Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.8% → 55.9%

Time: 21.9s

Alternatives: 6

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 7.716698143871379e+247
  2. w = -2.3858013024334675e-268
  3. h = 4.403106321412911e-35
  4. D = -4.0439280330998776e-258
  5. d = 2.712846893445323e+286
  6. M = -9.336415912711112e-200

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: 24.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \left(w \cdot h\right) \cdot \left(D \cdot D\right)\\ t_2 := \left(d \cdot d\right) \cdot \frac{c0}{t_1}\\ t_3 := {t_2}^{2}\\ t_4 := \frac{c0 \cdot \left(d \cdot d\right)}{t_1}\\ t_5 := t_0 \cdot \left(t_4 + \sqrt{t_4 \cdot t_4 - M \cdot M}\right)\\ t_6 := {\left(\frac{d}{D}\right)}^{2}\\ \mathbf{if}\;t_5 \leq -2 \cdot 10^{-251}:\\ \;\;\;\;t_0 \cdot \frac{c0 \cdot \frac{t_6}{w} - t_6 \cdot \frac{c0}{w \cdot \sqrt[3]{-1}}}{h}\\ \mathbf{elif}\;t_5 \leq 0:\\ \;\;\;\;t_0 \cdot \frac{M \cdot M + \left(t_3 - t_3\right)}{t_2 - \sqrt{t_3 - M \cdot M}}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(0, c0, \frac{0.5}{c0} \cdot \frac{w \cdot \left(M \cdot \left(h \cdot M\right)\right)}{t_6}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* (* w h) (* D D)))
        (t_2 (* (* d d) (/ c0 t_1)))
        (t_3 (pow t_2 2.0))
        (t_4 (/ (* c0 (* d d)) t_1))
        (t_5 (* t_0 (+ t_4 (sqrt (- (* t_4 t_4) (* M M))))))
        (t_6 (pow (/ d D) 2.0)))
   (if (<= t_5 -2e-251)
     (* t_0 (/ (- (* c0 (/ t_6 w)) (* t_6 (/ c0 (* w (cbrt -1.0))))) h))
     (if (<= t_5 0.0)
       (* t_0 (/ (+ (* M M) (- t_3 t_3)) (- t_2 (sqrt (- t_3 (* M M))))))
       (if (<= t_5 INFINITY)
         (* (/ c0 w) (* (* d d) (/ c0 (* D (* (* w h) D)))))
         (* t_0 (fma 0.0 c0 (* (/ 0.5 c0) (/ (* w (* M (* h M))) t_6)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (w * h) * (D * D);
	double t_2 = (d * d) * (c0 / t_1);
	double t_3 = pow(t_2, 2.0);
	double t_4 = (c0 * (d * d)) / t_1;
	double t_5 = t_0 * (t_4 + sqrt(((t_4 * t_4) - (M * M))));
	double t_6 = pow((d / D), 2.0);
	double tmp;
	if (t_5 <= -2e-251) {
		tmp = t_0 * (((c0 * (t_6 / w)) - (t_6 * (c0 / (w * cbrt(-1.0))))) / h);
	} else if (t_5 <= 0.0) {
		tmp = t_0 * (((M * M) + (t_3 - t_3)) / (t_2 - sqrt((t_3 - (M * M)))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (c0 / w) * ((d * d) * (c0 / (D * ((w * h) * D))));
	} else {
		tmp = t_0 * fma(0.0, c0, ((0.5 / c0) * ((w * (M * (h * M))) / t_6)));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(w * h) * Float64(D * D))
	t_2 = Float64(Float64(d * d) * Float64(c0 / t_1))
	t_3 = t_2 ^ 2.0
	t_4 = Float64(Float64(c0 * Float64(d * d)) / t_1)
	t_5 = Float64(t_0 * Float64(t_4 + sqrt(Float64(Float64(t_4 * t_4) - Float64(M * M)))))
	t_6 = Float64(d / D) ^ 2.0
	tmp = 0.0
	if (t_5 <= -2e-251)
		tmp = Float64(t_0 * Float64(Float64(Float64(c0 * Float64(t_6 / w)) - Float64(t_6 * Float64(c0 / Float64(w * cbrt(-1.0))))) / h));
	elseif (t_5 <= 0.0)
		tmp = Float64(t_0 * Float64(Float64(Float64(M * M) + Float64(t_3 - t_3)) / Float64(t_2 - sqrt(Float64(t_3 - Float64(M * M))))));
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(d * d) * Float64(c0 / Float64(D * Float64(Float64(w * h) * D)))));
	else
		tmp = Float64(t_0 * fma(0.0, c0, Float64(Float64(0.5 / c0) * Float64(Float64(w * Float64(M * Float64(h * M))) / t_6))));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(d * d), $MachinePrecision] * N[(c0 / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * N[(t$95$4 + N[Sqrt[N[(N[(t$95$4 * t$95$4), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$5, -2e-251], N[(t$95$0 * N[(N[(N[(c0 * N[(t$95$6 / w), $MachinePrecision]), $MachinePrecision] - N[(t$95$6 * N[(c0 / N[(w * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(t$95$0 * N[(N[(N[(M * M), $MachinePrecision] + N[(t$95$3 - t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - N[Sqrt[N[(t$95$3 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(c0 / w), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] * N[(c0 / N[(D * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.0 * c0 + N[(N[(0.5 / c0), $MachinePrecision] * N[(N[(w * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -2.00000000000000003e-251

