Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.0% → 53.9%

Time: 28.1s

Alternatives: 6

Speedup: 151.0×

• Could not converge on a ground truth

  1. c0 = -1.2203120602553686e-219
  2. w = -2.4415060354141265e-96
  3. h = 1.6026677064141885e-113
  4. D = 2.400299656630984e-246
  5. d = 1.3547871265691598e+254
  6. M = 2.910444620065119e+262

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 53.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c0}{w}}{h}\\ t_1 := {\left(\frac{d}{D}\right)}^{2}\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \frac{c0}{2 \cdot w}\\ t_4 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_3 \cdot \left(t\_4 + \sqrt{t\_4 \cdot t\_4 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_3 \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(t\_0, t\_1, M\right)}, \sqrt{t\_2 - M}, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (/ c0 w) h))
        (t_1 (pow (/ d D) 2.0))
        (t_2 (* t_0 t_1))
        (t_3 (/ c0 (* 2.0 w)))
        (t_4 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_3 (+ t_4 (sqrt (- (* t_4 t_4) (* M M))))) INFINITY)
     (* t_3 (fma (sqrt (fma t_0 t_1 M)) (sqrt (- t_2 M)) t_2))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / w) / h;
	double t_1 = pow((d / D), 2.0);
	double t_2 = t_0 * t_1;
	double t_3 = c0 / (2.0 * w);
	double t_4 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_3 * (t_4 + sqrt(((t_4 * t_4) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_3 * fma(sqrt(fma(t_0, t_1, M)), sqrt((t_2 - M)), t_2);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / w) / h)
	t_1 = Float64(d / D) ^ 2.0
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(c0 / Float64(2.0 * w))
	t_4 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_3 * Float64(t_4 + sqrt(Float64(Float64(t_4 * t_4) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_3 * fma(sqrt(fma(t_0, t_1, M)), sqrt(Float64(t_2 - M)), t_2));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$3 * N[(t$95$4 + N[Sqrt[N[(N[(t$95$4 * t$95$4), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$3 * N[(N[Sqrt[N[(t$95$0 * t$95$1 + M), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$2 - M), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.8%

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

   3. Simplified 67.1%

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

   6. Applied egg-rr 66.9%

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

      1. associate-/r* 66.9%

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

      2. associate-*l/ 66.9%

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

      3. associate-/r* 66.9%

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

      4. associate-*l/ 69.4%

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

      5. associate-/r* 71.9%

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

   8. Simplified 71.9%

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

   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 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

   6. Applied egg-rr 18.4%

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

      1. associate-/r* 17.8%

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

      2. associate-*l/ 17.8%

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

      3. associate-/r* 17.8%

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

      4. associate-*l/ 18.0%

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

      5. associate-/r* 20.6%

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

   8. Simplified 20.6%

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

      1. div-inv 20.6%

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

      2. *-commutative 20.6%

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

   10. Applied egg-rr 20.6%

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

   11. Taylor expanded in c0 around -inf 0.1%

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

       1. associate-*r/ 0.1%

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

   13. Simplified 47.0%

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

Alternative 2: 53.9% accurate, 0.4× speedup

Math

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

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 (* (/ (/ c0 w) h) (/ (pow d 2.0) (pow D 2.0)))) (* 2.0 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 * (((c0 / w) / h) * (pow(d, 2.0) / pow(D, 2.0)))) / (2.0 * 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 * (((c0 / w) / h) * (Math.pow(d, 2.0) / Math.pow(D, 2.0)))) / (2.0 * 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 * (((c0 / w) / h) * (math.pow(d, 2.0) / math.pow(D, 2.0)))) / (2.0 * 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(c0 * Float64(Float64(2.0 * Float64(Float64(Float64(c0 / w) / h) * Float64((d ^ 2.0) / (D ^ 2.0)))) / Float64(2.0 * 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 * (((c0 / w) / h) * ((d ^ 2.0) / (D ^ 2.0)))) / (2.0 * 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[(c0 * N[(N[(2.0 * N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * N[(N[Power[d, 2.0], $MachinePrecision] / N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.8%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in c0 around inf 69.9%

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

      1. *-commutative 69.9%

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

      2. associate-/l/ 68.5%

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

      3. associate-*l/ 71.0%

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

      4. associate-/l* 70.0%

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

      5. associate-/r* 71.3%

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

   7. Simplified 71.3%

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

   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 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

