Henrywood and Agarwal, Equation (13)

Percentage Accurate: 23.9% → 53.9%

Time: 25.7s

Alternatives: 6

Speedup: 11.6×

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: 23.9% 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

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: 53.9% accurate, 0.5× speedup

Math

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

FPCore

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

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));
	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 = c0 * (0.0 / w);
	}
	return tmp;
}

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));
	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 = c0 * (0.0 / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	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 = c0 * (0.0 / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	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 = Float64(c0 * Float64(0.0 / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	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 = c0 * (0.0 / w);
	end
	tmp_2 = tmp;
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]}, 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, N[(c0 * N[(0.0 / 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 74.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 22.6%

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

   5. Taylor expanded in w around 0 21.7%

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

      1. expm1-log1p-u 18.3%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(h \cdot D\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      2. expm1-undefine 8.7%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(w \cdot \left(h \cdot D\right)\right)} - 1\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      3. *-commutative 8.7%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(e^{\mathsf{log1p}\left(w \cdot \color{blue}{\left(D \cdot h\right)}\right)} - 1\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   7. Applied egg-rr 8.7%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)} - 1\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]
   8. Step-by-step derivation

      1. expm1-define 18.3%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   9. Simplified 18.3%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   10. Taylor expanded in c0 around -inf 1.5%

       \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 1.5%

          \[\leadsto c0 \cdot \color{blue}{\frac{-0.5 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]

       2. distribute-lft-in 0.8%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) + c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w} \]

       3. mul-1-neg 0.8%

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

       4. distribute-rgt-neg-in 0.8%

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

       5. associate-/l* 0.1%

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

       6. mul-1-neg 0.1%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(\color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}} + c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \]

       7. associate-/l* 0.1%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \color{blue}{\frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)}{w} \]

       8. distribute-lft1-in 0.1%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w} \]

       9. metadata-eval 0.1%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \]

       10. mul0-lft 43.4%

           \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{0}}{w} \]

       11. metadata-eval 43.4%

           \[\leadsto c0 \cdot \frac{\color{blue}{0}}{w} \]

   12. Simplified 43.4%

       \[\leadsto c0 \cdot \color{blue}{\frac{0}{w}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 36.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -8 \cdot 10^{-145} \lor \neg \left(w \leq 4.4 \cdot 10^{+50}\right):\\ \;\;\;\;c0 \cdot \frac{0}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} - M\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= w -8e-145) (not (<= w 4.4e+50)))
   (* c0 (/ 0.0 w))
   (* (/ c0 (* 2.0 w)) (- (* (/ c0 (* w h)) (/ (* d d) (* D D))) M))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((w <= -8e-145) || !(w <= 4.4e+50)) {
		tmp = c0 * (0.0 / w);
	} else {
		tmp = (c0 / (2.0 * w)) * (((c0 / (w * h)) * ((d * d) / (D * D))) - M);
	}
	return tmp;
}

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) :: tmp
    if ((w <= (-8d-145)) .or. (.not. (w <= 4.4d+50))) then
        tmp = c0 * (0.0d0 / w)
    else
        tmp = (c0 / (2.0d0 * w)) * (((c0 / (w * h)) * ((d_1 * d_1) / (d * d))) - m)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((w <= -8e-145) || !(w <= 4.4e+50)) {
		tmp = c0 * (0.0 / w);
	} else {
		tmp = (c0 / (2.0 * w)) * (((c0 / (w * h)) * ((d * d) / (D * D))) - M);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (w <= -8e-145) or not (w <= 4.4e+50):
		tmp = c0 * (0.0 / w)
	else:
		tmp = (c0 / (2.0 * w)) * (((c0 / (w * h)) * ((d * d) / (D * D))) - M)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((w <= -8e-145) || !(w <= 4.4e+50))
		tmp = Float64(c0 * Float64(0.0 / w));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D))) - M));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((w <= -8e-145) || ~((w <= 4.4e+50)))
		tmp = c0 * (0.0 / w);
	else
		tmp = (c0 / (2.0 * w)) * (((c0 / (w * h)) * ((d * d) / (D * D))) - M);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[w, -8e-145], N[Not[LessEqual[w, 4.4e+50]], $MachinePrecision]], N[(c0 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -7.99999999999999932e-145 or 4.40000000000000034e50 < w