   1. Initial program 74.2%

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

      1. times-frac 70.1%

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

      2. fma-def 70.1%

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

      3. associate-/r* 70.1%

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

      4. difference-of-squares 70.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in c0 around inf 72.3%

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

      1. times-frac 72.3%

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

      2. associate-*l/ 72.3%

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

      3. associate-*r/ 72.3%

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

      4. unpow2 72.3%

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

      5. associate-/l/ 72.3%

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

      6. unpow2 72.3%

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

   6. Simplified 72.3%

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

      1. add-cbrt-cube 66.1%

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

      2. *-commutative 66.1%

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

      3. *-commutative 66.1%

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

      4. *-commutative 66.1%

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

   8. Applied egg-rr 66.1%

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

      1. associate-*l* 66.1%

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

      2. cube-unmult 66.1%

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

      3. associate-*r* 66.1%

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

      4. *-commutative 66.1%

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

      5. associate-*l* 66.1%

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

      6. *-commutative 66.1%

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

   10. Simplified 66.1%

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

   11. Taylor expanded in h around -inf 78.3%

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

       1. associate-*r/ 78.3%

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

   13. Simplified 80.5%

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

   if -2.00000000000000003e-251 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -0.0

   1. Initial program 52.3%

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

      1. times-frac 37.3%

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

      2. fma-def 26.4%

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

      3. associate-/r* 26.4%

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

      4. difference-of-squares 26.4%

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

   3. Simplified 26.1%

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

      1. fma-udef 52.4%

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

      2. associate-/l/ 52.4%

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

      3. times-frac 36.5%

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

      4. fma-udef 36.5%

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

      5. associate-/l/ 36.5%

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

      6. times-frac 41.1%

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

      7. associate-/l/ 41.1%

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

      8. times-frac 52.3%

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

   5. Applied egg-rr 51.0%

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

   7. Simplified 77.6%

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

   if -0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 72.0%

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

      1. times-frac 69.0%

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

      2. fma-def 69.0%

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

      3. associate-/r* 69.0%

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

      4. difference-of-squares 69.0%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in c0 around inf 74.2%

