64(d * d) / Float64(Float64(Float64(w * h) * D) * D))) - M_m)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M_m * M_m)) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Inf)
		tmp = t_0 * (t_1 + sqrt((M_m * ((c0 * ((d * d) / (((w * h) * D) * D))) - M_m))));
	else
		tmp = 0.5 * ((c0 * (-(M_m * M_m) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(M$95$m * N[(N[(c0 * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \color{blue}{M \cdot M}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      4. difference-of-squares N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Applied rewrites 30.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}}\right) \]

   4. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{M} \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 28.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{M} \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-*.f64 23.1

         \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 23.1%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 39.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{M\_m \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(-M\_m \cdot M\_m\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m))))) INFINITY)
     (*
      t_0
      (fma
       (/ c0 (* w h))
       (/ (* d d) (* D D))
       (sqrt (* M_m (- (* c0 (/ (* d d) (* (* (* w h) D) D))) M_m)))))
     (* 0.5 (/ (* c0 (pow (- (* M_m M_m)) 0.5)) w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_0 * fma((c0 / (w * h)), ((d * d) / (D * D)), sqrt((M_m * ((c0 * ((d * d) / (((w * h) * D) * D))) - M_m))));
	} else {
		tmp = 0.5 * ((c0 * pow(-(M_m * M_m), 0.5)) / w);
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_0 * fma(Float64(c0 / Float64(w * h)), Float64(Float64(d * d) / Float64(D * D)), sqrt(Float64(M_m * Float64(Float64(c0 * Float64(Float64(d * d) / Float64(Float64(Float64(w * h) * D) * D))) - M_m)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M_m * M_m)) ^ 0.5)) / w));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(M$95$m * N[(N[(c0 * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \color{blue}{M \cdot M}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}} - M \cdot M}\right) \]

      4. difference-of-squares N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + M\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M\right)}}\right) \]

   3. Applied rewrites 30.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}}\right) \]

   5. Applied rewrites 29.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D}, M\right) \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right)} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\color{blue}{M} \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 27.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{d \cdot d}{D \cdot D}, \sqrt{\color{blue}{M} \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right) \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-*.f64 23.1