   6. Applied egg-rr 18.4%

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

      1. associate-/r* 17.8%

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

      2. associate-*l/ 17.8%

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

      3. associate-/r* 17.8%

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

      4. associate-*l/ 18.0%

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

      5. associate-/r* 20.6%

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

   8. Simplified 20.6%

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

      1. div-inv 20.6%

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

      2. *-commutative 20.6%

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

   10. Applied egg-rr 20.6%

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

   11. Taylor expanded in c0 around -inf 0.1%

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

       1. associate-*r/ 0.1%

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

   13. Simplified 47.0%

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

Alternative 3: 55.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.8%

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

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

   1. Initial program 0.0%

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

   3. Simplified 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

   6. Applied egg-rr 18.4%

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

      1. associate-/r* 17.8%

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

      2. associate-*l/ 17.8%

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

      3. associate-/r* 17.8%

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

      4. associate-*l/ 18.0%

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

      5. associate-/r* 20.6%

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

   8. Simplified 20.6%

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

      1. div-inv 20.6%

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

      2. *-commutative 20.6%

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

   10. Applied egg-rr 20.6%

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

   11. Taylor expanded in c0 around -inf 0.1%

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

       1. associate-*r/ 0.1%

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

   13. Simplified 47.0%

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

Alternative 4: 33.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\\ \mathbf{if}\;w \leq -1.9 \cdot 10^{-131}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq -2.6 \cdot 10^{-228}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (/ (* d d) (* D D)))))
   (if (<= w -1.9e-131)
     0.0
     (if (<= w -2.6e-228)
       (*
        (/ c0 (* 2.0 w))
        (+
         (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))
         (sqrt (- (* t_0 t_0) (* M M)))))
       0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double tmp;
	if (w <= -1.9e-131) {
		tmp = 0.0;
	} else if (w <= -2.6e-228) {
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * ((d / D) * (d / D))) + sqrt(((t_0 * t_0) - (M * M))));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (c0 / (w * h)) * ((d_1 * d_1) / (d * d))
    if (w <= (-1.9d-131)) then
        tmp = 0.0d0
    else if (w <= (-2.6d-228)) then
        tmp = (c0 / (2.0d0 * w)) * ((((c0 / w) / h) * ((d_1 / d) * (d_1 / d))) + sqrt(((t_0 * t_0) - (m * m))))
    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 t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double tmp;
	if (w <= -1.9e-131) {
		tmp = 0.0;
	} else if (w <= -2.6e-228) {
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * ((d / D) * (d / D))) + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D))
	tmp = 0
	if w <= -1.9e-131:
		tmp = 0.0
	elif w <= -2.6e-228:
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * ((d / D) * (d / D))) + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D)))
	tmp = 0.0
	if (w <= -1.9e-131)
		tmp = 0.0;
	elseif (w <= -2.6e-228)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(Float64(c0 / w) / h) * Float64(Float64(d / D) * Float64(d / D))) + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	tmp = 0.0;
	if (w <= -1.9e-131)
		tmp = 0.0;
	elseif (w <= -2.6e-228)
		tmp = (c0 / (2.0 * w)) * ((((c0 / w) / h) * ((d / D) * (d / D))) + sqrt(((t_0 * t_0) - (M * M))));
	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[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.9e-131], 0.0, If[LessEqual[w, -2.6e-228], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 2 regimes

