Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.3% → 54.5%

Time: 36.6s

Alternatives: 7

Speedup: 12.6×

• 21.9% of points produce a very large (infinite) output. You may want to add a precondition.

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: 54.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left({D}^{2} \cdot \left(h \cdot {\left(\frac{M}{d}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

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

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 = 0.25 * (pow(D, 2.0) * (h * pow((M / 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 * (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 = 0.25 * (Math.pow(D, 2.0) * (h * Math.pow((M / d), 2.0)));
	}
	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 = 0.25 * (math.pow(D, 2.0) * (h * math.pow((M / d), 2.0)))
	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 = Float64(0.25 * Float64((D ^ 2.0) * Float64(h * (Float64(M / d) ^ 2.0))));
	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 = 0.25 * ((D ^ 2.0) * (h * ((M / d) ^ 2.0)));
	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, N[(0.25 * N[(N[Power[D, 2.0], $MachinePrecision] * N[(h * N[Power[N[(M / d), $MachinePrecision], 2.0], $MachinePrecision]), $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 71.9%

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

   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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* 0.0%

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

      2. associate-*r* 0.0%

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

      3. associate-*r* 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-*r/ 0.0%

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

      6. *-commutative 0.0%

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

   4. Applied egg-rr 0.6%

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

   5. Taylor expanded in c0 around -inf 1.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) + 0.5 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}\right)} \]
   6. 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-define 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 2.8%

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

      4. associate-*l/ 3.4%

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

      5. *-commutative 3.4%

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

      6. *-commutative 3.4%

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

      7. associate-/l* 2.9%

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

      8. *-commutative 2.9%

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

      9. associate-*r* 2.9%

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

   7. Simplified 22.4%

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

   8. Taylor expanded in c0 around 0 31.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 31.6%

         \[\leadsto 0.25 \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

      2. *-commutative 31.6%

         \[\leadsto 0.25 \cdot \left({D}^{2} \cdot \frac{\color{blue}{h \cdot {M}^{2}}}{{d}^{2}}\right) \]

      3. associate-/l* 31.2%

         \[\leadsto 0.25 \cdot \left({D}^{2} \cdot \color{blue}{\left(h \cdot \frac{{M}^{2}}{{d}^{2}}\right)}\right) \]

      4. unpow2 31.2%

         \[\leadsto 0.25 \cdot \left({D}^{2} \cdot \left(h \cdot \frac{\color{blue}{M \cdot M}}{{d}^{2}}\right)\right) \]

      5. unpow2 31.2%

         \[\leadsto 0.25 \cdot \left({D}^{2} \cdot \left(h \cdot \frac{M \cdot M}{\color{blue}{d \cdot d}}\right)\right) \]

      6. times-frac 43.3%

         \[\leadsto 0.25 \cdot \left({D}^{2} \cdot \left(h \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{d}\right)}\right)\right) \]

      7. unpow2 43.3%

         \[\leadsto 0.25 \cdot \left({D}^{2} \cdot \left(h \cdot \color{blue}{{\left(\frac{M}{d}\right)}^{2}}\right)\right) \]