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

      1. times-frac 74.4%

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

      2. unpow2 74.4%

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

      3. unpow2 74.4%

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

      4. *-commutative 74.4%

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

      5. times-frac 77.1%

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

      6. unpow2 77.1%

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

   6. Simplified 77.1%

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

      1. expm1-log1p-u 76.4%

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

      2. expm1-udef 71.6%

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

      3. *-commutative 71.6%

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

      4. associate-*r* 71.6%

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

   8. Applied egg-rr 71.6%

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

      1. expm1-def 76.4%

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

      2. expm1-log1p 77.1%

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

      3. associate-*l/ 75.1%

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

      4. times-frac 77.2%

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

      5. associate-*l* 77.2%

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

      6. *-commutative 77.2%

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

      7. unpow2 77.2%

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

      8. times-frac 74.5%

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

      9. unpow2 74.5%

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

      10. times-frac 74.3%

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

      11. unpow2 74.3%

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

      12. *-commutative 74.3%

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

      13. associate-/l* 74.3%

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

   10. Simplified 74.6%

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

   11. Taylor expanded in d around 0 74.3%

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

       1. unpow2 74.3%

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

       2. associate-*r/ 76.9%

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

       3. unpow2 76.9%

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

       4. associate-*r* 79.6%

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

   13. Simplified 79.6%

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

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

   1. Initial program 0.0%

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

      1. associate-*l/ 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*l* 0.0%

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

      5. *-commutative 0.0%

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

      6. associate-*r* 0.0%

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

      7. associate-*l* 0.1%

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

      8. *-commutative 0.1%

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

   3. Simplified 4.0%

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

      1. fma-udef 4.0%

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

      2. associate-*l* 3.3%

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

      3. associate-/l/ 2.6%

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

      4. associate-/l/ 2.6%

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

      5. *-commutative 2.6%

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

   5. Applied egg-rr 2.6%

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

      1. fma-def 2.7%

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

      2. associate-/r* 2.7%

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

      3. *-commutative 2.7%

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

      4. fma-def 2.7%

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

   7. Simplified 4.1%

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

   8. Taylor expanded in c0 around -inf 2.0%

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

      1. +-commutative 2.0%

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

      2. associate-*r* 2.0%

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

      3. *-commutative 2.0%

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

      4. fma-def 2.0%

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

   10. Simplified 51.1%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0, c0, \frac{0.5}{c0} \cdot \frac{w \cdot \left(\left(h \cdot M\right) \cdot M\right)}{{\left(\frac{d}{D}\right)}^{2}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -2 \cdot 10^{-251}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{c0 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w} - {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot \sqrt[3]{-1}}}{h}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{M \cdot M + \left({\left(\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)}^{2} - {\left(\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)}^{2}\right)}{\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{{\left(\left(d \cdot d\right) \cdot \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)}^{2} - M \cdot M}}\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\left(d \cdot d\right) \cdot \frac{c0}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0, c0, \frac{0.5}{c0} \cdot \frac{w \cdot \left(M \cdot \left(h \cdot M\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}\right)\\ \end{array} \]

Alternative 2: 55.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t_1 \cdot \left(t_2 + \sqrt{t_2 \cdot t_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t_1 \cdot \frac{c0 \cdot \frac{t_0}{w} - t_0 \cdot \frac{c0}{w \cdot \sqrt[3]{-1}}}{h}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \mathsf{fma}\left(0, c0, \frac{0.5}{c0} \cdot \frac{w \cdot \left(M \cdot \left(h \cdot M\right)\right)}{t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (/ d D) 2.0))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (/ (- (* c0 (/ t_0 w)) (* t_0 (/ c0 (* w (cbrt -1.0))))) h))
     (* t_1 (fma 0.0 c0 (* (/ 0.5 c0) (/ (* w (* M (* h M))) t_0)))))))

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 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * (((c0 * (t_0 / w)) - (t_0 * (c0 / (w * cbrt(-1.0))))) / h);
	} else {
		tmp = t_1 * fma(0.0, c0, ((0.5 / c0) * ((w * (M * (h * M))) / t_0)));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / D) ^ 2.0
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(Float64(c0 * Float64(t_0 / w)) - Float64(t_0 * Float64(c0 / Float64(w * cbrt(-1.0))))) / h));
	else
		tmp = Float64(t_1 * fma(0.0, c0, Float64(Float64(0.5 / c0) * Float64(Float64(w * Float64(M * Float64(h * M))) / t_0))));
	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[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(N[(c0 * N[(t$95$0 / w), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(c0 / N[(w * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(0.0 * c0 + N[(N[(0.5 / c0), $MachinePrecision] * N[(N[(w * N[(M * N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 71.2%

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

      1. times-frac 66.5%

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

      2. fma-def 65.4%

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

      3. associate-/r* 65.4%

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

      4. difference-of-squares 65.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in c0 around inf 66.6%