   1. Initial program 16.9%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in w around 0 29.6%

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

      1. expm1-log1p-u 22.6%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(h \cdot D\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      2. expm1-undefine 13.4%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(w \cdot \left(h \cdot D\right)\right)} - 1\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      3. *-commutative 13.4%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(e^{\mathsf{log1p}\left(w \cdot \color{blue}{\left(D \cdot h\right)}\right)} - 1\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   7. Applied egg-rr 13.4%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)} - 1\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]
   8. Step-by-step derivation

      1. expm1-define 22.6%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   9. Simplified 22.6%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   10. Taylor expanded in c0 around -inf 5.2%

       \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 5.2%

          \[\leadsto c0 \cdot \color{blue}{\frac{-0.5 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]

       2. distribute-lft-in 5.2%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) + c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w} \]

       3. mul-1-neg 5.2%

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

       4. distribute-rgt-neg-in 5.2%

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

       5. associate-/l* 3.5%

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

       6. mul-1-neg 3.5%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(\color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}} + c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \]

       7. associate-/l* 4.3%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \color{blue}{\frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)}{w} \]

       8. distribute-lft1-in 4.3%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w} \]

       9. metadata-eval 4.3%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \]

       10. mul0-lft 42.8%

           \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{0}}{w} \]

       11. metadata-eval 42.8%

           \[\leadsto c0 \cdot \frac{\color{blue}{0}}{w} \]

   12. Simplified 42.8%

       \[\leadsto c0 \cdot \color{blue}{\frac{0}{w}} \]

   if -7.99999999999999932e-145 < w < 4.40000000000000034e50

   1. Initial program 35.1%

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

   3. Simplified 35.3%

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

   5. Taylor expanded in c0 around 0 13.2%

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

      1. neg-mul-1 13.2%

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

   7. Simplified 13.2%

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

   9. Applied egg-rr 42.7%

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

       1. *-commutative 42.7%

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

       2. neg-mul-1 42.7%

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

   11. Simplified 42.7%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \color{blue}{\left(-M\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -8 \cdot 10^{-145} \lor \neg \left(w \leq 4.4 \cdot 10^{+50}\right):\\ \;\;\;\;c0 \cdot \frac{0}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} - M\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 33.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.45 \cdot 10^{-216} \lor \neg \left(w \leq 2.6 \cdot 10^{-182}\right):\\ \;\;\;\;c0 \cdot \frac{0}{w}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c0}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= w -1.45e-216) (not (<= w 2.6e-182)))
   (* c0 (/ 0.0 w))
   (* 2.0 (/ c0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((w <= -1.45e-216) || !(w <= 2.6e-182)) {
		tmp = c0 * (0.0 / w);
	} else {
		tmp = 2.0 * (c0 / w);
	}
	return tmp;
}

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) :: tmp
    if ((w <= (-1.45d-216)) .or. (.not. (w <= 2.6d-182))) then
        tmp = c0 * (0.0d0 / w)
    else
        tmp = 2.0d0 * (c0 / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((w <= -1.45e-216) || !(w <= 2.6e-182)) {
		tmp = c0 * (0.0 / w);
	} else {
		tmp = 2.0 * (c0 / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (w <= -1.45e-216) or not (w <= 2.6e-182):
		tmp = c0 * (0.0 / w)
	else:
		tmp = 2.0 * (c0 / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((w <= -1.45e-216) || !(w <= 2.6e-182))
		tmp = Float64(c0 * Float64(0.0 / w));
	else
		tmp = Float64(2.0 * Float64(c0 / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((w <= -1.45e-216) || ~((w <= 2.6e-182)))
		tmp = c0 * (0.0 / w);
	else
		tmp = 2.0 * (c0 / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[w, -1.45e-216], N[Not[LessEqual[w, 2.6e-182]], $MachinePrecision]], N[(c0 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(c0 / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.45e-216 or 2.60000000000000006e-182 < w

   1. Initial program 23.4%

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

   3. Simplified 35.5%

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

   5. Taylor expanded in w around 0 35.2%

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

      1. expm1-log1p-u 30.1%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(h \cdot D\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      2. expm1-undefine 18.6%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(w \cdot \left(h \cdot D\right)\right)} - 1\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      3. *-commutative 18.6%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(e^{\mathsf{log1p}\left(w \cdot \color{blue}{\left(D \cdot h\right)}\right)} - 1\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   7. Applied egg-rr 18.6%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)} - 1\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]
   8. Step-by-step derivation