         \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 23.1%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 33.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-197}:\\ \;\;\;\;0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + t\_0}{D \cdot h}}{w \cdot w}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;0.5 \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{h \cdot h}} + \frac{d \cdot d}{h}\right)\right)}{D \cdot D}}{w \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(-M\_m \cdot M\_m\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* c0 (* d d)))
        (t_1 (/ t_0 (* (* w h) (* D D))))
        (t_2 (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m)))))))
   (if (<= t_2 -5e-197)
     (*
      0.5
      (/
       (* (/ c0 D) (/ (+ (sqrt (* (* c0 c0) (pow d 4.0))) t_0) (* D h)))
       (* w w)))
     (if (<= t_2 INFINITY)
       (*
        0.5
        (/
         (/
          (* c0 (* c0 (+ (sqrt (/ (pow d 4.0) (* h h))) (/ (* d d) h))))
          (* D D))
         (* w w)))
       (* 0.5 (/ (* c0 (pow (- (* M_m M_m)) 0.5)) w))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))));
	double tmp;
	if (t_2 <= -5e-197) {
		tmp = 0.5 * (((c0 / D) * ((sqrt(((c0 * c0) * pow(d, 4.0))) + t_0) / (D * h))) / (w * w));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = 0.5 * (((c0 * (c0 * (sqrt((pow(d, 4.0) / (h * h))) + ((d * d) / h)))) / (D * D)) / (w * w));
	} else {
		tmp = 0.5 * ((c0 * pow(-(M_m * M_m), 0.5)) / w);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))));
	double tmp;
	if (t_2 <= -5e-197) {
		tmp = 0.5 * (((c0 / D) * ((Math.sqrt(((c0 * c0) * Math.pow(d, 4.0))) + t_0) / (D * h))) / (w * w));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (((c0 * (c0 * (Math.sqrt((Math.pow(d, 4.0) / (h * h))) + ((d * d) / h)))) / (D * D)) / (w * w));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(-(M_m * M_m), 0.5)) / w);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 * (d * d)
	t_1 = t_0 / ((w * h) * (D * D))
	t_2 = (c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))
	tmp = 0
	if t_2 <= -5e-197:
		tmp = 0.5 * (((c0 / D) * ((math.sqrt(((c0 * c0) * math.pow(d, 4.0))) + t_0) / (D * h))) / (w * w))
	elif t_2 <= math.inf:
		tmp = 0.5 * (((c0 * (c0 * (math.sqrt((math.pow(d, 4.0) / (h * h))) + ((d * d) / h)))) / (D * D)) / (w * w))
	else:
		tmp = 0.5 * ((c0 * math.pow(-(M_m * M_m), 0.5)) / w)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m)))))
	tmp = 0.0
	if (t_2 <= -5e-197)
		tmp = Float64(0.5 * Float64(Float64(Float64(c0 / D) * Float64(Float64(sqrt(Float64(Float64(c0 * c0) * (d ^ 4.0))) + t_0) / Float64(D * h))) / Float64(w * w)));
	elseif (t_2 <= Inf)
		tmp = Float64(0.5 * Float64(Float64(Float64(c0 * Float64(c0 * Float64(sqrt(Float64((d ^ 4.0) / Float64(h * h))) + Float64(Float64(d * d) / h)))) / Float64(D * D)) / Float64(w * w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M_m * M_m)) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 * (d * d);
	t_1 = t_0 / ((w * h) * (D * D));
	t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))));
	tmp = 0.0;
	if (t_2 <= -5e-197)
		tmp = 0.5 * (((c0 / D) * ((sqrt(((c0 * c0) * (d ^ 4.0))) + t_0) / (D * h))) / (w * w));
	elseif (t_2 <= Inf)
		tmp = 0.5 * (((c0 * (c0 * (sqrt(((d ^ 4.0) / (h * h))) + ((d * d) / h)))) / (D * D)) / (w * w));
	else
		tmp = 0.5 * ((c0 * (-(M_m * M_m) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-197], N[(0.5 * N[(N[(N[(c0 / D), $MachinePrecision] * N[(N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] * N[Power[d, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(0.5 * N[(N[(N[(c0 * N[(c0 * N[(N[Sqrt[N[(N[Power[d, 4.0], $MachinePrecision] / N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(d * d), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -5.0000000000000002e-197

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{D}^{4} \cdot {h}^{2}}} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}\right)}{{w}^{2}}} \]

   4. Applied rewrites 8.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(\sqrt{\left(c0 \cdot c0\right) \cdot \frac{\left(d \cdot d\right) \cdot \left(d \cdot d\right)}{\left(\left(D \cdot D\right) \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot h\right)}} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot h}\right)}{w \cdot w}} \]

   5. Taylor expanded in D around 0

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2}}} + \frac{c0 \cdot {d}^{2}}{h}\right)}{{D}^{2}}}{\color{blue}{w} \cdot w} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2}}} + \frac{c0 \cdot {d}^{2}}{h}\right)}{{D}^{2}}}{w \cdot w} \]

   7. Applied rewrites 10.6%

      \[\leadsto 0.5 \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{D \cdot D}}{\color{blue}{w} \cdot w} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{D \cdot D}}{w \cdot w} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{D \cdot D}}{w \cdot w} \]

   9. Applied rewrites 13.0%

      \[\leadsto 0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}}{D}}{w \cdot w} \]

   10. Taylor expanded in h around 0

       \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{{c0}^{2} \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{{c0}^{2} \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{{c0}^{2} \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{{c0}^{2} \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       4. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       6. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       8. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot \left(d \cdot d\right)}{D \cdot h}}{w \cdot w} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot \left(d \cdot d\right)}{D \cdot h}}{w \cdot w} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot \left(d \cdot d\right)}{D \cdot h}}{w \cdot w} \]

       11. lower-*.f64 15.3

           \[\leadsto 0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot \left(d \cdot d\right)}{D \cdot h}}{w \cdot w} \]

   12. Applied rewrites 15.3%

       \[\leadsto 0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot \left(d \cdot d\right)}{D \cdot h}}{w \cdot w} \]

   if -5.0000000000000002e-197 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{D}^{4} \cdot {h}^{2}}} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}\right)}{{w}^{2}}} \]