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

      1. times-frac 68.6%

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

      2. associate-*l/ 67.6%

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

      3. associate-*r/ 68.6%

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

      4. unpow2 68.6%

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

      5. associate-/l/ 67.6%

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

      6. unpow2 67.6%

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

   6. Simplified 67.6%

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

      1. add-cbrt-cube 63.5%

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

      2. *-commutative 63.5%

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

      3. *-commutative 63.5%

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

      4. *-commutative 63.5%

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

   8. Applied egg-rr 63.5%

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

      1. associate-*l* 63.5%

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

      2. cube-unmult 63.5%

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

      3. associate-*r* 63.5%

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

      4. *-commutative 63.5%

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

      5. associate-*l* 63.5%

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

      6. *-commutative 63.5%

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

   10. Simplified 63.5%

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

   11. Taylor expanded in h around -inf 73.1%

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

       1. associate-*r/ 73.1%

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

   13. Simplified 74.0%

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

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

   1. Initial program 0.0%

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

      1. associate-*l/ 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*l* 0.0%

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

      5. *-commutative 0.0%

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

      6. associate-*r* 0.0%

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

      7. associate-*l* 0.1%

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

      8. *-commutative 0.1%

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

   3. Simplified 4.0%

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

      1. fma-udef 4.0%

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

      2. associate-*l* 3.3%

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

      3. associate-/l/ 2.6%

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

      4. associate-/l/ 2.6%

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

      5. *-commutative 2.6%

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

   5. Applied egg-rr 2.6%

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

      1. fma-def 2.7%

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

      2. associate-/r* 2.7%

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

      3. *-commutative 2.7%

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

      4. fma-def 2.7%

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

   7. Simplified 4.1%

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

   8. Taylor expanded in c0 around -inf 2.0%

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

      1. +-commutative 2.0%

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

      2. associate-*r* 2.0%

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

      3. *-commutative 2.0%

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

      4. fma-def 2.0%

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

   10. Simplified 51.1%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0, c0, \frac{0.5}{c0} \cdot \frac{w \cdot \left(\left(h \cdot M\right) \cdot M\right)}{{\left(\frac{d}{D}\right)}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

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

Alternative 3: 55.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 71.2%

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

      1. times-frac 66.5%

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

      2. fma-def 65.4%

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

      3. associate-/r* 65.4%

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

      4. difference-of-squares 65.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in c0 around inf 70.7%

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

      1. times-frac 68.6%

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

      2. unpow2 68.6%

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

      3. unpow2 68.6%

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

      4. *-commutative 68.6%

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

      5. times-frac 72.8%

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

      6. unpow2 72.8%

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

   6. Simplified 72.8%

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

      1. expm1-log1p-u 34.1%

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

      2. expm1-udef 31.6%

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

      3. *-commutative 31.6%

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

      4. associate-*r* 31.6%

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

   8. Applied egg-rr 31.6%

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

      1. expm1-def 34.1%

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

      2. expm1-log1p 72.8%

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

      3. associate-*l/ 71.0%

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

      4. times-frac 72.9%

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

      5. associate-*l* 72.9%

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

      6. *-commutative 72.9%

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

      7. unpow2 72.9%

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

      8. times-frac 68.7%

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

      9. unpow2 68.7%

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

      10. times-frac 70.7%

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

      11. unpow2 70.7%

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

      12. *-commutative 70.7%

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

      13. associate-/l* 70.7%

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

   10. Simplified 70.9%

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

   11. Taylor expanded in d around 0 70.7%

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

       1. unpow2 70.7%

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

       2. associate-*r/ 71.6%

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

       3. unpow2 71.6%

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

       4. associate-*r* 73.7%

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

   13. Simplified 73.7%

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

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

   1. Initial program 0.0%

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

      1. associate-*l/ 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*l* 0.0%

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

      5. *-commutative 0.0%

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

      6. associate-*r* 0.0%

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

      7. associate-*l* 0.1%

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

      8. *-commutative 0.1%

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

   3. Simplified 4.0%

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

      1. fma-udef 4.0%

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

      2. associate-*l* 3.3%

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

      3. associate-/l/ 2.6%

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

      4. associate-/l/ 2.6%

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

      5. *-commutative 2.6%

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

   5. Applied egg-rr 2.6%

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

      1. fma-def 2.7%

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

      2. associate-/r* 2.7%

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

      3. *-commutative 2.7%

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

      4. fma-def 2.7%

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

   7. Simplified 4.1%

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

   8. Taylor expanded in c0 around -inf 2.0%

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

      1. +-commutative 2.0%

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

      2. associate-*r* 2.0%

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

      3. *-commutative 2.0%

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

      4. fma-def 2.0%

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

   10. Simplified 51.1%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0, c0, \frac{0.5}{c0} \cdot \frac{w \cdot \left(\left(h \cdot M\right) \cdot M\right)}{{\left(\frac{d}{D}\right)}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

