Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.2% → 53.3%

Time: 33.5s

Alternatives: 4

Speedup: 30.2×

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: 24.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 53.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \left(\frac{-0.5}{h} \cdot \frac{c0 \cdot \left(\frac{t\_0}{w \cdot \sqrt[3]{-1}} - \frac{t\_0}{w}\right)}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0)) (t_1 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (* c0 (* (/ -0.5 h) (/ (* c0 (- (/ t_0 (* w (cbrt -1.0))) (/ t_0 w))) w)))
     (* c0 (/ 0.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((d / D), 2.0);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * ((-0.5 / h) * ((c0 * ((t_0 / (w * cbrt(-1.0))) - (t_0 / w))) / w));
	} else {
		tmp = c0 * (0.0 / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow((d / D), 2.0);
	double t_1 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * ((-0.5 / h) * ((c0 * ((t_0 / (w * Math.cbrt(-1.0))) - (t_0 / w))) / w));
	} else {
		tmp = c0 * (0.0 / w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ 2.0
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(Float64(-0.5 / h) * Float64(Float64(c0 * Float64(Float64(t_0 / Float64(w * cbrt(-1.0))) - Float64(t_0 / w))) / w)));
	else
		tmp = Float64(c0 * Float64(0.0 / w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(N[(-0.5 / h), $MachinePrecision] * N[(N[(c0 * N[(N[(t$95$0 / N[(w * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 73.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in c0 around inf 74.9%

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

      1. *-commutative 74.9%

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

      2. add-cbrt-cube 64.1%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \color{blue}{\sqrt[3]{\left(\left(w \cdot h\right) \cdot \left(w \cdot h\right)\right) \cdot \left(w \cdot h\right)}}}\right)}{2 \cdot w} \]

      3. pow1/3 37.7%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \color{blue}{{\left(\left(\left(w \cdot h\right) \cdot \left(w \cdot h\right)\right) \cdot \left(w \cdot h\right)\right)}^{0.3333333333333333}}}\right)}{2 \cdot w} \]

      4. pow3 37.7%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot {\color{blue}{\left({\left(w \cdot h\right)}^{3}\right)}}^{0.3333333333333333}}\right)}{2 \cdot w} \]

      5. *-commutative 37.7%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot {\left({\color{blue}{\left(h \cdot w\right)}}^{3}\right)}^{0.3333333333333333}}\right)}{2 \cdot w} \]

   7. Applied egg-rr 37.7%

      \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \color{blue}{{\left({\left(h \cdot w\right)}^{3}\right)}^{0.3333333333333333}}}\right)}{2 \cdot w} \]

   8. Taylor expanded in h around -inf 75.7%

      \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(w \cdot \sqrt[3]{-1}\right)}}{h \cdot w}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 75.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{\color{blue}{\frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(w \cdot \sqrt[3]{-1}\right)} + -1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}}{h \cdot w}\right) \]

      2. mul-1-neg 75.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{\frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(w \cdot \sqrt[3]{-1}\right)} + \color{blue}{\left(-\frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}\right)}}{h \cdot w}\right) \]

      3. unsub-neg 75.7%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{\color{blue}{\frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(w \cdot \sqrt[3]{-1}\right)} - \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}}{h \cdot w}\right) \]

      4. associate-/l* 74.2%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{\color{blue}{c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot \sqrt[3]{-1}\right)}} - \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}{h \cdot w}\right) \]

      5. associate-/l* 76.4%

         \[\leadsto c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot \sqrt[3]{-1}\right)} - \color{blue}{c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot w}}}{h \cdot w}\right) \]

   10. Simplified 76.4%

       \[\leadsto c0 \cdot \color{blue}{\left(-0.5 \cdot \frac{c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot \sqrt[3]{-1}\right)} - c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot w}}{h \cdot w}\right)} \]
   11. Step-by-step derivation

       1. pow1 76.4%

          \[\leadsto \color{blue}{{\left(c0 \cdot \left(-0.5 \cdot \frac{c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot \sqrt[3]{-1}\right)} - c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot w}}{h \cdot w}\right)\right)}^{1}} \]

   12. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{{\left(c0 \cdot \frac{-0.5 \cdot \left(c0 \cdot \left(\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot \sqrt[3]{-1}} - \frac{{\left(\frac{d}{D}\right)}^{2}}{w}\right)\right)}{h \cdot w}\right)}^{1}} \]
   13. Step-by-step derivation

       1. unpow1 80.0%

          \[\leadsto \color{blue}{c0 \cdot \frac{-0.5 \cdot \left(c0 \cdot \left(\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot \sqrt[3]{-1}} - \frac{{\left(\frac{d}{D}\right)}^{2}}{w}\right)\right)}{h \cdot w}} \]

       2. times-frac 80.1%

          \[\leadsto c0 \cdot \color{blue}{\left(\frac{-0.5}{h} \cdot \frac{c0 \cdot \left(\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot \sqrt[3]{-1}} - \frac{{\left(\frac{d}{D}\right)}^{2}}{w}\right)}{w}\right)} \]

   14. Simplified 80.1%

       \[\leadsto \color{blue}{c0 \cdot \left(\frac{-0.5}{h} \cdot \frac{c0 \cdot \left(\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot \sqrt[3]{-1}} - \frac{{\left(\frac{d}{D}\right)}^{2}}{w}\right)}{w}\right)} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Simplified 11.1%