   4. Applied rewrites 8.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(\sqrt{\left(c0 \cdot c0\right) \cdot \frac{\left(d \cdot d\right) \cdot \left(d \cdot d\right)}{\left(\left(D \cdot D\right) \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot h\right)}} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot h}\right)}{w \cdot w}} \]

   5. Taylor expanded in D around 0

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2}}} + \frac{c0 \cdot {d}^{2}}{h}\right)}{{D}^{2}}}{\color{blue}{w} \cdot w} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2}}} + \frac{c0 \cdot {d}^{2}}{h}\right)}{{D}^{2}}}{w \cdot w} \]

   7. Applied rewrites 10.6%

      \[\leadsto 0.5 \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{D \cdot D}}{\color{blue}{w} \cdot w} \]

   8. Taylor expanded in c0 around 0

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{{h}^{2}}} + \frac{{d}^{2}}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{{h}^{2}}} + \frac{{d}^{2}}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{{h}^{2}}} + \frac{{d}^{2}}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{{h}^{2}}} + \frac{{d}^{2}}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{{h}^{2}}} + \frac{{d}^{2}}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{{h}^{2}}} + \frac{{d}^{2}}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{h \cdot h}} + \frac{{d}^{2}}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{h \cdot h}} + \frac{{d}^{2}}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{h \cdot h}} + \frac{{d}^{2}}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{h \cdot h}} + \frac{d \cdot d}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      10. lift-*.f64 13.3

          \[\leadsto 0.5 \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{h \cdot h}} + \frac{d \cdot d}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

   10. Applied rewrites 13.3%

       \[\leadsto 0.5 \cdot \frac{\frac{c0 \cdot \left(c0 \cdot \left(\sqrt{\frac{{d}^{4}}{h \cdot h}} + \frac{d \cdot d}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-*.f64 23.1

         \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 23.1%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 31.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := -M\_m \cdot M\_m\\ \mathbf{if}\;M\_m \leq 1.05 \cdot 10^{-163}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{t\_0}}{w}\\ \mathbf{elif}\;M\_m \leq 1.45 \cdot 10^{+155}:\\ \;\;\;\;0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{D}}{w \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_0}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (- (* M_m M_m))))
   (if (<= M_m 1.05e-163)
     (* 0.5 (/ (* c0 (sqrt t_0)) w))
     (if (<= M_m 1.45e+155)
       (*
        0.5
        (/
         (*
          (/ c0 D)
          (/ (* (* d d) (+ (sqrt (/ (* c0 c0) (* h h))) (/ c0 h))) D))
         (* w w)))
       (* 0.5 (/ (* c0 (pow t_0 0.5)) w))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = -(M_m * M_m);
	double tmp;
	if (M_m <= 1.05e-163) {
		tmp = 0.5 * ((c0 * sqrt(t_0)) / w);
	} else if (M_m <= 1.45e+155) {
		tmp = 0.5 * (((c0 / D) * (((d * d) * (sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / D)) / (w * w));
	} else {
		tmp = 0.5 * ((c0 * pow(t_0, 0.5)) / w);
	}
	return tmp;
}

