Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.3% → 54.6%

Time: 33.2s

Alternatives: 9

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 3.963337777373968e+242
  2. w = 3.3637458133512446e-304
  3. h = -5.912281085489983e-199
  4. D = -9.14124892420494e-307
  5. d = -7.94087797825461e+273
  6. M = 2.037844172402592e-38

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: 24.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 54.6% accurate, 0.6× speedup

Math

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

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) (pow (/ d D) 2.0)) (* h (/ w c0)))
     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) * pow((d / D), 2.0)) / (h * (w / c0));
	} 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) * Math.pow((d / D), 2.0)) / (h * (w / c0));
	} 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) * math.pow((d / D), 2.0)) / (h * (w / c0))
	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(Float64(c0 / w) * (Float64(d / D) ^ 2.0)) / Float64(h * Float64(w / c0)));
	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) ^ 2.0)) / (h * (w / c0));
	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[(N[(c0 / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(h * N[(w / c0), $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 78.0%

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

      1. +-commutative 78.0%

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

      2. +-commutative 78.0%

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

      3. times-frac 76.7%

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

      4. fma-neg 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in c0 around inf 83.9%

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

      1. expm1-log1p-u 39.3%

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

      2. expm1-udef 36.8%

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

   7. Applied egg-rr 36.8%

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

      1. expm1-def 39.3%

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

      2. expm1-log1p 81.3%

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

      3. associate-*l/ 78.8%

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

      4. associate-/l* 82.5%

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

      5. *-commutative 82.5%

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

      6. times-frac 82.5%

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

      7. metadata-eval 82.5%

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

      8. associate-*l/ 81.3%

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

      9. unpow2 81.3%

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

      10. unpow2 81.3%

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

      11. times-frac 86.7%

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

      12. unpow2 86.7%

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

      13. times-frac 86.1%

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

   9. Simplified 86.1%

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

       1. expm1-log1p-u 41.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 36.8%

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

       3. *-un-lft-identity 36.8%

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

   11. Applied egg-rr 36.8%

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

       1. expm1-def 41.6%

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

       2. expm1-log1p 86.1%

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

       3. associate-/r/ 87.3%

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

   13. Simplified 87.3%

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

       1. *-commutative 87.3%

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

       2. pow2 87.3%

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

       3. associate-*r/ 89.0%

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

       4. clear-num 89.0%

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

       5. frac-times 88.9%

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

       6. pow2 88.9%

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

   15. Applied egg-rr 88.9%

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

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

   1. Initial program 0.0%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. fma-neg 0.0%

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

   3. Simplified 1.3%

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

   5. Taylor expanded in c0 around -inf 1.8%

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

      1. associate-*r* 1.8%

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

      2. neg-mul-1 1.8%

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

      3. distribute-lft1-in 1.8%

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

      4. metadata-eval 1.8%

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

      5. mul0-lft 40.4%

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

      6. distribute-lft-neg-in 40.4%

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

      7. distribute-rgt-neg-in 40.4%

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

      8. metadata-eval 40.4%

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

   7. Simplified 40.4%

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

   8. Taylor expanded in c0 around 0 46.3%

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

4. Final simplification 59.1%

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

Alternative 2: 41.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 9.5 \cdot 10^{-219}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0}{w} \cdot \left(d \cdot \frac{d}{D}\right)}{h \cdot D}\\ \mathbf{elif}\;D \leq 1.12 \cdot 10^{-122}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.55 \cdot 10^{-74}:\\ \;\;\;\;\frac{c0}{\frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\\ \mathbf{elif}\;D \leq 2.4 \cdot 10^{-61}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 9.5e-219)
   (* (/ c0 w) (/ (* (/ c0 w) (* d (/ d D))) (* h D)))
   (if (<= D 1.12e-122)
     0.0
     (if (<= D 1.55e-74)
       (/ c0 (/ w (* (/ c0 w) (/ (pow (/ d D) 2.0) h))))
       (if (<= D 2.4e-61)