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

      1. associate-*r/ 11.1%

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

   6. Applied egg-rr 11.6%

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

      1. associate-/l* 11.6%

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

      2. fma-define 11.6%

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

      3. unsub-neg 11.6%

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

      4. associate-*l* 10.5%

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

   8. Simplified 10.5%

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

   9. Taylor expanded in c0 around -inf 3.4%

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

   11. Simplified 49.1%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \left(\frac{-0.5}{h} \cdot \frac{c0 \cdot \left(\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot \sqrt[3]{-1}} - \frac{{\left(\frac{d}{D}\right)}^{2}}{w}\right)}{w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 53.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	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(c0 * Float64(fma(c0, Float64(d * Float64(d / Float64(D * Float64(w * Float64(h * D))))), Float64(c0 * Float64(d * Float64(d / Float64(Float64(w * h) * (D ^ 2.0)))))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(0.0 / w));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 73.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in c0 around inf 74.9%

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

      1. associate-/l* 75.9%

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

      2. associate-/r* 72.1%

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

   7. Simplified 72.1%

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

      1. associate-/l/ 75.9%

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

      2. unpow2 75.9%

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

      3. *-commutative 75.9%

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

      4. unpow2 75.9%

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

      5. associate-*r* 77.0%

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

      6. associate-*r* 77.0%

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

      7. *-commutative 77.0%

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

      8. rem-cube-cbrt 76.9%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, c0 \cdot \frac{d \cdot d}{D \cdot \left(w \cdot \color{blue}{{\left(\sqrt[3]{D \cdot h}\right)}^{3}}\right)}\right)}{2 \cdot w} \]

      9. associate-*r/ 79.4%

         \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, c0 \cdot \color{blue}{\left(d \cdot \frac{d}{D \cdot \left(w \cdot {\left(\sqrt[3]{D \cdot h}\right)}^{3}\right)}\right)}\right)}{2 \cdot w} \]

      10. *-commutative 79.4%

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, c0 \cdot \color{blue}{\left(\frac{d}{D \cdot \left(w \cdot {\left(\sqrt[3]{D \cdot h}\right)}^{3}\right)} \cdot d\right)}\right)}{2 \cdot w} \]

      11. associate-/r* 79.6%

          \[\leadsto c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(w \cdot \left(h \cdot D\right)\right)}, c0 \cdot \left(\color{blue}{\frac{\frac{d}{D}}{w \cdot {\left(\sqrt[3]{D \cdot h}\right)}^{3}}} \cdot d\right)\right)}{2 \cdot w} \]

      12. rem-cube-cbrt 79.7%

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

      13. *-commutative 79.7%

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

      14. associate-*r* 79.7%

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

      15. associate-/r* 79.5%

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

      16. associate-*r* 78.5%

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

      17. unpow2 78.5%

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

      18. *-commutative 78.5%

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

   9. Applied egg-rr 78.5%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Simplified 11.1%

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

      1. associate-*r/ 11.1%

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

   6. Applied egg-rr 11.6%

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

      1. associate-/l* 11.6%

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

      2. fma-define 11.6%

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

      3. unsub-neg 11.6%

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

      4. associate-*l* 10.5%

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

   8. Simplified 10.5%

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

   9. Taylor expanded in c0 around -inf 3.4%

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

   11. Simplified 49.1%

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

4. Final simplification 57.9%

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

Alternative 3: 53.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	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(c0 * Float64(Float64(2.0 * Float64((Float64(d / D) ^ 2.0) * Float64(c0 / Float64(w * h)))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(0.0 / w));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 73.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in c0 around inf 74.9%

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

      1. associate-/l* 75.9%

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

      2. associate-/r* 72.1%

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

   7. Simplified 72.1%

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

   9. Applied egg-rr 44.0%

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

       1. rem-exp-log 75.3%

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

   11. Applied egg-rr 77.4%

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

       1. fma-define 77.4%

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

       2. count-2 77.4%

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

       3. associate-/l/ 77.9%

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

       4. *-commutative 77.9%

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

   13. Simplified 77.9%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Simplified 11.1%

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

      1. associate-*r/ 11.1%

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

   6. Applied egg-rr 11.6%

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

      1. associate-/l* 11.6%

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

      2. fma-define 11.6%

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

      3. unsub-neg 11.6%

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

      4. associate-*l* 10.5%

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

   8. Simplified 10.5%

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

   9. Taylor expanded in c0 around -inf 3.4%

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

   11. Simplified 49.1%

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

4. Final simplification 57.7%

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

Alternative 4: 33.6% accurate, 30.2× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \frac{0}{w} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

3. Simplified 30.3%

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

   1. associate-*r/ 29.6%

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

6. Applied egg-rr 30.3%

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

   1. associate-/l* 31.1%

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

   2. fma-define 31.1%

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

   3. unsub-neg 31.1%

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

   4. associate-*l* 29.9%

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

8. Simplified 29.9%

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

9. Taylor expanded in c0 around -inf 3.4%

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

11. Simplified 35.7%

    \[\leadsto c0 \cdot \color{blue}{\frac{0}{w}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))))))