   10. Simplified 43.3%

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

4. Final simplification 52.1%

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

Alternative 2: 46.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \frac{0}{2 \cdot w}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := c0 \cdot \frac{\frac{t\_1}{h} \cdot \frac{c0}{w}}{w}\\ \mathbf{if}\;c0 \leq -3.8 \cdot 10^{-180}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c0 \leq 1.85 \cdot 10^{-128}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c0 \leq 1.05 \cdot 10^{+74}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(c0 \cdot \frac{t\_1}{w \cdot h}\right)}{2 \cdot w}\\ \mathbf{elif}\;c0 \leq 1.85 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (/ 0.0 (* 2.0 w))))
        (t_1 (pow (/ d D) 2.0))
        (t_2 (* c0 (/ (* (/ t_1 h) (/ c0 w)) w))))
   (if (<= c0 -3.8e-180)
     t_2
     (if (<= c0 1.85e-128)
       t_0
       (if (<= c0 1.05e+74)
         (* c0 (/ (* 2.0 (* c0 (/ t_1 (* w h)))) (* 2.0 w)))
         (if (<= c0 1.85e+124) t_0 t_2))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (0.0 / (2.0 * w));
	double t_1 = pow((d / D), 2.0);
	double t_2 = c0 * (((t_1 / h) * (c0 / w)) / w);
	double tmp;
	if (c0 <= -3.8e-180) {
		tmp = t_2;
	} else if (c0 <= 1.85e-128) {
		tmp = t_0;
	} else if (c0 <= 1.05e+74) {
		tmp = c0 * ((2.0 * (c0 * (t_1 / (w * h)))) / (2.0 * w));
	} else if (c0 <= 1.85e+124) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = c0 * (0.0d0 / (2.0d0 * w))
    t_1 = (d_1 / d) ** 2.0d0
    t_2 = c0 * (((t_1 / h) * (c0 / w)) / w)
    if (c0 <= (-3.8d-180)) then
        tmp = t_2
    else if (c0 <= 1.85d-128) then
        tmp = t_0
    else if (c0 <= 1.05d+74) then
        tmp = c0 * ((2.0d0 * (c0 * (t_1 / (w * h)))) / (2.0d0 * w))
    else if (c0 <= 1.85d+124) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (0.0 / (2.0 * w));
	double t_1 = Math.pow((d / D), 2.0);
	double t_2 = c0 * (((t_1 / h) * (c0 / w)) / w);
	double tmp;
	if (c0 <= -3.8e-180) {
		tmp = t_2;
	} else if (c0 <= 1.85e-128) {
		tmp = t_0;
	} else if (c0 <= 1.05e+74) {
		tmp = c0 * ((2.0 * (c0 * (t_1 / (w * h)))) / (2.0 * w));
	} else if (c0 <= 1.85e+124) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 * (0.0 / (2.0 * w))
	t_1 = math.pow((d / D), 2.0)
	t_2 = c0 * (((t_1 / h) * (c0 / w)) / w)
	tmp = 0
	if c0 <= -3.8e-180:
		tmp = t_2
	elif c0 <= 1.85e-128:
		tmp = t_0
	elif c0 <= 1.05e+74:
		tmp = c0 * ((2.0 * (c0 * (t_1 / (w * h)))) / (2.0 * w))
	elif c0 <= 1.85e+124:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(0.0 / Float64(2.0 * w)))
	t_1 = Float64(d / D) ^ 2.0
	t_2 = Float64(c0 * Float64(Float64(Float64(t_1 / h) * Float64(c0 / w)) / w))
	tmp = 0.0
	if (c0 <= -3.8e-180)
		tmp = t_2;
	elseif (c0 <= 1.85e-128)
		tmp = t_0;
	elseif (c0 <= 1.05e+74)
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(c0 * Float64(t_1 / Float64(w * h)))) / Float64(2.0 * w)));
	elseif (c0 <= 1.85e+124)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 * (0.0 / (2.0 * w));
	t_1 = (d / D) ^ 2.0;
	t_2 = c0 * (((t_1 / h) * (c0 / w)) / w);
	tmp = 0.0;
	if (c0 <= -3.8e-180)
		tmp = t_2;
	elseif (c0 <= 1.85e-128)
		tmp = t_0;
	elseif (c0 <= 1.05e+74)
		tmp = c0 * ((2.0 * (c0 * (t_1 / (w * h)))) / (2.0 * w));
	elseif (c0 <= 1.85e+124)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(c0 * N[(N[(N[(t$95$1 / h), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -3.8e-180], t$95$2, If[LessEqual[c0, 1.85e-128], t$95$0, If[LessEqual[c0, 1.05e+74], N[(c0 * N[(N[(2.0 * N[(c0 * N[(t$95$1 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 1.85e+124], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -3.79999999999999999e-180 or 1.85000000000000004e124 < c0