      1. expm1-define 30.1%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   9. Simplified 30.1%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   10. Taylor expanded in c0 around -inf 5.2%

       \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 5.2%

          \[\leadsto c0 \cdot \color{blue}{\frac{-0.5 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]

       2. distribute-lft-in 4.6%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) + c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w} \]

       3. mul-1-neg 4.6%

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

       4. distribute-rgt-neg-in 4.6%

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

       5. associate-/l* 3.0%

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

       6. mul-1-neg 3.0%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(\color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}} + c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \]

       7. associate-/l* 4.0%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \color{blue}{\frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)}{w} \]

       8. distribute-lft1-in 4.0%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w} \]

       9. metadata-eval 4.0%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \]

       10. mul0-lft 37.6%

           \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{0}}{w} \]

       11. metadata-eval 37.6%

           \[\leadsto c0 \cdot \frac{\color{blue}{0}}{w} \]

   12. Simplified 37.6%

       \[\leadsto c0 \cdot \color{blue}{\frac{0}{w}} \]

   if -1.45e-216 < w < 2.60000000000000006e-182

   1. Initial program 35.1%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in M around -inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

   7. Simplified 0.0%

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

   9. Applied egg-rr 28.9%

      \[\leadsto c0 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(M \cdot -0.5\right)} - -1}}{w} \]
   10. Step-by-step derivation

       1. sub-neg 28.9%

          \[\leadsto c0 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(M \cdot -0.5\right)} + \left(--1\right)}}{w} \]

       2. log1p-undefine 28.9%

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

       3. rem-exp-log 31.0%

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

       4. metadata-eval 31.0%

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

   11. Simplified 31.0%

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

   12. Taylor expanded in M around 0 32.8%

       \[\leadsto \color{blue}{2 \cdot \frac{c0}{w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.45 \cdot 10^{-216} \lor \neg \left(w \leq 2.6 \cdot 10^{-182}\right):\\ \;\;\;\;c0 \cdot \frac{0}{w}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c0}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 35.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.75 \cdot 10^{+76}:\\ \;\;\;\;c0 \cdot \frac{0}{w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left(-M\right) \cdot \frac{-0.5}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 2.75e+76) (* c0 (/ 0.0 w)) (* c0 (* (- M) (/ -0.5 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2.75e+76) {
		tmp = c0 * (0.0 / w);
	} else {
		tmp = c0 * (-M * (-0.5 / w));
	}
	return tmp;
}

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) :: tmp
    if (m <= 2.75d+76) then
        tmp = c0 * (0.0d0 / w)
    else
        tmp = c0 * (-m * ((-0.5d0) / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2.75e+76) {
		tmp = c0 * (0.0 / w);
	} else {
		tmp = c0 * (-M * (-0.5 / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 2.75e+76:
		tmp = c0 * (0.0 / w)
	else:
		tmp = c0 * (-M * (-0.5 / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 2.75e+76)
		tmp = Float64(c0 * Float64(0.0 / w));
	else
		tmp = Float64(c0 * Float64(Float64(-M) * Float64(-0.5 / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 2.75e+76)
		tmp = c0 * (0.0 / w);
	else
		tmp = c0 * (-M * (-0.5 / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 2.75e+76], N[(c0 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision], N[(c0 * N[((-M) * N[(-0.5 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.75e76

   1. Initial program 28.3%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in w around 0 36.3%

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

      1. expm1-log1p-u 32.2%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(h \cdot D\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      2. expm1-undefine 20.4%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(w \cdot \left(h \cdot D\right)\right)} - 1\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

      3. *-commutative 20.4%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(e^{\mathsf{log1p}\left(w \cdot \color{blue}{\left(D \cdot h\right)}\right)} - 1\right)}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   7. Applied egg-rr 20.4%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)} - 1\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]
   8. Step-by-step derivation

      1. expm1-define 32.2%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   9. Simplified 32.2%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w \cdot \left(D \cdot h\right)\right)\right)}}, \sqrt{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(D \cdot \left(h \cdot w\right)\right)}, M\right) \cdot \left(c0 \cdot \left(d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}\right) - M\right)}\right)}{2 \cdot w} \]