         0.0
         (* (/ c0 w) (* (/ c0 w) (/ (/ d (* D (/ D d))) h))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 9.5e-219) {
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	} else if (D <= 1.12e-122) {
		tmp = 0.0;
	} else if (D <= 1.55e-74) {
		tmp = c0 / (w / ((c0 / w) * (pow((d / D), 2.0) / h)));
	} else if (D <= 2.4e-61) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	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 (d <= 9.5d-219) then
        tmp = (c0 / w) * (((c0 / w) * (d_1 * (d_1 / d))) / (h * d))
    else if (d <= 1.12d-122) then
        tmp = 0.0d0
    else if (d <= 1.55d-74) then
        tmp = c0 / (w / ((c0 / w) * (((d_1 / d) ** 2.0d0) / h)))
    else if (d <= 2.4d-61) then
        tmp = 0.0d0
    else
        tmp = (c0 / w) * ((c0 / w) * ((d_1 / (d * (d / d_1))) / h))
    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 (D <= 9.5e-219) {
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	} else if (D <= 1.12e-122) {
		tmp = 0.0;
	} else if (D <= 1.55e-74) {
		tmp = c0 / (w / ((c0 / w) * (Math.pow((d / D), 2.0) / h)));
	} else if (D <= 2.4e-61) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 9.5e-219:
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D))
	elif D <= 1.12e-122:
		tmp = 0.0
	elif D <= 1.55e-74:
		tmp = c0 / (w / ((c0 / w) * (math.pow((d / D), 2.0) / h)))
	elif D <= 2.4e-61:
		tmp = 0.0
	else:
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 9.5e-219)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(Float64(c0 / w) * Float64(d * Float64(d / D))) / Float64(h * D)));
	elseif (D <= 1.12e-122)
		tmp = 0.0;
	elseif (D <= 1.55e-74)
		tmp = Float64(c0 / Float64(w / Float64(Float64(c0 / w) * Float64((Float64(d / D) ^ 2.0) / h))));
	elseif (D <= 2.4e-61)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 / w) * Float64(Float64(d / Float64(D * Float64(D / d))) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 9.5e-219)
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	elseif (D <= 1.12e-122)
		tmp = 0.0;
	elseif (D <= 1.55e-74)
		tmp = c0 / (w / ((c0 / w) * (((d / D) ^ 2.0) / h)));
	elseif (D <= 2.4e-61)
		tmp = 0.0;
	else
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 9.5e-219], N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(c0 / w), $MachinePrecision] * N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 1.12e-122], 0.0, If[LessEqual[D, 1.55e-74], N[(c0 / N[(w / N[(N[(c0 / w), $MachinePrecision] * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 2.4e-61], 0.0, N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] * N[(N[(d / N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if D < 9.4999999999999997e-219

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

      2. +-commutative 23.7%

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

      3. times-frac 23.0%

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

      4. fma-neg 23.0%

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

   3. Simplified 23.8%

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

   5. Taylor expanded in c0 around inf 33.0%

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

      1. expm1-log1p-u 14.6%

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

      2. expm1-udef 14.6%

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

   7. Applied egg-rr 15.3%

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

      1. expm1-def 15.3%

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

      2. expm1-log1p 32.6%

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

      3. associate-*l/ 32.6%

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

      4. associate-/l* 32.6%

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

      5. *-commutative 32.6%

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

      6. times-frac 32.6%

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

      7. metadata-eval 32.6%

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

      8. associate-*l/ 34.0%

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

      9. unpow2 34.0%

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

      10. unpow2 34.0%

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

      11. times-frac 40.7%

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

      12. unpow2 40.7%

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

      13. times-frac 41.7%

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

   9. Simplified 41.7%

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

       1. expm1-log1p-u 20.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 20.9%

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

       3. *-un-lft-identity 20.9%

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

   11. Applied egg-rr 20.9%

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

       1. expm1-def 20.5%

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

       2. expm1-log1p 41.7%

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

       3. associate-/r/ 42.8%

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

   13. Simplified 42.8%

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

       1. pow2 42.8%

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

       2. frac-times 40.5%

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

       3. associate-*l/ 39.9%

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

       4. associate-/r* 42.3%

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

       5. associate-*r/ 41.1%

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

       6. frac-times 41.7%

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

   15. Applied egg-rr 41.7%

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

   if 9.4999999999999997e-219 < D < 1.12e-122 or 1.5500000000000001e-74 < D < 2.4000000000000001e-61

   1. Initial program 10.0%

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

      1. +-commutative 10.0%

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

      2. +-commutative 10.0%

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

      3. times-frac 10.0%

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

      4. fma-neg 10.0%

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

   3. Simplified 10.1%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. distribute-lft1-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. mul0-lft 53.1%