   1. Initial program 22.1%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in c0 around inf 36.4%

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

      1. *-commutative 36.4%

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

      2. associate-/l/ 37.0%

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

      3. associate-/r* 38.5%

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

      4. associate-*l/ 39.7%

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

      5. associate-/r* 39.1%

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

      6. associate-/l* 39.1%

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

      7. *-commutative 39.1%

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

   7. Simplified 39.1%

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

      1. times-frac 39.1%

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

      2. metadata-eval 39.1%

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

      3. associate-/r* 38.0%

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

      4. pow2 38.0%

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

      5. pow2 38.0%

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

      6. frac-times 48.9%

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

      7. pow2 48.9%

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

   9. Applied egg-rr 48.9%

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

   if -3.79999999999999999e-180 < c0 < 1.85e-128 or 1.0499999999999999e74 < c0 < 1.85000000000000004e124

   1. Initial program 14.0%

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

   3. Simplified 22.1%

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

   5. Taylor expanded in c0 around -inf 4.0%

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

      1. associate-*r* 4.0%

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

      2. neg-mul-1 4.0%

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

      3. distribute-lft1-in 4.0%

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

      4. metadata-eval 4.0%

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

      5. mul0-lft 51.4%

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

      6. distribute-lft-neg-in 51.4%

         \[\leadsto c0 \cdot \frac{\color{blue}{-c0 \cdot 0}}{2 \cdot w} \]

      7. distribute-rgt-neg-in 51.4%

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

      8. metadata-eval 51.4%

         \[\leadsto c0 \cdot \frac{c0 \cdot \color{blue}{0}}{2 \cdot w} \]

      9. mul0-lft 4.0%

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

      10. metadata-eval 4.0%

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

      11. distribute-lft1-in 4.0%

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

      12. distribute-lft-in 4.0%

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

      13. associate-/l* 3.9%

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

   7. Simplified 51.4%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   if 1.85e-128 < c0 < 1.0499999999999999e74

   1. Initial program 30.2%

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

   3. Simplified 53.9%

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

   6. Applied egg-rr 61.7%

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

      1. fma-define 61.7%

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

      2. associate-/r* 61.7%

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

      3. associate-/r* 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in d around inf 49.7%

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

       1. associate-/r* 51.6%

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

       2. associate-/l* 51.6%

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

       3. unpow2 51.6%

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

       4. unpow2 51.6%

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

       5. times-frac 61.7%

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

       6. unpow2 61.7%

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

       7. *-commutative 61.7%

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

       8. associate-*r/ 61.7%

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

       9. *-commutative 61.7%

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

   11. Simplified 61.7%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -3.8 \cdot 10^{-180}:\\ \;\;\;\;c0 \cdot \frac{\frac{{\left(\frac{d}{D}\right)}^{2}}{h} \cdot \frac{c0}{w}}{w}\\ \mathbf{elif}\;c0 \leq 1.85 \cdot 10^{-128}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{elif}\;c0 \leq 1.05 \cdot 10^{+74}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)}{2 \cdot w}\\ \mathbf{elif}\;c0 \leq 1.85 \cdot 10^{+124}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{{\left(\frac{d}{D}\right)}^{2}}{h} \cdot \frac{c0}{w}}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-244}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}{w}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 2e-244)
   (* c0 (/ 0.0 (* 2.0 w)))
   (* c0 (/ (* 2.0 (/ (* c0 (/ (pow (/ d D) 2.0) h)) w)) (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 2e-244) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * ((2.0 * ((c0 * (pow((d / D), 2.0) / h)) / w)) / (2.0 * w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m * m) <= 2d-244) then
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    else
        tmp = c0 * ((2.0d0 * ((c0 * (((d_1 / d) ** 2.0d0) / h)) / w)) / (2.0d0 * w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 2e-244) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * ((2.0 * ((c0 * (Math.pow((d / D), 2.0) / h)) / w)) / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 2e-244:
		tmp = c0 * (0.0 / (2.0 * w))
	else:
		tmp = c0 * ((2.0 * ((c0 * (math.pow((d / D), 2.0) / h)) / w)) / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 2e-244)
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(c0 * Float64((Float64(d / D) ^ 2.0) / h)) / w)) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 2e-244)
		tmp = c0 * (0.0 / (2.0 * w));
	else
		tmp = c0 * ((2.0 * ((c0 * (((d / D) ^ 2.0) / h)) / w)) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 2e-244], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(2.0 * N[(N[(c0 * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 1.9999999999999999e-244