Fortran

M_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(m_m * m_m)
    if (m_m <= 1.05d-163) then
        tmp = 0.5d0 * ((c0 * sqrt(t_0)) / w)
    else if (m_m <= 1.45d+155) then
        tmp = 0.5d0 * (((c0 / d) * (((d_1 * d_1) * (sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / d)) / (w * w))
    else
        tmp = 0.5d0 * ((c0 * (t_0 ** 0.5d0)) / w)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = -(M_m * M_m);
	double tmp;
	if (M_m <= 1.05e-163) {
		tmp = 0.5 * ((c0 * Math.sqrt(t_0)) / w);
	} else if (M_m <= 1.45e+155) {
		tmp = 0.5 * (((c0 / D) * (((d * d) * (Math.sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / D)) / (w * w));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(t_0, 0.5)) / w);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = -(M_m * M_m)
	tmp = 0
	if M_m <= 1.05e-163:
		tmp = 0.5 * ((c0 * math.sqrt(t_0)) / w)
	elif M_m <= 1.45e+155:
		tmp = 0.5 * (((c0 / D) * (((d * d) * (math.sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / D)) / (w * w))
	else:
		tmp = 0.5 * ((c0 * math.pow(t_0, 0.5)) / w)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(-Float64(M_m * M_m))
	tmp = 0.0
	if (M_m <= 1.05e-163)
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(t_0)) / w));
	elseif (M_m <= 1.45e+155)
		tmp = Float64(0.5 * Float64(Float64(Float64(c0 / D) * Float64(Float64(Float64(d * d) * Float64(sqrt(Float64(Float64(c0 * c0) / Float64(h * h))) + Float64(c0 / h))) / D)) / Float64(w * w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_0 ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = -(M_m * M_m);
	tmp = 0.0;
	if (M_m <= 1.05e-163)
		tmp = 0.5 * ((c0 * sqrt(t_0)) / w);
	elseif (M_m <= 1.45e+155)
		tmp = 0.5 * (((c0 / D) * (((d * d) * (sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / D)) / (w * w));
	else
		tmp = 0.5 * ((c0 * (t_0 ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = (-N[(M$95$m * M$95$m), $MachinePrecision])}, If[LessEqual[M$95$m, 1.05e-163], N[(0.5 * N[(N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 1.45e+155], N[(0.5 * N[(N[(N[(c0 / D), $MachinePrecision] * N[(N[(N[(d * d), $MachinePrecision] * N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$0, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.04999999999999999e-163

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]

   if 1.04999999999999999e-163 < M < 1.45e155

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{D}^{4} \cdot {h}^{2}}} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}\right)}{{w}^{2}}} \]

   4. Applied rewrites 8.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(\sqrt{\left(c0 \cdot c0\right) \cdot \frac{\left(d \cdot d\right) \cdot \left(d \cdot d\right)}{\left(\left(D \cdot D\right) \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot h\right)}} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot h}\right)}{w \cdot w}} \]

   5. Taylor expanded in D around 0

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2}}} + \frac{c0 \cdot {d}^{2}}{h}\right)}{{D}^{2}}}{\color{blue}{w} \cdot w} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2}}} + \frac{c0 \cdot {d}^{2}}{h}\right)}{{D}^{2}}}{w \cdot w} \]

   7. Applied rewrites 10.6%

      \[\leadsto 0.5 \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{D \cdot D}}{\color{blue}{w} \cdot w} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{D \cdot D}}{w \cdot w} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{D \cdot D}}{w \cdot w} \]

   9. Applied rewrites 13.0%

      \[\leadsto 0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}}{D}}{w \cdot w} \]

   10. Taylor expanded in d around 0

       \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

       2. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

       4. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

       7. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2}}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2}}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

       9. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

       11. lower-/.f64 18.1

           \[\leadsto 0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

   12. Applied rewrites 18.1%

       \[\leadsto 0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{D}}{w \cdot w} \]

   if 1.45e155 < M

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-*.f64 23.1

         \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 23.1%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 30.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + t\_0}{D \cdot h}}{w \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(-M\_m \cdot M\_m\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* c0 (* d d))) (t_1 (/ t_0 (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m)))))
        INFINITY)
     (*
      0.5
      (/
       (* (/ c0 D) (/ (+ (sqrt (* (* c0 c0) (pow d 4.0))) t_0) (* D h)))
       (* w w)))
     (* 0.5 (/ (* c0 (pow (- (* M_m M_m)) 0.5)) w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = 0.5 * (((c0 / D) * ((sqrt(((c0 * c0) * pow(d, 4.0))) + t_0) / (D * h))) / (w * w));
	} else {
		tmp = 0.5 * ((c0 * pow(-(M_m * M_m), 0.5)) / w);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (((c0 / D) * ((Math.sqrt(((c0 * c0) * Math.pow(d, 4.0))) + t_0) / (D * h))) / (w * w));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(-(M_m * M_m), 0.5)) / w);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 * (d * d)
	t_1 = t_0 / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= math.inf:
		tmp = 0.5 * (((c0 / D) * ((math.sqrt(((c0 * c0) * math.pow(d, 4.0))) + t_0) / (D * h))) / (w * w))
	else:
		tmp = 0.5 * ((c0 * math.pow(-(M_m * M_m), 0.5)) / w)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(0.5 * Float64(Float64(Float64(c0 / D) * Float64(Float64(sqrt(Float64(Float64(c0 * c0) * (d ^ 4.0))) + t_0) / Float64(D * h))) / Float64(w * w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M_m * M_m)) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 * (d * d);
	t_1 = t_0 / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Inf)
		tmp = 0.5 * (((c0 / D) * ((sqrt(((c0 * c0) * (d ^ 4.0))) + t_0) / (D * h))) / (w * w));
	else
		tmp = 0.5 * ((c0 * (-(M_m * M_m) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(0.5 * N[(N[(N[(c0 / D), $MachinePrecision] * N[(N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] * N[Power[d, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / N[(D * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{D}^{4} \cdot {h}^{2}}} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}\right)}{{w}^{2}}} \]

