Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.0% → 53.4%

Time: 20.2s

Alternatives: 4

Speedup: 16.8×

• could not determine a ground truth

  1. c0 = -1.3121003706514074e+307
  2. w = 2.9008713245619006e-218
  3. h = 6.961483678821475e-173
  4. D = -4.5430181298137214e-305
  5. d = -1.5038528650280284e+279
  6. M = 6.9350214187604715e+155

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: 25.0% 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: 53.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\\ \end{array} \end{array} \]

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 w) h) (pow (/ d D) 2.0))))
     (* 0.25 (/ (* (pow D 2.0) (* h (pow M 2.0))) (pow d 2.0))))))

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 / w) / h) * pow((d / D), 2.0)));
	} else {
		tmp = 0.25 * ((pow(D, 2.0) * (h * pow(M, 2.0))) / pow(d, 2.0));
	}
	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 / w) / h) * Math.pow((d / D), 2.0)));
	} else {
		tmp = 0.25 * ((Math.pow(D, 2.0) * (h * Math.pow(M, 2.0))) / Math.pow(d, 2.0));
	}
	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 / w) / h) * math.pow((d / D), 2.0)))
	else:
		tmp = 0.25 * ((math.pow(D, 2.0) * (h * math.pow(M, 2.0))) / math.pow(d, 2.0))
	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(Float64(c0 / w) / h) * (Float64(d / D) ^ 2.0))));
	else
		tmp = Float64(0.25 * Float64(Float64((D ^ 2.0) * Float64(h * (M ^ 2.0))) / (d ^ 2.0)));
	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 / w) / h) * ((d / D) ^ 2.0)));
	else
		tmp = 0.25 * (((D ^ 2.0) * (h * (M ^ 2.0))) / (d ^ 2.0));
	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[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[Power[D, 2.0], $MachinePrecision] * N[(h * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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 75.0%

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

      1. times-frac 68.4%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in c0 around inf 71.7%

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

      1. associate-*r/ 71.7%

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

   6. Simplified 71.7%

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

   7. Taylor expanded in c0 around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. times-frac 72.7%

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

      3. associate-/l/ 72.8%

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

      4. unpow2 72.8%

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

      5. unpow2 72.8%

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

      6. times-frac 76.3%

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

      7. unpow2 76.3%

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

   9. Simplified 76.3%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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%

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

      1. times-frac 0.0%

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

   3. Simplified 0.8%

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

   4. Taylor expanded in c0 around -inf 1.8%

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

      1. +-commutative 1.8%

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

      2. fma-def 1.8%

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

      3. times-frac 1.8%

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

      4. *-commutative 1.8%

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

      5. associate-*r* 1.8%

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

      6. neg-mul-1 1.8%

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

      7. distribute-lft1-in 1.8%

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

      8. metadata-eval 1.8%

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

      9. mul0-lft 32.6%

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

      10. distribute-lft-neg-in 32.6%

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

      11. distribute-rgt-neg-in 32.6%

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

      12. metadata-eval 32.6%

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

   6. Simplified 32.6%

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

   7. Taylor expanded in c0 around 0 48.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\\ \end{array} \]

Alternative 2: 50.1% accurate, 0.6× speedup

Math

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

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 w) h) (pow (/ d D) 2.0))))
     (* t_0 (* c0 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 / w) / h) * pow((d / D), 2.0)));
	} else {
		tmp = t_0 * (c0 * 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 / w) / h) * Math.pow((d / D), 2.0)));
	} else {
		tmp = t_0 * (c0 * 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 / w) / h) * math.pow((d / D), 2.0)))
	else:
		tmp = t_0 * (c0 * 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(Float64(c0 / w) / h) * (Float64(d / D) ^ 2.0))));
	else
		tmp = Float64(t_0 * Float64(c0 * 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 / w) / h) * ((d / D) ^ 2.0)));
	else
		tmp = t_0 * (c0 * 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[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 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 75.0%

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

      1. times-frac 68.4%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in c0 around inf 71.7%

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

      1. associate-*r/ 71.7%

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

   6. Simplified 71.7%

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

   7. Taylor expanded in c0 around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. times-frac 72.7%

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

      3. associate-/l/ 72.8%

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

      4. unpow2 72.8%

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

      5. unpow2 72.8%

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

      6. times-frac 76.3%

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

      7. unpow2 76.3%

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

   9. Simplified 76.3%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 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%

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

      1. times-frac 0.0%

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

   3. Simplified 0.8%

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

   4. Taylor expanded in c0 around -inf 0.7%

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

      1. associate-*r* 0.7%

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

      2. neg-mul-1 0.7%

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

      3. distribute-lft1-in 0.7%

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

      4. metadata-eval 0.7%

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

      5. mul0-lft 40.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-c0\right) \cdot \color{blue}{0}\right) \]

      6. distribute-lft-neg-in 40.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-c0 \cdot 0\right)} \]

      7. distribute-rgt-neg-in 40.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-0\right)\right)} \]

      8. metadata-eval 40.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 40.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