   1. Initial program 28.8%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in c0 around -inf 12.4%

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

      1. associate-*r* 12.4%

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

      2. neg-mul-1 12.4%

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

      3. distribute-lft1-in 12.4%

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

      4. metadata-eval 12.4%

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

      5. mul0-lft 45.2%

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

      6. distribute-lft-neg-in 45.2%

         \[\leadsto c0 \cdot \frac{\color{blue}{-c0 \cdot 0}}{2 \cdot w} \]

      7. distribute-rgt-neg-in 45.2%

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

      8. metadata-eval 45.2%

         \[\leadsto c0 \cdot \frac{c0 \cdot \color{blue}{0}}{2 \cdot w} \]

      9. mul0-lft 12.4%

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

      10. metadata-eval 12.4%

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

      11. distribute-lft1-in 12.4%

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

      12. distribute-lft-in 12.4%

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

      13. associate-/l* 8.1%

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

   7. Simplified 45.2%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   if 1.9999999999999999e-244 < (*.f64 M M)

   1. Initial program 18.0%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in c0 around inf 39.2%

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

      1. *-commutative 39.2%

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

      2. associate-/l/ 39.2%

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

      3. associate-/r* 41.0%

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

      4. associate-*l/ 42.2%

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

      5. associate-/r* 41.6%

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

      6. associate-/l* 41.0%

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

      7. *-commutative 41.0%

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

   7. Simplified 41.0%

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

      1. associate-*l/ 41.1%

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

      2. associate-/r* 40.0%

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

      3. pow2 40.0%

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

      4. pow2 40.0%

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

      5. frac-times 50.0%

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

      6. pow2 50.0%

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

   9. Applied egg-rr 50.0%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-244}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}{w}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -2.7 \cdot 10^{-177} \lor \neg \left(c0 \leq 2.3 \cdot 10^{-125}\right):\\ \;\;\;\;c0 \cdot \frac{\frac{{\left(\frac{d}{D}\right)}^{2}}{h} \cdot \frac{c0}{w}}{w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= c0 -2.7e-177) (not (<= c0 2.3e-125)))
   (* c0 (/ (* (/ (pow (/ d D) 2.0) h) (/ c0 w)) w))
   (* c0 (/ 0.0 (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -2.7e-177) || !(c0 <= 2.3e-125)) {
		tmp = c0 * (((pow((d / D), 2.0) / h) * (c0 / w)) / w);
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((c0 <= (-2.7d-177)) .or. (.not. (c0 <= 2.3d-125))) then
        tmp = c0 * (((((d_1 / d) ** 2.0d0) / h) * (c0 / w)) / w)
    else
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((c0 <= -2.7e-177) || !(c0 <= 2.3e-125)) {
		tmp = c0 * (((Math.pow((d / D), 2.0) / h) * (c0 / w)) / w);
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (c0 <= -2.7e-177) or not (c0 <= 2.3e-125):
		tmp = c0 * (((math.pow((d / D), 2.0) / h) * (c0 / w)) / w)
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((c0 <= -2.7e-177) || !(c0 <= 2.3e-125))
		tmp = Float64(c0 * Float64(Float64(Float64((Float64(d / D) ^ 2.0) / h) * Float64(c0 / w)) / w));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((c0 <= -2.7e-177) || ~((c0 <= 2.3e-125)))
		tmp = c0 * (((((d / D) ^ 2.0) / h) * (c0 / w)) / w);
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[c0, -2.7e-177], N[Not[LessEqual[c0, 2.3e-125]], $MachinePrecision]], N[(c0 * N[(N[(N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c0 < -2.7000000000000002e-177 or 2.2999999999999999e-125 < c0

