Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.3% → 55.3%

Time: 27.3s

Alternatives: 6

Speedup: 151.0×

• Rival profile vector overflowed, profile may not be complete

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

real(8) function code(c0, w, h, d, d_1, m)
    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: 24.3% 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

real(8) function code(c0, w, h, d, d_1, m)
    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: 55.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ t_3 := t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right)\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{+256}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot \left(\frac{d\_m}{D\_m} \cdot \frac{d\_m}{D\_m}\right) + \sqrt{\mathsf{fma}\left({\left(\frac{d\_m}{D\_m}\right)}^{2}, t\_0, M\right)} \cdot \left(\frac{d\_m}{D\_m} \cdot \sqrt{t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))))
   (if (<= t_3 5e+256)
     t_3
     (if (<= t_3 INFINITY)
       (*
        t_1
        (+
         (* t_0 (* (/ d_m D_m) (/ d_m D_m)))
         (*
          (sqrt (fma (pow (/ d_m D_m) 2.0) t_0 M))
          (* (/ d_m D_m) (sqrt t_0)))))
       0.0))))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= 5e+256) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 * ((t_0 * ((d_m / D_m) * (d_m / D_m))) + (sqrt(fma(pow((d_m / D_m), 2.0), t_0, M)) * ((d_m / D_m) * sqrt(t_0))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	t_3 = Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	tmp = 0.0
	if (t_3 <= 5e+256)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = Float64(t_1 * Float64(Float64(t_0 * Float64(Float64(d_m / D_m) * Float64(d_m / D_m))) + Float64(sqrt(fma((Float64(d_m / D_m) ^ 2.0), t_0, M)) * Float64(Float64(d_m / D_m) * sqrt(t_0)))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e+256], t$95$3, If[LessEqual[t$95$3, Infinity], N[(t$95$1 * N[(N[(t$95$0 * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + M), $MachinePrecision]], $MachinePrecision] * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]

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))))) < 5.00000000000000015e256

   1. Initial program 82.3%

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

   if 5.00000000000000015e256 < (*.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 74.4%

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

   3. Simplified 74.3%

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

      1. times-frac 74.3%

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

   6. Applied egg-rr 74.3%

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

      1. pow1/2 74.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \color{blue}{{\left(\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) - M \cdot M\right)}^{0.5}}\right) \]

      2. difference-of-squares 74.3%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + {\color{blue}{\left(\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + M\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} - M\right)\right)}}^{0.5}\right) \]

      3. unpow-prod-down 80.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \color{blue}{{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + M\right)}^{0.5} \cdot {\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} - M\right)}^{0.5}}\right) \]

      4. times-frac 80.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + {\left(\frac{c0}{w \cdot h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)} + M\right)}^{0.5} \cdot {\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} - M\right)}^{0.5}\right) \]

      5. unpow2 80.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + {\left(\frac{c0}{w \cdot h} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{2}} + M\right)}^{0.5} \cdot {\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} - M\right)}^{0.5}\right) \]

      6. *-commutative 80.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + {\left(\color{blue}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}} + M\right)}^{0.5} \cdot {\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} - M\right)}^{0.5}\right) \]

      7. fma-define 80.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + {\color{blue}{\left(\mathsf{fma}\left({\left(\frac{d}{D}\right)}^{2}, \frac{c0}{w \cdot h}, M\right)\right)}}^{0.5} \cdot {\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} - M\right)}^{0.5}\right) \]

      8. *-commutative 80.6%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + {\left(\mathsf{fma}\left({\left(\frac{d}{D}\right)}^{2}, \frac{c0}{\color{blue}{h \cdot w}}, M\right)\right)}^{0.5} \cdot {\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} - M\right)}^{0.5}\right) \]

   8. Applied egg-rr 90.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \color{blue}{{\left(\mathsf{fma}\left({\left(\frac{d}{D}\right)}^{2}, \frac{c0}{h \cdot w}, M\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left({\left(\frac{d}{D}\right)}^{2}, \frac{c0}{h \cdot w}, -M\right)\right)}^{0.5}}\right) \]
   9. Step-by-step derivation

      1. unpow1/2 90.0%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \color{blue}{\sqrt{\mathsf{fma}\left({\left(\frac{d}{D}\right)}^{2}, \frac{c0}{h \cdot w}, M\right)}} \cdot {\left(\mathsf{fma}\left({\left(\frac{d}{D}\right)}^{2}, \frac{c0}{h \cdot w}, -M\right)\right)}^{0.5}\right) \]