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

      6. distribute-lft-neg-in 53.1%

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

      7. distribute-rgt-neg-in 53.1%

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

      8. metadata-eval 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in c0 around 0 59.9%

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

   if 1.12e-122 < D < 1.5500000000000001e-74

   1. Initial program 54.6%

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

      1. +-commutative 54.6%

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

      2. +-commutative 54.6%

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

      3. times-frac 54.6%

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

      4. fma-neg 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in c0 around inf 62.6%

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

      1. expm1-log1p-u 31.8%

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

      2. expm1-udef 31.8%

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

   7. Applied egg-rr 31.8%

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

      1. expm1-def 31.8%

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

      2. expm1-log1p 62.6%

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

      3. associate-*l/ 69.5%

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

      4. associate-/l* 69.5%

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

      5. *-commutative 69.5%

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

      6. times-frac 69.5%

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

      7. metadata-eval 69.5%

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

      8. associate-*l/ 69.5%

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

      9. unpow2 69.5%

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

      10. unpow2 69.5%

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

      11. times-frac 69.7%

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

      12. unpow2 69.7%

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

      13. times-frac 61.9%

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

   9. Simplified 61.9%

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

   if 2.4000000000000001e-61 < D

   1. Initial program 22.9%

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

      1. +-commutative 22.9%

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

      2. +-commutative 22.9%

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

      3. times-frac 22.9%

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

      4. fma-neg 22.9%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in c0 around inf 40.2%