   4. Applied rewrites 8.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(\sqrt{\left(c0 \cdot c0\right) \cdot \frac{\left(d \cdot d\right) \cdot \left(d \cdot d\right)}{\left(\left(D \cdot D\right) \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot h\right)}} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot h}\right)}{w \cdot w}} \]

   5. Taylor expanded in D around 0

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2}}} + \frac{c0 \cdot {d}^{2}}{h}\right)}{{D}^{2}}}{\color{blue}{w} \cdot w} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2}}} + \frac{c0 \cdot {d}^{2}}{h}\right)}{{D}^{2}}}{w \cdot w} \]

   7. Applied rewrites 10.6%

      \[\leadsto 0.5 \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{D \cdot D}}{\color{blue}{w} \cdot w} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{D \cdot D}}{w \cdot w} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{D \cdot D}}{w \cdot w} \]

   9. Applied rewrites 13.0%

      \[\leadsto 0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}}{D}}{w \cdot w} \]

   10. Taylor expanded in h around 0

       \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{{c0}^{2} \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{{c0}^{2} \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{{c0}^{2} \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{{c0}^{2} \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       4. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       6. lift-pow.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot {d}^{2}}{D \cdot h}}{w \cdot w} \]

       8. pow2 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot \left(d \cdot d\right)}{D \cdot h}}{w \cdot w} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot \left(d \cdot d\right)}{D \cdot h}}{w \cdot w} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot \left(d \cdot d\right)}{D \cdot h}}{w \cdot w} \]

       11. lower-*.f64 15.3

           \[\leadsto 0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot \left(d \cdot d\right)}{D \cdot h}}{w \cdot w} \]

   12. Applied rewrites 15.3%

       \[\leadsto 0.5 \cdot \frac{\frac{c0}{D} \cdot \frac{\sqrt{\left(c0 \cdot c0\right) \cdot {d}^{4}} + c0 \cdot \left(d \cdot d\right)}{D \cdot h}}{w \cdot w} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-*.f64 23.1

         \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 23.1%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 29.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := -M\_m \cdot M\_m\\ \mathbf{if}\;M\_m \leq 1.05 \cdot 10^{-163}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \sqrt{t\_0}}{w}\\ \mathbf{elif}\;M\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_0}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (- (* M_m M_m))))
   (if (<= M_m 1.05e-163)
     (* 0.5 (/ (* c0 (sqrt t_0)) w))
     (if (<= M_m 1.35e+154)
       (*
        0.5
        (/
         (/
          (* c0 (* (* d d) (+ (sqrt (/ (* c0 c0) (* h h))) (/ c0 h))))
          (* D D))
         (* w w)))
       (* 0.5 (/ (* c0 (pow t_0 0.5)) w))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = -(M_m * M_m);
	double tmp;
	if (M_m <= 1.05e-163) {
		tmp = 0.5 * ((c0 * sqrt(t_0)) / w);
	} else if (M_m <= 1.35e+154) {
		tmp = 0.5 * (((c0 * ((d * d) * (sqrt(((c0 * c0) / (h * h))) + (c0 / h)))) / (D * D)) / (w * w));
	} else {
		tmp = 0.5 * ((c0 * pow(t_0, 0.5)) / w);
	}
	return tmp;
}