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

Alternative 4: 56.1% accurate, 0.9× speedup

Math

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

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 w) (* (* d d) (/ c0 (* D (* (* w h) D)))))
     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 / w) * ((d * d) * (c0 / (D * ((w * h) * D))));
	} 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 / w) * ((d * d) * (c0 / (D * ((w * h) * D))));
	} 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 / w) * ((d * d) * (c0 / (D * ((w * h) * D))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(d * d) * Float64(c0 / Float64(D * Float64(Float64(w * h) * D)))));
	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 / w) * ((d * d) * (c0 / (D * ((w * h) * D))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 71.2%

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

      1. times-frac 66.5%

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

      2. fma-def 65.4%

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

      3. associate-/r* 65.4%

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

      4. difference-of-squares 65.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in c0 around inf 70.7%

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

      1. times-frac 68.6%

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

      2. unpow2 68.6%

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

      3. unpow2 68.6%

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

      4. *-commutative 68.6%

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

      5. times-frac 72.8%

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

      6. unpow2 72.8%

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

   6. Simplified 72.8%

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

      1. expm1-log1p-u 34.1%

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

      2. expm1-udef 31.6%

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

      3. *-commutative 31.6%

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

      4. associate-*r* 31.6%

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

   8. Applied egg-rr 31.6%

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

      1. expm1-def 34.1%

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

      2. expm1-log1p 72.8%

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

      3. associate-*l/ 71.0%

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

      4. times-frac 72.9%

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

      5. associate-*l* 72.9%

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

      6. *-commutative 72.9%

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

      7. unpow2 72.9%

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

      8. times-frac 68.7%

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

      9. unpow2 68.7%

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

      10. times-frac 70.7%

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

      11. unpow2 70.7%

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

      12. *-commutative 70.7%

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

      13. associate-/l* 70.7%

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

   10. Simplified 70.9%

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

   11. Taylor expanded in d around 0 70.7%

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

       1. unpow2 70.7%

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

       2. associate-*r/ 71.6%

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

       3. unpow2 71.6%

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

       4. associate-*r* 73.7%

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

   13. Simplified 73.7%

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

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

   1. Initial program 0.0%

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

      1. times-frac 0.0%

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

      2. fma-def 0.0%

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

      3. associate-/r* 0.1%

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

      4. difference-of-squares 6.8%

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

   3. Simplified 15.8%

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

   4. Taylor expanded in c0 around -inf 1.9%

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

      1. associate-*r* 1.9%

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

      2. distribute-rgt1-in 1.9%

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

      3. metadata-eval 1.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 45.7%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 45.7%

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

      6. mul0-lft 2.5%

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

      7. metadata-eval 2.5%

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

      8. distribute-lft1-in 2.5%

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

      9. *-commutative 2.5%

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

      10. distribute-lft1-in 2.5%

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

      11. metadata-eval 2.5%

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

      12. mul0-lft 45.7%

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

   6. Simplified 45.7%

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

   7. Taylor expanded in c0 around 0 49.6%

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

4. Final simplification 58.2%

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

Alternative 5: 36.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -6 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 1.22 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w -6e-150)
   0.0
   (if (<= w 1.22e-18)
     (/ (* (* d d) (* c0 c0)) (* (* D D) (* h (* w w))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -6e-150) {
		tmp = 0.0;
	} else if (w <= 1.22e-18) {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (w <= (-6d-150)) then
        tmp = 0.0d0
    else if (w <= 1.22d-18) then
        tmp = ((d_1 * d_1) * (c0 * c0)) / ((d * d) * (h * (w * w)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -6e-150) {
		tmp = 0.0;
	} else if (w <= 1.22e-18) {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= -6e-150:
		tmp = 0.0
	elif w <= 1.22e-18:
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= -6e-150)
		tmp = 0.0;
	elseif (w <= 1.22e-18)
		tmp = Float64(Float64(Float64(d * d) * Float64(c0 * c0)) / Float64(Float64(D * D) * Float64(h * Float64(w * w))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= -6e-150)
		tmp = 0.0;
	elseif (w <= 1.22e-18)
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, -6e-150], 0.0, If[LessEqual[w, 1.22e-18], N[(N[(N[(d * d), $MachinePrecision] * N[(c0 * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if w < -6.0000000000000003e-150 or 1.2200000000000001e-18 < w