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

      1. expm1-log1p-u 19.9%

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

      2. expm1-udef 16.3%

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

   7. Applied egg-rr 18.1%

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

      1. expm1-def 21.8%

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

      2. expm1-log1p 42.3%

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

      3. associate-*l/ 36.7%

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

      4. associate-/l* 42.3%

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

      5. *-commutative 42.3%

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

      6. times-frac 42.3%

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

      7. metadata-eval 42.3%

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

      8. associate-*l/ 38.6%

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

      9. unpow2 38.6%

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

      10. unpow2 38.6%

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

      11. times-frac 56.1%

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

      12. unpow2 56.1%

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

      13. times-frac 59.9%

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

   9. Simplified 59.9%

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

       1. expm1-log1p-u 30.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 24.4%

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

       3. *-un-lft-identity 24.4%

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

   11. Applied egg-rr 24.4%

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

       1. expm1-def 30.0%

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

       2. expm1-log1p 59.9%

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

       3. associate-/r/ 59.9%

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

   13. Simplified 59.9%

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

       1. pow2 59.9%

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

       2. clear-num 59.9%

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

       3. frac-times 60.0%

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

       4. *-un-lft-identity 60.0%

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

   15. Applied egg-rr 60.0%

       \[\leadsto \frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{\frac{d}{\frac{D}{d} \cdot D}}}{h}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 9.5 \cdot 10^{-219}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0}{w} \cdot \left(d \cdot \frac{d}{D}\right)}{h \cdot D}\\ \mathbf{elif}\;D \leq 1.12 \cdot 10^{-122}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.55 \cdot 10^{-74}:\\ \;\;\;\;\frac{c0}{\frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\\ \mathbf{elif}\;D \leq 2.4 \cdot 10^{-61}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 1.3 \cdot 10^{-218}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0}{w} \cdot \left(d \cdot \frac{d}{D}\right)}{h \cdot D}\\ \mathbf{elif}\;D \leq 1.7 \cdot 10^{-131}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 2.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{c0}{\frac{w}{\frac{c0}{w \cdot \frac{h}{{\left(\frac{d}{D}\right)}^{2}}}}}\\ \mathbf{elif}\;D \leq 8 \cdot 10^{-62}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 1.3e-218)
   (* (/ c0 w) (/ (* (/ c0 w) (* d (/ d D))) (* h D)))
   (if (<= D 1.7e-131)
     0.0
     (if (<= D 2.5e-74)
       (/ c0 (/ w (/ c0 (* w (/ h (pow (/ d D) 2.0))))))
       (if (<= D 8e-62)
         0.0
         (* (/ c0 w) (* (/ c0 w) (/ (/ d (* D (/ D d))) h))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 1.3e-218) {
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	} else if (D <= 1.7e-131) {
		tmp = 0.0;
	} else if (D <= 2.5e-74) {
		tmp = c0 / (w / (c0 / (w * (h / pow((d / D), 2.0)))));
	} else if (D <= 8e-62) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	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 (d <= 1.3d-218) then
        tmp = (c0 / w) * (((c0 / w) * (d_1 * (d_1 / d))) / (h * d))
    else if (d <= 1.7d-131) then
        tmp = 0.0d0
    else if (d <= 2.5d-74) then
        tmp = c0 / (w / (c0 / (w * (h / ((d_1 / d) ** 2.0d0)))))
    else if (d <= 8d-62) then
        tmp = 0.0d0
    else
        tmp = (c0 / w) * ((c0 / w) * ((d_1 / (d * (d / d_1))) / h))
    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 (D <= 1.3e-218) {
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	} else if (D <= 1.7e-131) {
		tmp = 0.0;
	} else if (D <= 2.5e-74) {
		tmp = c0 / (w / (c0 / (w * (h / Math.pow((d / D), 2.0)))));
	} else if (D <= 8e-62) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 1.3e-218:
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D))
	elif D <= 1.7e-131:
		tmp = 0.0
	elif D <= 2.5e-74:
		tmp = c0 / (w / (c0 / (w * (h / math.pow((d / D), 2.0)))))
	elif D <= 8e-62:
		tmp = 0.0
	else:
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 1.3e-218)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(Float64(c0 / w) * Float64(d * Float64(d / D))) / Float64(h * D)));
	elseif (D <= 1.7e-131)
		tmp = 0.0;
	elseif (D <= 2.5e-74)
		tmp = Float64(c0 / Float64(w / Float64(c0 / Float64(w * Float64(h / (Float64(d / D) ^ 2.0))))));
	elseif (D <= 8e-62)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 / w) * Float64(Float64(d / Float64(D * Float64(D / d))) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 1.3e-218)
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	elseif (D <= 1.7e-131)
		tmp = 0.0;
	elseif (D <= 2.5e-74)
		tmp = c0 / (w / (c0 / (w * (h / ((d / D) ^ 2.0)))));
	elseif (D <= 8e-62)
		tmp = 0.0;
	else
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 1.3e-218], N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(c0 / w), $MachinePrecision] * N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 1.7e-131], 0.0, If[LessEqual[D, 2.5e-74], N[(c0 / N[(w / N[(c0 / N[(w * N[(h / N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 8e-62], 0.0, N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] * N[(N[(d / N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if D < 1.29999999999999992e-218

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

      2. +-commutative 23.7%

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

      3. times-frac 23.0%

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

      4. fma-neg 23.0%

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

   3. Simplified 23.8%

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

   5. Taylor expanded in c0 around inf 33.0%

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

      1. expm1-log1p-u 14.6%

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

      2. expm1-udef 14.6%

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

   7. Applied egg-rr 15.3%

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

      1. expm1-def 15.3%

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

      2. expm1-log1p 32.6%

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

      3. associate-*l/ 32.6%

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

      4. associate-/l* 32.6%

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

      5. *-commutative 32.6%

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

      6. times-frac 32.6%

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

      7. metadata-eval 32.6%

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

      8. associate-*l/ 34.0%

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

      9. unpow2 34.0%

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

      10. unpow2 34.0%

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

      11. times-frac 40.7%

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

      12. unpow2 40.7%

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

      13. times-frac 41.7%

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

   9. Simplified 41.7%

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

       1. expm1-log1p-u 20.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 20.9%

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

       3. *-un-lft-identity 20.9%

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

   11. Applied egg-rr 20.9%

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

       1. expm1-def 20.5%

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

       2. expm1-log1p 41.7%

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

       3. associate-/r/ 42.8%

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

   13. Simplified 42.8%

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

       1. pow2 42.8%

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

       2. frac-times 40.5%

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

       3. associate-*l/ 39.9%

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

       4. associate-/r* 42.3%

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

       5. associate-*r/ 41.1%

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

       6. frac-times 41.7%

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

   15. Applied egg-rr 41.7%

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

   if 1.29999999999999992e-218 < D < 1.69999999999999998e-131 or 2.49999999999999999e-74 < D < 8.0000000000000003e-62