Fortran

M_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(m_m * m_m)
    if (m_m <= 1.05d-163) then
        tmp = 0.5d0 * ((c0 * sqrt(t_0)) / w)
    else if (m_m <= 1.35d+154) then
        tmp = 0.5d0 * (((c0 * ((d_1 * d_1) * (sqrt(((c0 * c0) / (h * h))) + (c0 / h)))) / (d * d)) / (w * w))
    else
        tmp = 0.5d0 * ((c0 * (t_0 ** 0.5d0)) / w)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = -(M_m * M_m);
	double tmp;
	if (M_m <= 1.05e-163) {
		tmp = 0.5 * ((c0 * Math.sqrt(t_0)) / w);
	} else if (M_m <= 1.35e+154) {
		tmp = 0.5 * (((c0 * ((d * d) * (Math.sqrt(((c0 * c0) / (h * h))) + (c0 / h)))) / (D * D)) / (w * w));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(t_0, 0.5)) / w);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = -(M_m * M_m)
	tmp = 0
	if M_m <= 1.05e-163:
		tmp = 0.5 * ((c0 * math.sqrt(t_0)) / w)
	elif M_m <= 1.35e+154:
		tmp = 0.5 * (((c0 * ((d * d) * (math.sqrt(((c0 * c0) / (h * h))) + (c0 / h)))) / (D * D)) / (w * w))
	else:
		tmp = 0.5 * ((c0 * math.pow(t_0, 0.5)) / w)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(-Float64(M_m * M_m))
	tmp = 0.0
	if (M_m <= 1.05e-163)
		tmp = Float64(0.5 * Float64(Float64(c0 * sqrt(t_0)) / w));
	elseif (M_m <= 1.35e+154)
		tmp = Float64(0.5 * Float64(Float64(Float64(c0 * Float64(Float64(d * d) * Float64(sqrt(Float64(Float64(c0 * c0) / Float64(h * h))) + Float64(c0 / h)))) / Float64(D * D)) / Float64(w * w)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_0 ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = -(M_m * M_m);
	tmp = 0.0;
	if (M_m <= 1.05e-163)
		tmp = 0.5 * ((c0 * sqrt(t_0)) / w);
	elseif (M_m <= 1.35e+154)
		tmp = 0.5 * (((c0 * ((d * d) * (sqrt(((c0 * c0) / (h * h))) + (c0 / h)))) / (D * D)) / (w * w));
	else
		tmp = 0.5 * ((c0 * (t_0 ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = (-N[(M$95$m * M$95$m), $MachinePrecision])}, If[LessEqual[M$95$m, 1.05e-163], N[(0.5 * N[(N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 1.35e+154], N[(0.5 * N[(N[(N[(c0 * N[(N[(d * d), $MachinePrecision] * N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$0, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.04999999999999999e-163

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]

   if 1.04999999999999999e-163 < M < 1.35000000000000003e154

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{D}^{4} \cdot {h}^{2}}} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}\right)}{{w}^{2}}} \]

   4. Applied rewrites 8.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(\sqrt{\left(c0 \cdot c0\right) \cdot \frac{\left(d \cdot d\right) \cdot \left(d \cdot d\right)}{\left(\left(D \cdot D\right) \cdot \left(D \cdot D\right)\right) \cdot \left(h \cdot h\right)}} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot h}\right)}{w \cdot w}} \]

   5. Taylor expanded in D around 0

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2}}} + \frac{c0 \cdot {d}^{2}}{h}\right)}{{D}^{2}}}{\color{blue}{w} \cdot w} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2}}} + \frac{c0 \cdot {d}^{2}}{h}\right)}{{D}^{2}}}{w \cdot w} \]

   7. Applied rewrites 10.6%

      \[\leadsto 0.5 \cdot \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{D \cdot D}}{\color{blue}{w} \cdot w} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left({d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left({d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      2. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2}}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2}}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      9. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