2. if w < -1.89999999999999997e-131 or -2.6e-228 < w

   1. Initial program 17.0%

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

   3. Simplified 16.6%

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

   6. Applied egg-rr 29.6%

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

      1. associate-/r* 29.1%

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

      2. associate-*l/ 29.2%

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

      3. associate-/r* 29.2%

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

      4. associate-*l/ 30.1%

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

      5. associate-/r* 32.5%

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

   8. Simplified 32.5%

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

      1. div-inv 32.5%

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

      2. *-commutative 32.5%

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

   10. Applied egg-rr 32.5%

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

   11. Taylor expanded in c0 around -inf 2.4%

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

       1. associate-*r/ 2.4%

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

   13. Simplified 39.0%

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

   if -1.89999999999999997e-131 < w < -2.6e-228

   1. Initial program 52.5%

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

   3. Simplified 52.2%

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

      1. times-frac 52.2%

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

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in c0 around 0 52.2%

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

      1. *-commutative 52.2%

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

      2. associate-/r* 56.2%

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

   9. Simplified 56.2%

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

Alternative 5: 33.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := t\_0 \cdot \frac{d \cdot d}{D \cdot D}\\ \mathbf{if}\;w \leq -2.9 \cdot 10^{-131}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq -2.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{t\_1 \cdot t\_1 - M \cdot M} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (* t_0 (/ (* d d) (* D D)))))
   (if (<= w -2.9e-131)
     0.0
     (if (<= w -2.5e-228)
       (*
        (/ c0 (* 2.0 w))
        (+ (sqrt (- (* t_1 t_1) (* M M))) (* (* (/ d D) (/ d D)) t_0)))
       0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double tmp;
	if (w <= -2.9e-131) {
		tmp = 0.0;
	} else if (w <= -2.5e-228) {
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (((d / D) * (d / D)) * t_0));
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = t_0 * ((d_1 * d_1) / (d * d))
    if (w <= (-2.9d-131)) then
        tmp = 0.0d0
    else if (w <= (-2.5d-228)) then
        tmp = (c0 / (2.0d0 * w)) * (sqrt(((t_1 * t_1) - (m * m))) + (((d_1 / d) * (d_1 / d)) * t_0))
    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 t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double tmp;
	if (w <= -2.9e-131) {
		tmp = 0.0;
	} else if (w <= -2.5e-228) {
		tmp = (c0 / (2.0 * w)) * (Math.sqrt(((t_1 * t_1) - (M * M))) + (((d / D) * (d / D)) * t_0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = t_0 * ((d * d) / (D * D))
	tmp = 0
	if w <= -2.9e-131:
		tmp = 0.0
	elif w <= -2.5e-228:
		tmp = (c0 / (2.0 * w)) * (math.sqrt(((t_1 * t_1) - (M * M))) + (((d / D) * (d / D)) * t_0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(t_0 * Float64(Float64(d * d) / Float64(D * D)))
	tmp = 0.0
	if (w <= -2.9e-131)
		tmp = 0.0;
	elseif (w <= -2.5e-228)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))) + Float64(Float64(Float64(d / D) * Float64(d / D)) * t_0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = t_0 * ((d * d) / (D * D));
	tmp = 0.0;
	if (w <= -2.9e-131)
		tmp = 0.0;
	elseif (w <= -2.5e-228)
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (((d / D) * (d / D)) * t_0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -2.9e-131], 0.0, If[LessEqual[w, -2.5e-228], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 2 regimes

2. if w < -2.9000000000000002e-131 or -2.49999999999999986e-228 < w

   1. Initial program 17.0%

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

   3. Simplified 16.6%

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

   6. Applied egg-rr 29.6%

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

      1. associate-/r* 29.1%

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

      2. associate-*l/ 29.2%

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

      3. associate-/r* 29.2%

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

      4. associate-*l/ 30.1%

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

      5. associate-/r* 32.5%

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

   8. Simplified 32.5%

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

      1. div-inv 32.5%

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

      2. *-commutative 32.5%

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

   10. Applied egg-rr 32.5%

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

   11. Taylor expanded in c0 around -inf 2.4%

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

       1. associate-*r/ 2.4%

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

   13. Simplified 39.0%

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

   if -2.9000000000000002e-131 < w < -2.49999999999999986e-228

   1. Initial program 52.5%

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

   3. Simplified 52.2%

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

      1. times-frac 52.2%

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

   6. Applied egg-rr 52.2%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.9 \cdot 10^{-131}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq -2.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) - M \cdot M} + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.5% 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 20.4%

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

3. Simplified 20.1%

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

6. Applied egg-rr 32.6%

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

   1. associate-/r* 32.2%

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

   2. associate-*l/ 32.2%

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

   3. associate-/r* 32.2%

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

   4. associate-*l/ 33.0%

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

   5. associate-/r* 35.6%

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

8. Simplified 35.6%

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

   1. div-inv 35.6%

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

   2. *-commutative 35.6%

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

10. Applied egg-rr 35.6%

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

11. Taylor expanded in c0 around -inf 2.1%

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

    1. associate-*r/ 2.1%

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

13. Simplified 36.1%

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

Reproduce

herbie shell --seed 2024191 
(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))))))