   1. Initial program 10.0%

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

      1. +-commutative 10.0%

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

      2. +-commutative 10.0%

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

      3. times-frac 10.0%

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

      4. fma-neg 10.0%

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

   3. Simplified 10.1%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. distribute-lft1-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. mul0-lft 53.1%

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

      6. distribute-lft-neg-in 53.1%

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

      7. distribute-rgt-neg-in 53.1%

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

      8. metadata-eval 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in c0 around 0 59.9%

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

   if 1.69999999999999998e-131 < D < 2.49999999999999999e-74

   1. Initial program 54.6%

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

      1. +-commutative 54.6%

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

      2. +-commutative 54.6%

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

      3. times-frac 54.6%

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

      4. fma-neg 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in c0 around inf 62.6%

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

      1. expm1-log1p-u 31.8%

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

      2. expm1-udef 31.8%

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

   7. Applied egg-rr 31.8%

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

      1. expm1-def 31.8%

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

      2. expm1-log1p 62.6%

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

      3. associate-*l/ 69.5%

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

      4. associate-/l* 69.5%

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

      5. *-commutative 69.5%

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

      6. times-frac 69.5%

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

      7. metadata-eval 69.5%

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

      8. associate-*l/ 69.5%

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

      9. unpow2 69.5%

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

      10. unpow2 69.5%

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

      11. times-frac 69.7%

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

      12. unpow2 69.7%

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

      13. times-frac 61.9%

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

   9. Simplified 61.9%

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

       1. *-commutative 61.9%

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

       2. pow2 61.9%

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

       3. clear-num 61.9%

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

       4. frac-times 62.1%

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

       5. *-un-lft-identity 62.1%

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

       6. pow2 62.1%

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

   11. Applied egg-rr 62.1%

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

   if 8.0000000000000003e-62 < D

   1. Initial program 22.9%

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

      1. +-commutative 22.9%

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

      2. +-commutative 22.9%

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

      3. times-frac 22.9%

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

      4. fma-neg 22.9%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in c0 around inf 40.2%