      11. lower-/.f64 15.0

          \[\leadsto 0.5 \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

   10. Applied rewrites 15.0%

       \[\leadsto 0.5 \cdot \frac{\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)\right)}{D \cdot D}}{w \cdot w} \]

   if 1.35000000000000003e154 < M

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-*.f64 23.1

         \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 23.1%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 27.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := -M\_m \cdot M\_m\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M\_m \cdot M\_m}\right) \leq -5 \cdot 10^{-197}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 + \sqrt{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_0}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (- (* M_m M_m)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M_m M_m))))) -5e-197)
     (* t_1 (+ t_2 (sqrt t_0)))
     (* 0.5 (/ (* c0 (pow t_0 0.5)) w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = -(M_m * M_m);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))))) <= -5e-197) {
		tmp = t_1 * (t_2 + sqrt(t_0));
	} else {
		tmp = 0.5 * ((c0 * pow(t_0, 0.5)) / w);
	}
	return tmp;
}

Fortran

M_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = -(m_m * m_m)
    t_1 = c0 / (2.0d0 * w)
    t_2 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (m_m * m_m))))) <= (-5d-197)) then
        tmp = t_1 * (t_2 + sqrt(t_0))
    else
        tmp = 0.5d0 * ((c0 * (t_0 ** 0.5d0)) / w)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = -(M_m * M_m);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M_m * M_m))))) <= -5e-197) {
		tmp = t_1 * (t_2 + Math.sqrt(t_0));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(t_0, 0.5)) / w);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = -(M_m * M_m)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M_m * M_m))))) <= -5e-197:
		tmp = t_1 * (t_2 + math.sqrt(t_0))
	else:
		tmp = 0.5 * ((c0 * math.pow(t_0, 0.5)) / w)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(-Float64(M_m * M_m))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M_m * M_m))))) <= -5e-197)
		tmp = Float64(t_1 * Float64(t_2 + sqrt(t_0)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_0 ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = -(M_m * M_m);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))))) <= -5e-197)
		tmp = t_1 * (t_2 + sqrt(t_0));
	else
		tmp = 0.5 * ((c0 * (t_0 ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = (-N[(M$95$m * M$95$m), $MachinePrecision])}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-197], N[(t$95$1 * N[(t$95$2 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$0, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -5.0000000000000002e-197

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{-1 \cdot {M}^{2}}}\right) \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\mathsf{neg}\left({M}^{2}\right)}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{-{M}^{2}}\right) \]

      3. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{-M \cdot M}\right) \]

      4. lift-*.f64 8.2

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{-M \cdot M}\right) \]

   4. Applied rewrites 8.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{-M \cdot M}}\right) \]

   if -5.0000000000000002e-197 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-*.f64 23.1

         \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 23.1%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 26.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := -M\_m \cdot M\_m\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{t\_0} + c0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {t\_0}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (- (* M_m M_m)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M_m M_m))))) INFINITY)
     (* t_1 (+ (sqrt t_0) (* c0 (/ (* d d) (* (* (* D D) h) w)))))
     (* 0.5 (/ (* c0 (pow t_0 0.5)) w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = -(M_m * M_m);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_1 * (sqrt(t_0) + (c0 * ((d * d) / (((D * D) * h) * w))));
	} else {
		tmp = 0.5 * ((c0 * pow(t_0, 0.5)) / w);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = -(M_m * M_m);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (Math.sqrt(t_0) + (c0 * ((d * d) / (((D * D) * h) * w))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(t_0, 0.5)) / w);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = -(M_m * M_m)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M_m * M_m))))) <= math.inf:
		tmp = t_1 * (math.sqrt(t_0) + (c0 * ((d * d) / (((D * D) * h) * w))))
	else:
		tmp = 0.5 * ((c0 * math.pow(t_0, 0.5)) / w)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(-Float64(M_m * M_m))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_1 * Float64(sqrt(t_0) + Float64(c0 * Float64(Float64(d * d) / Float64(Float64(Float64(D * D) * h) * w)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (t_0 ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = -(M_m * M_m);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))))) <= Inf)
		tmp = t_1 * (sqrt(t_0) + (c0 * ((d * d) / (((D * D) * h) * w))));
	else
		tmp = 0.5 * ((c0 * (t_0 ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = (-N[(M$95$m * M$95$m), $MachinePrecision])}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[Sqrt[t$95$0], $MachinePrecision] + N[(c0 * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[t$95$0, 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left({M}^{2}\right)} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{\mathsf{neg}\left({M}^{2}\right)} + \color{blue}{\frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{\mathsf{neg}\left({M}^{2}\right)} + \frac{\color{blue}{c0 \cdot {d}^{2}}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-{M}^{2}} + \frac{\color{blue}{c0} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \]