      2. unpow1/2 90.0%

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

      3. fmm-undef 90.0%

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

   10. Simplified 90.0%

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

   11. Taylor expanded in d around inf 50.3%

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

   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 0.0%

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

   3. Simplified 19.4%

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

   5. Taylor expanded in M around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w} \cdot -0.5} \]

      2. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(M \cdot c0\right) \cdot \sqrt{-1}}}{w} \cdot -0.5 \]

   7. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\left(M \cdot c0\right) \cdot \sqrt{-1}}{w} \cdot -0.5} \]
   8. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto \frac{\color{blue}{\log \left(e^{\left(M \cdot c0\right) \cdot \sqrt{-1}}\right)}}{w} \cdot -0.5 \]

      2. exp-prod 25.8%

         \[\leadsto \frac{\log \color{blue}{\left({\left(e^{M \cdot c0}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

      3. *-commutative 25.8%

         \[\leadsto \frac{\log \left({\left(e^{\color{blue}{c0 \cdot M}}\right)}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

      4. exp-prod 19.2%

         \[\leadsto \frac{\log \left({\color{blue}{\left({\left(e^{c0}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

   9. Applied egg-rr 19.2%

      \[\leadsto \frac{\color{blue}{\log \left({\left({\left(e^{c0}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

   10. Taylor expanded in c0 around 0 44.4%

       \[\leadsto \frac{\log \color{blue}{1}}{w} \cdot -0.5 \]

   11. Taylor expanded in w around 0 44.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 5 \cdot 10^{+256}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \sqrt{\mathsf{fma}\left({\left(\frac{d}{D}\right)}^{2}, \frac{c0}{w \cdot h}, M\right)} \cdot \left(\frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 INFINITY) t_1 0.0)))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
def code(c0, w, h, D_m, d_m, M):
	t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, Infinity], t$95$1, 0.0]]]

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 79.3%

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

   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 0.0%

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

   3. Simplified 19.4%

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

   5. Taylor expanded in M around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w} \cdot -0.5} \]

      2. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(M \cdot c0\right) \cdot \sqrt{-1}}}{w} \cdot -0.5 \]

   7. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\left(M \cdot c0\right) \cdot \sqrt{-1}}{w} \cdot -0.5} \]
   8. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto \frac{\color{blue}{\log \left(e^{\left(M \cdot c0\right) \cdot \sqrt{-1}}\right)}}{w} \cdot -0.5 \]

      2. exp-prod 25.8%

         \[\leadsto \frac{\log \color{blue}{\left({\left(e^{M \cdot c0}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

      3. *-commutative 25.8%

         \[\leadsto \frac{\log \left({\left(e^{\color{blue}{c0 \cdot M}}\right)}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

      4. exp-prod 19.2%

         \[\leadsto \frac{\log \left({\color{blue}{\left({\left(e^{c0}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

   9. Applied egg-rr 19.2%

      \[\leadsto \frac{\color{blue}{\log \left({\left({\left(e^{c0}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

   10. Taylor expanded in c0 around 0 44.4%

       \[\leadsto \frac{\log \color{blue}{1}}{w} \cdot -0.5 \]

   11. Taylor expanded in w around 0 44.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 32.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot \frac{d\_m \cdot d\_m}{D\_m \cdot D\_m}\\ \mathbf{if}\;h \leq -2.05 \cdot 10^{+38}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (/ (* d_m d_m) (* D_m D_m)))))
   (if (<= h -2.05e+38)
     (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
     0.0)))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = (c0 / (w * h)) * ((d_m * d_m) / (D_m * D_m));
	double tmp;
	if (h <= -2.05e+38) {
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
real(8) function code(c0, w, h, d_m, d_m_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c0 / (w * h)) * ((d_m_1 * d_m_1) / (d_m * d_m))
    if (h <= (-2.05d+38)) then
        tmp = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = (c0 / (w * h)) * ((d_m * d_m) / (D_m * D_m));
	double tmp;
	if (h <= -2.05e+38) {
		tmp = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
def code(c0, w, h, D_m, d_m, M):
	t_0 = (c0 / (w * h)) * ((d_m * d_m) / (D_m * D_m))
	tmp = 0
	if h <= -2.05e+38:
		tmp = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = 0.0
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d_m * d_m) / Float64(D_m * D_m)))
	tmp = 0.0
	if (h <= -2.05e+38)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
function tmp_2 = code(c0, w, h, D_m, d_m, M)
	t_0 = (c0 / (w * h)) * ((d_m * d_m) / (D_m * D_m));
	tmp = 0.0;
	if (h <= -2.05e+38)
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d$95$m * d$95$m), $MachinePrecision] / N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.05e+38], 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], 0.0]]

