Henrywood and Agarwal, Equation (13)

Percentage Accurate: 23.8% → 54.9%

Time: 25.1s

Alternatives: 3

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = -2.5452327659057073e+275
  2. w = 8.629091808066478e-110
  3. h = 6.484487600740385e-299
  4. D = 8.80995789492975e-305
  5. d = 1.4547677776723708e+287
  6. M = 6.284999940521954e+149

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

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

Initial Program: 23.8% 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: 54.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(c0 / 2.0), $MachinePrecision] * N[(N[(2.0 * N[(N[(c0 * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], 0.0]]

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 83.4%

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

      1. +-commutative 83.4%

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

      2. +-commutative 83.4%

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

      3. times-frac 75.8%

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

      4. fma-neg 75.8%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in c0 around inf 86.1%

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

      1. *-commutative 86.1%

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

      2. *-commutative 86.1%

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

      3. associate-*r* 80.3%

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

      4. associate-/r* 80.3%

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

      5. associate-*l/ 80.2%

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

      6. times-frac 77.5%

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

      7. unpow2 77.5%

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

      8. associate-*r/ 81.0%

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

      9. unpow2 81.0%

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

      10. associate-/l/ 81.0%

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

      11. associate-*r/ 81.0%

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

      12. associate-/r* 82.3%

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

      13. associate-*l/ 82.3%

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

      14. unpow2 82.3%

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

      15. associate-*l/ 81.0%

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

      16. *-commutative 81.0%

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

   7. Simplified 81.0%

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

      1. associate-*l/ 81.0%

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

      2. pow2 81.0%

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

      3. *-commutative 81.0%

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

      4. frac-times 81.1%

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

      5. pow2 81.1%

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

   9. Applied egg-rr 81.1%

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

       1. times-frac 79.8%

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

   11. Simplified 79.8%

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

       1. pow2 79.8%

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

   13. Applied egg-rr 79.8%

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

       1. frac-times 79.8%

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

       2. associate-*l* 87.0%

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

       3. *-commutative 87.0%

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

       4. associate-/l* 89.7%

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

   15. Applied egg-rr 89.7%

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

   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. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.1%

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

      4. fma-neg 0.1%

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

   3. Simplified 0.6%

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

   5. Taylor expanded in c0 around -inf 1.2%

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

      1. associate-*r* 1.2%

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

      2. neg-mul-1 1.2%

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

      3. distribute-lft1-in 1.2%

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

      4. metadata-eval 1.2%

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

      5. mul0-lft 37.4%

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

      6. distribute-lft-neg-in 37.4%

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

      7. distribute-rgt-neg-in 37.4%

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

      8. metadata-eval 37.4%

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

      9. mul0-lft 1.2%

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

      10. metadata-eval 1.2%

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

      11. distribute-lft1-in 1.2%

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

      12. distribute-lft-in 0.7%

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

   7. Simplified 37.4%

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

   8. Taylor expanded in c0 around 0 42.8%

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

4. Final simplification 56.3%

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

Alternative 2: 38.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 7 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.8 \cdot 10^{-40} \lor \neg \left(M \leq 8.5 \cdot 10^{-11} \lor \neg \left(M \leq 3.6 \cdot 10^{+19}\right) \land \left(M \leq 1.36 \cdot 10^{+29} \lor \neg \left(M \leq 7.5 \cdot 10^{+85}\right) \land M \leq 9.8 \cdot 10^{+104}\right)\right):\\ \;\;\;\;\frac{c0}{2} \cdot \frac{2 \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)}{w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 7e-150)
   0.0
   (if (or (<= M 1.8e-40)
           (not
            (or (<= M 8.5e-11)
                (and (not (<= M 3.6e+19))
                     (or (<= M 1.36e+29)
                         (and (not (<= M 7.5e+85)) (<= M 9.8e+104)))))))
     (* (/ c0 2.0) (/ (* 2.0 (* (/ c0 w) (/ (* (/ d D) (/ d D)) h))) w))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7e-150) {
		tmp = 0.0;
	} else if ((M <= 1.8e-40) || !((M <= 8.5e-11) || (!(M <= 3.6e+19) && ((M <= 1.36e+29) || (!(M <= 7.5e+85) && (M <= 9.8e+104)))))) {
		tmp = (c0 / 2.0) * ((2.0 * ((c0 / w) * (((d / D) * (d / D)) / h))) / w);
	} else {
		tmp = 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 (m <= 7d-150) then
        tmp = 0.0d0
    else if ((m <= 1.8d-40) .or. (.not. (m <= 8.5d-11) .or. (.not. (m <= 3.6d+19)) .and. (m <= 1.36d+29) .or. (.not. (m <= 7.5d+85)) .and. (m <= 9.8d+104))) then
        tmp = (c0 / 2.0d0) * ((2.0d0 * ((c0 / w) * (((d_1 / d) * (d_1 / d)) / h))) / w)
    else
        tmp = 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 (M <= 7e-150) {
		tmp = 0.0;
	} else if ((M <= 1.8e-40) || !((M <= 8.5e-11) || (!(M <= 3.6e+19) && ((M <= 1.36e+29) || (!(M <= 7.5e+85) && (M <= 9.8e+104)))))) {
		tmp = (c0 / 2.0) * ((2.0 * ((c0 / w) * (((d / D) * (d / D)) / h))) / w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 7e-150:
		tmp = 0.0
	elif (M <= 1.8e-40) or not ((M <= 8.5e-11) or (not (M <= 3.6e+19) and ((M <= 1.36e+29) or (not (M <= 7.5e+85) and (M <= 9.8e+104))))):
		tmp = (c0 / 2.0) * ((2.0 * ((c0 / w) * (((d / D) * (d / D)) / h))) / w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 7e-150)
		tmp = 0.0;
	elseif ((M <= 1.8e-40) || !((M <= 8.5e-11) || (!(M <= 3.6e+19) && ((M <= 1.36e+29) || (!(M <= 7.5e+85) && (M <= 9.8e+104))))))
		tmp = Float64(Float64(c0 / 2.0) * Float64(Float64(2.0 * Float64(Float64(c0 / w) * Float64(Float64(Float64(d / D) * Float64(d / D)) / h))) / w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 7e-150)
		tmp = 0.0;
	elseif ((M <= 1.8e-40) || ~(((M <= 8.5e-11) || (~((M <= 3.6e+19)) && ((M <= 1.36e+29) || (~((M <= 7.5e+85)) && (M <= 9.8e+104)))))))
		tmp = (c0 / 2.0) * ((2.0 * ((c0 / w) * (((d / D) * (d / D)) / h))) / w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 7e-150], 0.0, If[Or[LessEqual[M, 1.8e-40], N[Not[Or[LessEqual[M, 8.5e-11], And[N[Not[LessEqual[M, 3.6e+19]], $MachinePrecision], Or[LessEqual[M, 1.36e+29], And[N[Not[LessEqual[M, 7.5e+85]], $MachinePrecision], LessEqual[M, 9.8e+104]]]]]], $MachinePrecision]], N[(N[(c0 / 2.0), $MachinePrecision] * N[(N[(2.0 * N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if M < 6.9999999999999996e-150 or 1.8e-40 < M < 8.50000000000000037e-11 or 3.6e19 < M < 1.36e29 or 7.49999999999999942e85 < M < 9.7999999999999997e104