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

Alternative 3: 45.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -7.6 \cdot 10^{-15} \lor \neg \left(c0 \leq 3.9 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{c0}{\frac{w}{\frac{\frac{c0}{w}}{h} \cdot {\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= c0 -7.6e-15) (not (<= c0 3.9e-12)))
   (/ c0 (/ w (* (/ (/ c0 w) h) (pow (/ d D) 2.0))))
   (* (/ c0 (* 2.0 w)) (* c0 0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -7.6e-15) || !(c0 <= 3.9e-12)) {
		tmp = c0 / (w / (((c0 / w) / h) * pow((d / D), 2.0)));
	} else {
		tmp = (c0 / (2.0 * w)) * (c0 * 0.0);
	}
	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 ((c0 <= (-7.6d-15)) .or. (.not. (c0 <= 3.9d-12))) then
        tmp = c0 / (w / (((c0 / w) / h) * ((d_1 / d) ** 2.0d0)))
    else
        tmp = (c0 / (2.0d0 * w)) * (c0 * 0.0d0)
    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 ((c0 <= -7.6e-15) || !(c0 <= 3.9e-12)) {
		tmp = c0 / (w / (((c0 / w) / h) * Math.pow((d / D), 2.0)));
	} else {
		tmp = (c0 / (2.0 * w)) * (c0 * 0.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (c0 <= -7.6e-15) or not (c0 <= 3.9e-12):
		tmp = c0 / (w / (((c0 / w) / h) * math.pow((d / D), 2.0)))
	else:
		tmp = (c0 / (2.0 * w)) * (c0 * 0.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((c0 <= -7.6e-15) || !(c0 <= 3.9e-12))
		tmp = Float64(c0 / Float64(w / Float64(Float64(Float64(c0 / w) / h) * (Float64(d / D) ^ 2.0))));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(c0 * 0.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((c0 <= -7.6e-15) || ~((c0 <= 3.9e-12)))
		tmp = c0 / (w / (((c0 / w) / h) * ((d / D) ^ 2.0)));
	else
		tmp = (c0 / (2.0 * w)) * (c0 * 0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[c0, -7.6e-15], N[Not[LessEqual[c0, 3.9e-12]], $MachinePrecision]], N[(c0 / N[(w / N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c0 < -7.6000000000000004e-15 or 3.89999999999999994e-12 < c0

   1. Initial program 27.7%

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

      1. times-frac 26.5%

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

   3. Simplified 29.0%

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

   4. Taylor expanded in c0 around inf 34.4%

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

      1. associate-*r/ 34.4%

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

   6. Simplified 34.4%

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

      1. expm1-log1p-u 22.2%

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

      2. expm1-udef 22.2%

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

      3. *-commutative 22.2%

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

      4. associate-/r* 22.2%

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

      5. associate-*r* 22.2%

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

   8. Applied egg-rr 22.2%

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

      1. expm1-def 22.2%

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

      2. expm1-log1p 35.1%

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

      3. associate-*r/ 35.1%

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

      4. *-commutative 35.1%

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

      5. associate-/l* 35.1%

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

      6. associate-/l/ 34.4%

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

      7. *-commutative 34.4%

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

      8. associate-*l* 34.4%

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

      9. associate-*r/ 34.4%

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

      10. times-frac 34.4%

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

      11. metadata-eval 34.4%

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

      12. *-commutative 34.4%

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

      13. times-frac 35.7%

          \[\leadsto \frac{c0}{1 \cdot \frac{w}{\color{blue}{\frac{c0}{h \cdot w} \cdot \frac{{d}^{2}}{{D}^{2}}}}} \]

   10. Simplified 47.9%

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

   if -7.6000000000000004e-15 < c0 < 3.89999999999999994e-12

   1. Initial program 21.6%

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

      1. times-frac 17.9%

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

   3. Simplified 21.0%

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

   4. Taylor expanded in c0 around -inf 4.9%

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

      1. associate-*r* 4.9%

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

      2. neg-mul-1 4.9%

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

      3. distribute-lft1-in 4.9%

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

      4. metadata-eval 4.9%

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

      5. mul0-lft 49.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-c0\right) \cdot \color{blue}{0}\right) \]

      6. distribute-lft-neg-in 49.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-c0 \cdot 0\right)} \]

      7. distribute-rgt-neg-in 49.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-0\right)\right)} \]

      8. metadata-eval 49.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

   6. Simplified 49.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -7.6 \cdot 10^{-15} \lor \neg \left(c0 \leq 3.9 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{c0}{\frac{w}{\frac{\frac{c0}{w}}{h} \cdot {\left(\frac{d}{D}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right)\\ \end{array} \]

Alternative 4: 28.8% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* (/ c0 (* 2.0 w)) (* c0 0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (c0 * 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 = (c0 / (2.0d0 * w)) * (c0 * 0.0d0)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (c0 * 0.0);
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (2.0 * w)) * (c0 * 0.0)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(c0 * 0.0))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 / (2.0 * w)) * (c0 * 0.0);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(c0 * 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.2%

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

   1. times-frac 23.0%

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

3. Simplified 25.7%

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

4. Taylor expanded in c0 around -inf 4.1%

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

   1. associate-*r* 4.1%

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

   2. neg-mul-1 4.1%

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

   3. distribute-lft1-in 4.1%

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

   4. metadata-eval 4.1%

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

   5. mul0-lft 31.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-c0\right) \cdot \color{blue}{0}\right) \]

   6. distribute-lft-neg-in 31.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-c0 \cdot 0\right)} \]

   7. distribute-rgt-neg-in 31.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-0\right)\right)} \]

   8. metadata-eval 31.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

6. Simplified 31.6%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

7. Final simplification 31.6%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot 0\right) \]

Reproduce

herbie shell --seed 2023300 
(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))))))