   10. Taylor expanded in c0 around -inf 5.1%

       \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 5.1%

          \[\leadsto c0 \cdot \color{blue}{\frac{-0.5 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]

       2. distribute-lft-in 4.6%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{\left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right) + c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w} \]

       3. mul-1-neg 4.6%

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

       4. distribute-rgt-neg-in 4.6%

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

       5. associate-/l* 3.2%

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

       6. mul-1-neg 3.2%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(\color{blue}{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}} + c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \]

       7. associate-/l* 4.0%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \color{blue}{\frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)}{w} \]

       8. distribute-lft1-in 4.0%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}}{w} \]

       9. metadata-eval 4.0%

          \[\leadsto c0 \cdot \frac{-0.5 \cdot \left(\color{blue}{0} \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w} \]

       10. mul0-lft 35.8%

           \[\leadsto c0 \cdot \frac{-0.5 \cdot \color{blue}{0}}{w} \]

       11. metadata-eval 35.8%

           \[\leadsto c0 \cdot \frac{\color{blue}{0}}{w} \]

   12. Simplified 35.8%

       \[\leadsto c0 \cdot \color{blue}{\frac{0}{w}} \]

   if 2.75e76 < M

   1. Initial program 13.5%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in M around -inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

   7. Simplified 0.0%

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

   9. Applied egg-rr 41.7%

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

       1. associate-*r* 41.7%

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

       2. neg-mul-1 41.7%

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

   11. Simplified 41.7%

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

Alternative 5: 15.7% accurate, 30.2× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{c0}{w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* 2.0 (/ c0 w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 2.0 * (c0 / w);
}

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
    code = 2.0d0 * (c0 / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 2.0 * (c0 / w);
}

Python

def code(c0, w, h, D, d, M):
	return 2.0 * (c0 / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(2.0 * Float64(c0 / w))
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(2.0 * N[(c0 / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.1%

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

3. Simplified 33.8%

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

5. Taylor expanded in M around -inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

7. Simplified 0.0%

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

9. Applied egg-rr 14.7%

   \[\leadsto c0 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(M \cdot -0.5\right)} - -1}}{w} \]
10. Step-by-step derivation

    1. sub-neg 14.7%

       \[\leadsto c0 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(M \cdot -0.5\right)} + \left(--1\right)}}{w} \]

    2. log1p-undefine 14.7%

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

    3. rem-exp-log 15.8%

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

    4. metadata-eval 15.8%

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

11. Simplified 15.8%

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

12. Taylor expanded in M around 0 17.2%

    \[\leadsto \color{blue}{2 \cdot \frac{c0}{w}} \]
13. Add Preprocessing

Alternative 6: 3.3% accurate, 50.3× speedup

Math

\[\begin{array}{l} \\ c0 \cdot 2 \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* c0 2.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * 2.0;
}

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
    code = c0 * 2.0d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * 2.0;
}

Python

def code(c0, w, h, D, d, M):
	return c0 * 2.0

Julia

function code(c0, w, h, D, d, M)
	return Float64(c0 * 2.0)
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = c0 * 2.0;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(c0 * 2.0), $MachinePrecision]

Derivation

1. Initial program 26.1%

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

3. Simplified 33.8%

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

5. Taylor expanded in M around -inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

7. Simplified 0.0%

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

9. Applied egg-rr 5.0%

   \[\leadsto c0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(M \cdot \frac{-0.5}{w}\right)} - -1\right)} \]
10. Step-by-step derivation

    1. sub-neg 5.0%

       \[\leadsto c0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(M \cdot \frac{-0.5}{w}\right)} + \left(--1\right)\right)} \]

    2. log1p-undefine 5.0%

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

    3. rem-exp-log 8.3%

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

    4. metadata-eval 8.3%

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

11. Simplified 8.3%

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

12. Taylor expanded in M around 0 3.7%

    \[\leadsto \color{blue}{2 \cdot c0} \]
13. Step-by-step derivation

    1. *-commutative 3.7%

       \[\leadsto \color{blue}{c0 \cdot 2} \]

14. Simplified 3.7%

    \[\leadsto \color{blue}{c0 \cdot 2} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(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))))))