   1. Initial program 20.4%

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

      1. +-commutative 20.4%

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

      2. +-commutative 20.4%

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

      3. times-frac 18.1%

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

      4. fma-neg 18.1%

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

   3. Simplified 19.3%

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

   5. Taylor expanded in c0 around -inf 3.2%

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

      1. associate-*r* 3.2%

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

      2. neg-mul-1 3.2%

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

      3. distribute-lft1-in 3.2%

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

      4. metadata-eval 3.2%

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

      5. mul0-lft 36.7%

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

      6. distribute-lft-neg-in 36.7%

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

      7. distribute-rgt-neg-in 36.7%

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

      8. metadata-eval 36.7%

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

      9. mul0-lft 3.2%

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

      10. metadata-eval 3.2%

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

      11. distribute-lft1-in 3.2%

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

      12. distribute-lft-in 2.5%

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

   7. Simplified 36.7%

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

   8. Taylor expanded in c0 around 0 42.4%

      \[\leadsto \color{blue}{0} \]

   if 6.9999999999999996e-150 < M < 1.8e-40 or 8.50000000000000037e-11 < M < 3.6e19 or 1.36e29 < M < 7.49999999999999942e85 or 9.7999999999999997e104 < M

   1. Initial program 31.3%

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

      1. +-commutative 31.3%

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

      2. +-commutative 31.3%

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

      3. times-frac 29.5%

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

      4. fma-neg 29.5%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in c0 around inf 48.3%

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

      1. *-commutative 48.3%

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

      2. *-commutative 48.3%

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

      3. associate-*r* 47.5%

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

      4. associate-/r* 47.6%

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

      5. associate-*l/ 48.7%

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

      6. times-frac 50.1%

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

      7. unpow2 50.1%

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

      8. associate-*r/ 52.2%

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

      9. unpow2 52.2%

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

      10. associate-/l/ 59.7%

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

      11. associate-*r/ 60.8%

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

      12. associate-/r* 57.3%

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

      13. associate-*l/ 58.5%

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

      14. unpow2 58.5%

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

      15. associate-*l/ 58.4%

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

      16. *-commutative 58.4%

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

   7. Simplified 58.4%

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

      1. associate-*l/ 58.5%

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

      2. pow2 58.5%

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

      3. *-commutative 58.5%

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

      4. frac-times 61.0%

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

      5. pow2 61.0%

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

   9. Applied egg-rr 61.0%

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

       1. times-frac 62.1%

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

   11. Simplified 62.1%

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

       1. pow2 62.1%

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

   13. Applied egg-rr 62.1%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.8 \cdot 10^{-40} \lor \neg \left(M \leq 8.5 \cdot 10^{-11} \lor \neg \left(M \leq 3.6 \cdot 10^{+19}\right) \land \left(M \leq 1.36 \cdot 10^{+29} \lor \neg \left(M \leq 7.5 \cdot 10^{+85}\right) \land M \leq 9.8 \cdot 10^{+104}\right)\right):\\ \;\;\;\;\frac{c0}{2} \cdot \frac{2 \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)}{w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 33.6% accurate, 151.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.1%

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

   1. +-commutative 24.1%

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

   2. +-commutative 24.1%

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

   3. times-frac 22.0%

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

   4. fma-neg 22.0%

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

3. Simplified 22.7%

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

5. Taylor expanded in c0 around -inf 2.2%

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

   1. associate-*r* 2.2%

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

   2. neg-mul-1 2.2%

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

   3. distribute-lft1-in 2.2%

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

   4. metadata-eval 2.2%

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

   5. mul0-lft 28.3%

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

   6. distribute-lft-neg-in 28.3%

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

   7. distribute-rgt-neg-in 28.3%

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

   8. metadata-eval 28.3%

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

   9. mul0-lft 2.2%

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

   10. metadata-eval 2.2%

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

   11. distribute-lft1-in 2.2%

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

   12. distribute-lft-in 1.8%

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

7. Simplified 28.3%

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

8. Taylor expanded in c0 around 0 32.5%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 32.5%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(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))))))