      4. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) \]

      6. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + c0 \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + c0 \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + c0 \cdot \frac{{d}^{2}}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}}\right) \]

      9. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + c0 \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + c0 \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + c0 \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{w}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + c0 \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{w}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + c0 \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot w}\right) \]

      14. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + c0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right) \]

      15. lift-*.f64 8.2

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-M \cdot M} + c0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right) \]

   4. Applied rewrites 8.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{-M \cdot M} + c0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 24.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.7

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lift-*.f64 23.1

         \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 23.1%

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 23.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ 0.5 \cdot \frac{c0 \cdot {\left(-M\_m \cdot M\_m\right)}^{0.5}}{w} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (* 0.5 (/ (* c0 (pow (- (* M_m M_m)) 0.5)) w)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.5 * ((c0 * pow(-(M_m * M_m), 0.5)) / w);
}

Fortran

M_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m_m
    code = 0.5d0 * ((c0 * (-(m_m * m_m) ** 0.5d0)) / w)
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.5 * ((c0 * Math.pow(-(M_m * M_m), 0.5)) / w);
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return 0.5 * ((c0 * math.pow(-(M_m * M_m), 0.5)) / w)

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	return Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M_m * M_m)) ^ 0.5)) / w))
end

MATLAB

M_m = abs(M);
function tmp = code(c0, w, h, D, d, M_m)
	tmp = 0.5 * ((c0 * (-(M_m * M_m) ^ 0.5)) / w);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := N[(0.5 * N[(N[(c0 * N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.9%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   7. lift-*.f64 15.7

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

4. Applied rewrites 15.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   2. pow1/2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

   4. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

   5. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

   7. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

   8. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{\frac{1}{2}}}{w} \]

   9. lift-*.f64 23.1

      \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]

6. Applied rewrites 23.1%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w} \]
7. Add Preprocessing

Alternative 13: 15.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ 0.5 \cdot \frac{c0 \cdot \sqrt{-M\_m \cdot M\_m}}{w} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (* 0.5 (/ (* c0 (sqrt (- (* M_m M_m)))) w)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.5 * ((c0 * sqrt(-(M_m * M_m))) / w);
}

Fortran

M_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m_m
    code = 0.5d0 * ((c0 * sqrt(-(m_m * m_m))) / w)
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.5 * ((c0 * Math.sqrt(-(M_m * M_m))) / w);
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return 0.5 * ((c0 * math.sqrt(-(M_m * M_m))) / w)

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	return Float64(0.5 * Float64(Float64(c0 * sqrt(Float64(-Float64(M_m * M_m)))) / w))
end

MATLAB

M_m = abs(M);
function tmp = code(c0, w, h, D, d, M_m)
	tmp = 0.5 * ((c0 * sqrt(-(M_m * M_m))) / w);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := N[(0.5 * N[(N[(c0 * N[Sqrt[(-N[(M$95$m * M$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.9%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{\color{blue}{w}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   7. lift-*.f64 15.7

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

4. Applied rewrites 15.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
5. Add Preprocessing

Alternative 14: 0.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \frac{c0}{2 \cdot w} \cdot \left(M\_m \cdot \sqrt{-1}\right) \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (* (/ c0 (* 2.0 w)) (* M_m (sqrt -1.0))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	return (c0 / (2.0 * w)) * (M_m * sqrt(-1.0));
}

Fortran

M_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m_m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m_m
    code = (c0 / (2.0d0 * w)) * (m_m * sqrt((-1.0d0)))
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	return (c0 / (2.0 * w)) * (M_m * Math.sqrt(-1.0));
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return (c0 / (2.0 * w)) * (M_m * math.sqrt(-1.0))

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(M_m * sqrt(-1.0)))
end

MATLAB

M_m = abs(M);
function tmp = code(c0, w, h, D, d, M_m)
	tmp = (c0 / (2.0 * w)) * (M_m * sqrt(-1.0));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[Sqrt[-1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.9%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in M around inf

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(M \cdot \color{blue}{\sqrt{-1}}\right) \]

   2. lower-sqrt.f64 0.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(M \cdot \sqrt{-1}\right) \]

4. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))