Derivation

1. Split input into 2 regimes

2. if h < -2.0500000000000002e38

   1. Initial program 37.9%

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

   3. Simplified 40.3%

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

   if -2.0500000000000002e38 < h

   1. Initial program 22.0%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in M around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w} \cdot -0.5} \]

      2. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(M \cdot c0\right) \cdot \sqrt{-1}}}{w} \cdot -0.5 \]

   7. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\left(M \cdot c0\right) \cdot \sqrt{-1}}{w} \cdot -0.5} \]
   8. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto \frac{\color{blue}{\log \left(e^{\left(M \cdot c0\right) \cdot \sqrt{-1}}\right)}}{w} \cdot -0.5 \]

      2. exp-prod 21.9%

         \[\leadsto \frac{\log \color{blue}{\left({\left(e^{M \cdot c0}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

      3. *-commutative 21.9%

         \[\leadsto \frac{\log \left({\left(e^{\color{blue}{c0 \cdot M}}\right)}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

      4. exp-prod 16.4%

         \[\leadsto \frac{\log \left({\color{blue}{\left({\left(e^{c0}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

   9. Applied egg-rr 16.4%

      \[\leadsto \frac{\color{blue}{\log \left({\left({\left(e^{c0}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

   10. Taylor expanded in c0 around 0 36.8%

       \[\leadsto \frac{\log \color{blue}{1}}{w} \cdot -0.5 \]

   11. Taylor expanded in w around 0 36.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 32.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := \frac{d\_m \cdot d\_m}{D\_m \cdot D\_m}\\ \mathbf{if}\;h \leq -1.75 \cdot 10^{+40}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 \cdot \left(\frac{d\_m}{D\_m} \cdot \frac{d\_m}{D\_m}\right) + \sqrt{\left(t\_0 \cdot t\_1\right) \cdot \left(t\_1 \cdot \frac{\frac{c0}{w}}{h}\right) - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (/ (* d_m d_m) (* D_m D_m))))
   (if (<= h -1.75e+40)
     (*
      (/ c0 (* 2.0 w))
      (+
       (* t_0 (* (/ d_m D_m) (/ d_m D_m)))
       (sqrt (- (* (* t_0 t_1) (* t_1 (/ (/ c0 w) h))) (* M M)))))
     0.0)))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = (d_m * d_m) / (D_m * D_m);
	double tmp;
	if (h <= -1.75e+40) {
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d_m / D_m) * (d_m / D_m))) + sqrt((((t_0 * t_1) * (t_1 * ((c0 / w) / h))) - (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
real(8) function code(c0, w, h, d_m, d_m_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = (d_m_1 * d_m_1) / (d_m * d_m)
    if (h <= (-1.75d+40)) then
        tmp = (c0 / (2.0d0 * w)) * ((t_0 * ((d_m_1 / d_m) * (d_m_1 / d_m))) + sqrt((((t_0 * t_1) * (t_1 * ((c0 / w) / h))) - (m * m))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = (d_m * d_m) / (D_m * D_m);
	double tmp;
	if (h <= -1.75e+40) {
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d_m / D_m) * (d_m / D_m))) + Math.sqrt((((t_0 * t_1) * (t_1 * ((c0 / w) / h))) - (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
def code(c0, w, h, D_m, d_m, M):
	t_0 = c0 / (w * h)
	t_1 = (d_m * d_m) / (D_m * D_m)
	tmp = 0
	if h <= -1.75e+40:
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d_m / D_m) * (d_m / D_m))) + math.sqrt((((t_0 * t_1) * (t_1 * ((c0 / w) / h))) - (M * M))))
	else:
		tmp = 0.0
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(Float64(d_m * d_m) / Float64(D_m * D_m))
	tmp = 0.0
	if (h <= -1.75e+40)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(t_0 * Float64(Float64(d_m / D_m) * Float64(d_m / D_m))) + sqrt(Float64(Float64(Float64(t_0 * t_1) * Float64(t_1 * Float64(Float64(c0 / w) / h))) - Float64(M * M)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
function tmp_2 = code(c0, w, h, D_m, d_m, M)
	t_0 = c0 / (w * h);
	t_1 = (d_m * d_m) / (D_m * D_m);
	tmp = 0.0;
	if (h <= -1.75e+40)
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d_m / D_m) * (d_m / D_m))) + sqrt((((t_0 * t_1) * (t_1 * ((c0 / w) / h))) - (M * M))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d$95$m * d$95$m), $MachinePrecision] / N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.75e+40], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(t$95$1 * N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 2 regimes

