Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.3% → 54.6%

Time: 28.8s

Alternatives: 4

Speedup: 151.0×

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: 25.3% 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: 54.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(2.0 * N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 81.3%

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

   3. Simplified 78.5%

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

   6. Applied egg-rr 80.4%

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

   7. Taylor expanded in c0 around inf 43.5%

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

   8. Taylor expanded in d around inf 81.9%

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

   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 28.1%

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

   5. Taylor expanded in c0 around -inf 2.0%

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

      1. associate-*r/ 2.0%

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

      2. neg-mul-1 2.0%

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

      3. distribute-lft-neg-in 2.0%

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

   7. Simplified 2.0%

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

      1. expm1-log1p-u 0.7%

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

      2. expm1-undefine 0.7%

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

   9. Applied egg-rr 0.2%

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

       1. expm1-define 0.2%

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

       2. associate-*l* 0.2%

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

       3. associate-*l* 0.8%

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

   11. Simplified 0.8%

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

   12. Taylor expanded in h around 0 1.9%

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

       1. associate-*r/ 1.9%

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

       2. distribute-lft1-in 1.9%

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

       3. metadata-eval 1.9%

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

       4. mul0-lft 32.5%

          \[\leadsto \frac{0.5 \cdot \left(c0 \cdot \color{blue}{0}\right)}{h \cdot w} \]

   14. Simplified 32.5%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c0 \cdot 0\right)}{h \cdot w}} \]

   15. Taylor expanded in c0 around 0 40.1%

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

4. Final simplification 55.1%

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

Alternative 2: 54.8% accurate, 0.5× speedup

Math

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

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 0.0)))

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 = 0.0;
	}
	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 = 0.0;
	}
	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 = 0.0
	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 = 0.0;
	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 = 0.0;
	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, 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 81.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 28.1%

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

   5. Taylor expanded in c0 around -inf 2.0%

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

      1. associate-*r/ 2.0%

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

      2. neg-mul-1 2.0%

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

      3. distribute-lft-neg-in 2.0%

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

   7. Simplified 2.0%

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

      1. expm1-log1p-u 0.7%

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

      2. expm1-undefine 0.7%

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

   9. Applied egg-rr 0.2%

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

       1. expm1-define 0.2%

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

       2. associate-*l* 0.2%

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

       3. associate-*l* 0.8%

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

   11. Simplified 0.8%

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

   12. Taylor expanded in h around 0 1.9%

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

       1. associate-*r/ 1.9%

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

       2. distribute-lft1-in 1.9%

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

       3. metadata-eval 1.9%

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

       4. mul0-lft 32.5%

          \[\leadsto \frac{0.5 \cdot \left(c0 \cdot \color{blue}{0}\right)}{h \cdot w} \]

   14. Simplified 32.5%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c0 \cdot 0\right)}{h \cdot w}} \]

   15. Taylor expanded in c0 around 0 40.1%

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

Alternative 3: 40.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, M\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w 2.65e-21)
   (* c0 (/ (fma c0 (* d (/ d (* D (* w (* h D))))) M) (* 2.0 w)))
   0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= 2.65e-21) {
		tmp = c0 * (fma(c0, (d * (d / (D * (w * (h * D))))), M) / (2.0 * w));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= 2.65e-21)
		tmp = Float64(c0 * Float64(fma(c0, Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D))))), M) / Float64(2.0 * w)));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, 2.65e-21], N[(c0 * N[(N[(c0 * N[(d * N[(d / N[(D * N[(w * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + M), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 2.65e-21

   1. Initial program 30.4%

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

   3. Simplified 49.0%

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

      1. fma-undefine 49.0%

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

      2. associate-*r/ 45.4%

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

      3. *-commutative 45.4%

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

      4. associate-*r* 44.2%

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

      5. associate-*r* 40.3%

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

      6. associate-/l* 40.3%

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

      7. frac-times 39.2%

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

   6. Applied egg-rr 36.7%

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

      1. *-lft-identity 36.7%

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

      2. associate-*l/ 36.3%

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

      3. times-frac 36.2%

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

   8. Simplified 36.2%

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

   9. Taylor expanded in c0 around 0 13.0%

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

       1. neg-mul-1 13.0%

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

   11. Simplified 13.0%

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

       1. add-sqr-sqrt 13.0%

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

       2. pow1/2 13.0%

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

       3. pow1/2 23.8%

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

       4. pow-prod-down 49.3%

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

       5. pow2 49.3%

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

   13. Applied egg-rr 49.3%

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

       1. unpow1/2 49.3%

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

       2. unpow2 49.3%

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

       3. sqr-neg 49.3%

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

       4. pow-sqr 49.3%

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

       5. metadata-eval 49.3%

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

   15. Simplified 49.3%

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

   16. Taylor expanded in M around 0 51.3%

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

   if 2.65e-21 < w

   1. Initial program 24.6%

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

   3. Simplified 35.2%

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

   5. Taylor expanded in c0 around -inf 2.8%

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

      1. associate-*r/ 2.8%

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

      2. neg-mul-1 2.8%

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

      3. distribute-lft-neg-in 2.8%

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

   7. Simplified 2.8%

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

      1. expm1-log1p-u 2.6%

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

      2. expm1-undefine 4.4%

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

   9. Applied egg-rr 4.6%

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

       1. expm1-define 2.8%

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

       2. associate-*l* 2.8%

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

       3. associate-*l* 2.8%

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

   11. Simplified 2.8%

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

   12. Taylor expanded in h around 0 12.2%

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

       1. associate-*r/ 12.2%

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

       2. distribute-lft1-in 12.2%

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

       3. metadata-eval 12.2%

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

       4. mul0-lft 48.3%

          \[\leadsto \frac{0.5 \cdot \left(c0 \cdot \color{blue}{0}\right)}{h \cdot w} \]

   14. Simplified 48.3%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c0 \cdot 0\right)}{h \cdot w}} \]

   15. Taylor expanded in c0 around 0 48.3%

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

Alternative 4: 33.2% accurate, 151.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 0.0)

C

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

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 = 0.0d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Python

def code(c0, w, h, D, d, M):
	return 0.0

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 29.2%

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

3. Simplified 46.2%

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

5. Taylor expanded in c0 around -inf 2.8%

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

   1. associate-*r/ 2.8%

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

   2. neg-mul-1 2.8%

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

   3. distribute-lft-neg-in 2.8%

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

7. Simplified 2.8%

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

   1. expm1-log1p-u 1.9%

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

   2. expm1-undefine 2.6%

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

9. Applied egg-rr 2.3%

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

    1. expm1-define 1.6%

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

    2. associate-*l* 2.0%

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

    3. associate-*l* 2.4%

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

11. Simplified 2.4%

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

12. Taylor expanded in h around 0 4.2%

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

    1. associate-*r/ 4.2%

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

    2. distribute-lft1-in 4.2%

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

    3. metadata-eval 4.2%

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

    4. mul0-lft 24.5%

       \[\leadsto \frac{0.5 \cdot \left(c0 \cdot \color{blue}{0}\right)}{h \cdot w} \]

14. Simplified 24.5%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c0 \cdot 0\right)}{h \cdot w}} \]

15. Taylor expanded in c0 around 0 29.5%

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

Reproduce

herbie shell --seed 2024137 
(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))))))