   1. Initial program 23.5%

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

   3. Simplified 40.4%

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

   5. Taylor expanded in c0 around inf 37.6%

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

      1. *-commutative 37.6%

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

      2. associate-/l/ 38.5%

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

      3. associate-/r* 40.0%

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

      4. associate-*l/ 40.9%

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

      5. associate-/r* 39.5%

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

      6. associate-/l* 39.6%

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

      7. *-commutative 39.6%

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

   7. Simplified 39.6%

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

      1. times-frac 39.6%

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

      2. metadata-eval 39.6%

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

      3. associate-/r* 38.8%

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

      4. pow2 38.8%

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

      5. pow2 38.8%

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

      6. frac-times 49.4%

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

      7. pow2 49.4%

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

   9. Applied egg-rr 49.4%

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

   if -2.7000000000000002e-177 < c0 < 2.2999999999999999e-125

   1. Initial program 13.4%

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

   3. Simplified 23.9%

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

   5. Taylor expanded in c0 around -inf 2.6%

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

      1. associate-*r* 2.6%

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

      2. neg-mul-1 2.6%

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

      3. distribute-lft1-in 2.6%

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

      4. metadata-eval 2.6%

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

      5. mul0-lft 50.4%

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

      6. distribute-lft-neg-in 50.4%

         \[\leadsto c0 \cdot \frac{\color{blue}{-c0 \cdot 0}}{2 \cdot w} \]

      7. distribute-rgt-neg-in 50.4%

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

      8. metadata-eval 50.4%

         \[\leadsto c0 \cdot \frac{c0 \cdot \color{blue}{0}}{2 \cdot w} \]

      9. mul0-lft 2.6%

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

      10. metadata-eval 2.6%

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

      11. distribute-lft1-in 2.6%

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

      12. distribute-lft-in 2.6%

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

      13. associate-/l* 2.6%

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

   7. Simplified 50.4%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -2.7 \cdot 10^{-177} \lor \neg \left(c0 \leq 2.3 \cdot 10^{-125}\right):\\ \;\;\;\;c0 \cdot \frac{\frac{{\left(\frac{d}{D}\right)}^{2}}{h} \cdot \frac{c0}{w}}{w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.5% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 3.05 \cdot 10^{+91}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(M \cdot \frac{c0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 3.05e+91) (* c0 (/ 0.0 (* 2.0 w))) (* 0.5 (* M (/ c0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 3.05e+91) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = 0.5 * (M * (c0 / w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 3.05d+91) then
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    else
        tmp = 0.5d0 * (m * (c0 / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 3.05e+91) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = 0.5 * (M * (c0 / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 3.05e+91:
		tmp = c0 * (0.0 / (2.0 * w))
	else:
		tmp = 0.5 * (M * (c0 / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 3.05e+91)
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	else
		tmp = Float64(0.5 * Float64(M * Float64(c0 / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 3.05e+91)
		tmp = c0 * (0.0 / (2.0 * w));
	else
		tmp = 0.5 * (M * (c0 / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 3.05e+91], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(M * N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.05e91

   1. Initial program 23.9%

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

   3. Simplified 39.0%

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

   5. Taylor expanded in c0 around -inf 5.4%

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

      1. associate-*r* 5.4%

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

      2. neg-mul-1 5.4%

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

      3. distribute-lft1-in 5.4%

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

      4. metadata-eval 5.4%

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

      5. mul0-lft 29.0%

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

      6. distribute-lft-neg-in 29.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{-c0 \cdot 0}}{2 \cdot w} \]