2. if h < -1.75e40

   1. Initial program 37.9%

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

   3. Simplified 40.3%

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

      1. times-frac 37.9%

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

   6. Applied egg-rr 37.9%

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

   7. Taylor expanded in c0 around 0 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-/r* 38.1%

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

   9. Simplified 38.1%

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

   if -1.75e40 < h

   1. Initial program 22.0%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in M around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w} \cdot -0.5} \]

      2. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(M \cdot c0\right) \cdot \sqrt{-1}}}{w} \cdot -0.5 \]

   7. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\left(M \cdot c0\right) \cdot \sqrt{-1}}{w} \cdot -0.5} \]
   8. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto \frac{\color{blue}{\log \left(e^{\left(M \cdot c0\right) \cdot \sqrt{-1}}\right)}}{w} \cdot -0.5 \]

      2. exp-prod 21.9%

         \[\leadsto \frac{\log \color{blue}{\left({\left(e^{M \cdot c0}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

      3. *-commutative 21.9%

         \[\leadsto \frac{\log \left({\left(e^{\color{blue}{c0 \cdot M}}\right)}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

      4. exp-prod 16.4%

         \[\leadsto \frac{\log \left({\color{blue}{\left({\left(e^{c0}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

   9. Applied egg-rr 16.4%

      \[\leadsto \frac{\color{blue}{\log \left({\left({\left(e^{c0}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

   10. Taylor expanded in c0 around 0 36.8%

       \[\leadsto \frac{\log \color{blue}{1}}{w} \cdot -0.5 \]

   11. Taylor expanded in w around 0 36.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.75 \cdot 10^{+40}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{\frac{c0}{w}}{h}\right) - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := t\_0 \cdot \frac{d\_m \cdot d\_m}{D\_m \cdot D\_m}\\ \mathbf{if}\;h \leq -2.3 \cdot 10^{+36}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 \cdot \left(\frac{d\_m}{D\_m} \cdot \frac{d\_m}{D\_m}\right) + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (* t_0 (/ (* d_m d_m) (* D_m D_m)))))
   (if (<= h -2.3e+36)
     (*
      (/ c0 (* 2.0 w))
      (+ (* t_0 (* (/ d_m D_m) (/ d_m D_m))) (sqrt (- (* t_1 t_1) (* M M)))))
     0.0)))

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d_m * d_m) / (D_m * D_m));
	double tmp;
	if (h <= -2.3e+36) {
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d_m / D_m) * (d_m / D_m))) + sqrt(((t_1 * t_1) - (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

D_m = abs(d)
d_m = abs(d)
real(8) function code(c0, w, h, d_m, d_m_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = t_0 * ((d_m_1 * d_m_1) / (d_m * d_m))
    if (h <= (-2.3d+36)) then
        tmp = (c0 / (2.0d0 * w)) * ((t_0 * ((d_m_1 / d_m) * (d_m_1 / d_m))) + sqrt(((t_1 * t_1) - (m * m))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d_m * d_m) / (D_m * D_m));
	double tmp;
	if (h <= -2.3e+36) {
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d_m / D_m) * (d_m / D_m))) + Math.sqrt(((t_1 * t_1) - (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
def code(c0, w, h, D_m, d_m, M):
	t_0 = c0 / (w * h)
	t_1 = t_0 * ((d_m * d_m) / (D_m * D_m))
	tmp = 0
	if h <= -2.3e+36:
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d_m / D_m) * (d_m / D_m))) + math.sqrt(((t_1 * t_1) - (M * M))))
	else:
		tmp = 0.0
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(t_0 * Float64(Float64(d_m * d_m) / Float64(D_m * D_m)))
	tmp = 0.0
	if (h <= -2.3e+36)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(t_0 * Float64(Float64(d_m / D_m) * Float64(d_m / D_m))) + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
function tmp_2 = code(c0, w, h, D_m, d_m, M)
	t_0 = c0 / (w * h);
	t_1 = t_0 * ((d_m * d_m) / (D_m * D_m));
	tmp = 0.0;
	if (h <= -2.3e+36)
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d_m / D_m) * (d_m / D_m))) + sqrt(((t_1 * t_1) - (M * M))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(d$95$m * d$95$m), $MachinePrecision] / N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.3e+36], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(N[(d$95$m / D$95$m), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 2 regimes