   1. Initial program 18.0%

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

      1. times-frac 16.5%

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

      2. fma-def 15.9%

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

      3. associate-/r* 15.9%

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

      4. difference-of-squares 19.1%

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

   3. Simplified 25.5%

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

   4. Taylor expanded in c0 around -inf 4.5%

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

      1. associate-*r* 4.5%

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

      2. distribute-rgt1-in 4.5%

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

      3. metadata-eval 4.5%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

      4. mul0-lft 42.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

      5. metadata-eval 42.9%

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

      6. mul0-lft 5.1%

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

      7. metadata-eval 5.1%

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

      8. distribute-lft1-in 5.1%

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

      9. *-commutative 5.1%

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

      10. distribute-lft1-in 5.1%

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

      11. metadata-eval 5.1%

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

      12. mul0-lft 42.9%

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

   6. Simplified 42.9%

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

   7. Taylor expanded in c0 around 0 44.2%

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

   if -6.0000000000000003e-150 < w < 1.2200000000000001e-18

   1. Initial program 37.7%

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

      1. times-frac 35.6%

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

      2. fma-def 35.6%

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

      3. associate-/r* 35.6%

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

      4. difference-of-squares 41.8%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in c0 around inf 45.8%

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

      1. times-frac 47.3%

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

      2. unpow2 47.3%

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

      3. unpow2 47.3%

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

      4. *-commutative 47.3%

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

      5. times-frac 59.0%

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

      6. unpow2 59.0%

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

   6. Simplified 59.0%

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

      1. expm1-log1p-u 27.7%

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

      2. expm1-udef 26.7%

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

      3. *-commutative 26.7%

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

      4. associate-*r* 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. expm1-def 27.7%

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

      2. expm1-log1p 59.0%

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

      3. associate-*l/ 58.0%

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

      4. times-frac 59.0%

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

      5. associate-*l* 59.0%

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

      6. *-commutative 59.0%

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

      7. unpow2 59.0%

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

      8. times-frac 47.3%

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

      9. unpow2 47.3%

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

      10. times-frac 45.8%

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

      11. unpow2 45.8%

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

      12. *-commutative 45.8%

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

      13. associate-/l* 45.8%

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

   10. Simplified 58.1%

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

   11. Taylor expanded in c0 around 0 38.1%

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

       1. unpow2 38.1%

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

       2. *-commutative 38.1%

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

       3. unpow2 38.1%

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

       4. unpow2 38.1%

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

       5. *-commutative 38.1%

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

       6. unpow2 38.1%

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

   13. Simplified 38.1%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -6 \cdot 10^{-150}:\\ \;\;\;\;0\\ \mathbf{elif}\;w \leq 1.22 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 34.3% 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 25.6%

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

   1. times-frac 23.9%

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

   2. fma-def 23.5%

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

   3. associate-/r* 23.5%

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

   4. difference-of-squares 27.9%

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

3. Simplified 34.7%

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

4. Taylor expanded in c0 around -inf 2.8%

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

   1. associate-*r* 2.8%

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

   2. distribute-rgt1-in 2.8%

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

   3. metadata-eval 2.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)\right) \cdot c0\right) \]

   4. mul0-lft 32.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot \color{blue}{0}\right) \cdot c0\right) \]

   5. metadata-eval 32.0%

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

   6. mul0-lft 3.6%

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

   7. metadata-eval 3.6%

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

   8. distribute-lft1-in 3.6%

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

   9. *-commutative 3.6%

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

   10. distribute-lft1-in 3.6%

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

   11. metadata-eval 3.6%

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

   12. mul0-lft 32.0%

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

6. Simplified 32.0%

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

7. Taylor expanded in c0 around 0 35.0%

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

8. Final simplification 35.0%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023182 
(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))))))