      7. distribute-rgt-neg-in 29.0%

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

      8. metadata-eval 29.0%

         \[\leadsto c0 \cdot \frac{c0 \cdot \color{blue}{0}}{2 \cdot w} \]

      9. mul0-lft 5.4%

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

      10. metadata-eval 5.4%

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

      11. distribute-lft1-in 5.4%

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

      12. distribute-lft-in 5.4%

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

      13. associate-/l* 4.0%

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

   7. Simplified 29.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   if 3.05e91 < M

   1. Initial program 10.5%

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

   3. Simplified 31.6%

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

   6. Applied egg-rr 48.6%

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

   7. Taylor expanded in c0 around 0 22.8%

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

      1. associate-/l* 25.5%

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

   9. Simplified 25.5%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.05 \cdot 10^{+91}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(M \cdot \frac{c0}{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.7% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{M \cdot 0.5}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 2.1e+91) (* c0 (/ 0.0 (* 2.0 w))) (* c0 (/ (* M 0.5) w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2.1e+91) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * ((M * 0.5) / w);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.1d+91) then
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    else
        tmp = c0 * ((m * 0.5d0) / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2.1e+91) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * ((M * 0.5) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 2.1e+91:
		tmp = c0 * (0.0 / (2.0 * w))
	else:
		tmp = c0 * ((M * 0.5) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 2.1e+91)
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(Float64(M * 0.5) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 2.1e+91)
		tmp = c0 * (0.0 / (2.0 * w));
	else
		tmp = c0 * ((M * 0.5) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 2.1e+91], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(M * 0.5), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.10000000000000008e91

   1. Initial program 23.9%

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

   3. Simplified 39.0%

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

   5. Taylor expanded in c0 around -inf 5.4%

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

      1. associate-*r* 5.4%

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

      2. neg-mul-1 5.4%

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

      3. distribute-lft1-in 5.4%

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

      4. metadata-eval 5.4%

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

      5. mul0-lft 29.0%

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

      6. distribute-lft-neg-in 29.0%

         \[\leadsto c0 \cdot \frac{\color{blue}{-c0 \cdot 0}}{2 \cdot w} \]

      7. distribute-rgt-neg-in 29.0%

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

      8. metadata-eval 29.0%

         \[\leadsto c0 \cdot \frac{c0 \cdot \color{blue}{0}}{2 \cdot w} \]

      9. mul0-lft 5.4%

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

      10. metadata-eval 5.4%

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

      11. distribute-lft1-in 5.4%

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

      12. distribute-lft-in 5.4%

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

      13. associate-/l* 4.0%

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

   7. Simplified 29.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   if 2.10000000000000008e91 < M

   1. Initial program 10.5%

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

   3. Simplified 31.6%

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

   6. Applied egg-rr 48.6%

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

      1. fma-define 48.6%

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

      2. associate-/r* 48.6%

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

      3. associate-/r* 48.6%

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

   8. Simplified 48.6%

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

   9. Taylor expanded in d around 0 27.7%

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

       1. associate-*r/ 27.7%

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

   11. Simplified 27.7%

       \[\leadsto c0 \cdot \color{blue}{\frac{0.5 \cdot M}{w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+91}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{M \cdot 0.5}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 19.2% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(M \cdot \frac{c0}{w}\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* 0.5 (* M (/ c0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * (M * (c0 / w));
}

Fortran

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

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * (M * (c0 / w));
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * (M * (c0 / w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(M * Float64(c0 / w)))
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(M * N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.9%

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

3. Simplified 37.9%

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

6. Applied egg-rr 43.9%

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

7. Taylor expanded in c0 around 0 15.0%

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

   1. associate-/l* 16.8%

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

9. Simplified 16.8%

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

10. Final simplification 16.8%

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

Reproduce

herbie shell --seed 2024052 
(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))))))