2. if h < -2.29999999999999996e36

   1. Initial program 37.9%

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

   3. Simplified 40.3%

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

      1. times-frac 37.9%

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

   6. Applied egg-rr 37.9%

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

   if -2.29999999999999996e36 < h

   1. Initial program 22.0%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in M around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{\frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w} \cdot -0.5} \]

      2. associate-*r* 0.0%

         \[\leadsto \frac{\color{blue}{\left(M \cdot c0\right) \cdot \sqrt{-1}}}{w} \cdot -0.5 \]

   7. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{\left(M \cdot c0\right) \cdot \sqrt{-1}}{w} \cdot -0.5} \]
   8. Step-by-step derivation

      1. add-log-exp 0.0%

         \[\leadsto \frac{\color{blue}{\log \left(e^{\left(M \cdot c0\right) \cdot \sqrt{-1}}\right)}}{w} \cdot -0.5 \]

      2. exp-prod 21.9%

         \[\leadsto \frac{\log \color{blue}{\left({\left(e^{M \cdot c0}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

      3. *-commutative 21.9%

         \[\leadsto \frac{\log \left({\left(e^{\color{blue}{c0 \cdot M}}\right)}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

      4. exp-prod 16.4%

         \[\leadsto \frac{\log \left({\color{blue}{\left({\left(e^{c0}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

   9. Applied egg-rr 16.4%

      \[\leadsto \frac{\color{blue}{\log \left({\left({\left(e^{c0}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

   10. Taylor expanded in c0 around 0 36.8%

       \[\leadsto \frac{\log \color{blue}{1}}{w} \cdot -0.5 \]

   11. Taylor expanded in w around 0 36.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 33.9% accurate, 151.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ 0 \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (c0 w h D_m d_m M) :precision binary64 0.0)

C

D_m = fabs(D);
d_m = fabs(d);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	return 0.0;
}

Fortran

D_m = abs(d)
d_m = abs(d)
real(8) function code(c0, w, h, d_m, d_m_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D_m, double d_m, double M) {
	return 0.0;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
def code(c0, w, h, D_m, d_m, M):
	return 0.0

Julia

D_m = abs(D)
d_m = abs(d)
function code(c0, w, h, D_m, d_m, M)
	return 0.0
end

MATLAB

D_m = abs(D);
d_m = abs(d);
function tmp = code(c0, w, h, D_m, d_m, M)
	tmp = 0.0;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := 0.0

Derivation

1. Initial program 24.5%

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

3. Simplified 35.3%

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

5. Taylor expanded in M around -inf 0.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w}} \]
6. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \color{blue}{\frac{M \cdot \left(c0 \cdot \sqrt{-1}\right)}{w} \cdot -0.5} \]

   2. associate-*r* 0.0%

      \[\leadsto \frac{\color{blue}{\left(M \cdot c0\right) \cdot \sqrt{-1}}}{w} \cdot -0.5 \]

7. Simplified 0.0%

   \[\leadsto \color{blue}{\frac{\left(M \cdot c0\right) \cdot \sqrt{-1}}{w} \cdot -0.5} \]
8. Step-by-step derivation

   1. add-log-exp 0.0%

      \[\leadsto \frac{\color{blue}{\log \left(e^{\left(M \cdot c0\right) \cdot \sqrt{-1}}\right)}}{w} \cdot -0.5 \]

   2. exp-prod 20.9%

      \[\leadsto \frac{\log \color{blue}{\left({\left(e^{M \cdot c0}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

   3. *-commutative 20.9%

      \[\leadsto \frac{\log \left({\left(e^{\color{blue}{c0 \cdot M}}\right)}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

   4. exp-prod 15.5%

      \[\leadsto \frac{\log \left({\color{blue}{\left({\left(e^{c0}\right)}^{M}\right)}}^{\left(\sqrt{-1}\right)}\right)}{w} \cdot -0.5 \]

9. Applied egg-rr 15.5%

   \[\leadsto \frac{\color{blue}{\log \left({\left({\left(e^{c0}\right)}^{M}\right)}^{\left(\sqrt{-1}\right)}\right)}}{w} \cdot -0.5 \]

10. Taylor expanded in c0 around 0 34.0%

    \[\leadsto \frac{\log \color{blue}{1}}{w} \cdot -0.5 \]

11. Taylor expanded in w around 0 34.0%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024156 
(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))))))