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

      1. expm1-log1p-u 19.9%

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

      2. expm1-udef 16.3%

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

   7. Applied egg-rr 18.1%

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

      1. expm1-def 21.8%

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

      2. expm1-log1p 42.3%

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

      3. associate-*l/ 36.7%

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

      4. associate-/l* 42.3%

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

      5. *-commutative 42.3%

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

      6. times-frac 42.3%

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

      7. metadata-eval 42.3%

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

      8. associate-*l/ 38.6%

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

      9. unpow2 38.6%

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

      10. unpow2 38.6%

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

      11. times-frac 56.1%

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

      12. unpow2 56.1%

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

      13. times-frac 59.9%

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

   9. Simplified 59.9%

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

       1. expm1-log1p-u 30.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 24.4%

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

       3. *-un-lft-identity 24.4%

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

   11. Applied egg-rr 24.4%

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

       1. expm1-def 30.0%

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

       2. expm1-log1p 59.9%

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

       3. associate-/r/ 59.9%

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

   13. Simplified 59.9%

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

       1. pow2 59.9%

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

       2. clear-num 59.9%

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

       3. frac-times 60.0%

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

       4. *-un-lft-identity 60.0%

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

   15. Applied egg-rr 60.0%

       \[\leadsto \frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{\frac{d}{\frac{D}{d} \cdot D}}}{h}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.3 \cdot 10^{-218}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0}{w} \cdot \left(d \cdot \frac{d}{D}\right)}{h \cdot D}\\ \mathbf{elif}\;D \leq 1.7 \cdot 10^{-131}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 2.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{c0}{\frac{w}{\frac{c0}{w \cdot \frac{h}{{\left(\frac{d}{D}\right)}^{2}}}}}\\ \mathbf{elif}\;D \leq 8 \cdot 10^{-62}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 2.5 \cdot 10^{-218}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0}{w} \cdot \left(d \cdot \frac{d}{D}\right)}{h \cdot D}\\ \mathbf{elif}\;D \leq 10^{-123}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.35 \cdot 10^{-74}:\\ \;\;\;\;\frac{c0}{\frac{w}{\frac{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}{h}}}\\ \mathbf{elif}\;D \leq 7.5 \cdot 10^{-62}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 2.5e-218)
   (* (/ c0 w) (/ (* (/ c0 w) (* d (/ d D))) (* h D)))
   (if (<= D 1e-123)
     0.0
     (if (<= D 1.35e-74)
       (/ c0 (/ w (/ (* (/ c0 w) (pow (/ d D) 2.0)) h)))
       (if (<= D 7.5e-62)
         0.0
         (* (/ c0 w) (* (/ c0 w) (/ (/ d (* D (/ D d))) h))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 2.5e-218) {
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	} else if (D <= 1e-123) {
		tmp = 0.0;
	} else if (D <= 1.35e-74) {
		tmp = c0 / (w / (((c0 / w) * pow((d / D), 2.0)) / h));
	} else if (D <= 7.5e-62) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	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 (d <= 2.5d-218) then
        tmp = (c0 / w) * (((c0 / w) * (d_1 * (d_1 / d))) / (h * d))
    else if (d <= 1d-123) then
        tmp = 0.0d0
    else if (d <= 1.35d-74) then
        tmp = c0 / (w / (((c0 / w) * ((d_1 / d) ** 2.0d0)) / h))
    else if (d <= 7.5d-62) then
        tmp = 0.0d0
    else
        tmp = (c0 / w) * ((c0 / w) * ((d_1 / (d * (d / d_1))) / h))
    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 (D <= 2.5e-218) {
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	} else if (D <= 1e-123) {
		tmp = 0.0;
	} else if (D <= 1.35e-74) {
		tmp = c0 / (w / (((c0 / w) * Math.pow((d / D), 2.0)) / h));
	} else if (D <= 7.5e-62) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 2.5e-218:
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D))
	elif D <= 1e-123:
		tmp = 0.0
	elif D <= 1.35e-74:
		tmp = c0 / (w / (((c0 / w) * math.pow((d / D), 2.0)) / h))
	elif D <= 7.5e-62:
		tmp = 0.0
	else:
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 2.5e-218)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(Float64(c0 / w) * Float64(d * Float64(d / D))) / Float64(h * D)));
	elseif (D <= 1e-123)
		tmp = 0.0;
	elseif (D <= 1.35e-74)
		tmp = Float64(c0 / Float64(w / Float64(Float64(Float64(c0 / w) * (Float64(d / D) ^ 2.0)) / h)));
	elseif (D <= 7.5e-62)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 / w) * Float64(Float64(d / Float64(D * Float64(D / d))) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 2.5e-218)
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	elseif (D <= 1e-123)
		tmp = 0.0;
	elseif (D <= 1.35e-74)
		tmp = c0 / (w / (((c0 / w) * ((d / D) ^ 2.0)) / h));
	elseif (D <= 7.5e-62)
		tmp = 0.0;
	else
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 2.5e-218], N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(c0 / w), $MachinePrecision] * N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 1e-123], 0.0, If[LessEqual[D, 1.35e-74], N[(c0 / N[(w / N[(N[(N[(c0 / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 7.5e-62], 0.0, N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] * N[(N[(d / N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if D < 2.50000000000000021e-218

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

      2. +-commutative 23.7%

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

      3. times-frac 23.0%

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

      4. fma-neg 23.0%

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

   3. Simplified 23.8%

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

   5. Taylor expanded in c0 around inf 33.0%

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

      1. expm1-log1p-u 14.6%

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

      2. expm1-udef 14.6%

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

   7. Applied egg-rr 15.3%

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

      1. expm1-def 15.3%

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

      2. expm1-log1p 32.6%

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

      3. associate-*l/ 32.6%

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

      4. associate-/l* 32.6%

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

      5. *-commutative 32.6%

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

      6. times-frac 32.6%

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

      7. metadata-eval 32.6%

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

      8. associate-*l/ 34.0%

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

      9. unpow2 34.0%

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

      10. unpow2 34.0%

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

      11. times-frac 40.7%

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

      12. unpow2 40.7%

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

      13. times-frac 41.7%

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

   9. Simplified 41.7%

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

       1. expm1-log1p-u 20.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 20.9%

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

       3. *-un-lft-identity 20.9%

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

   11. Applied egg-rr 20.9%

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

       1. expm1-def 20.5%

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

       2. expm1-log1p 41.7%

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

       3. associate-/r/ 42.8%

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

   13. Simplified 42.8%

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

       1. pow2 42.8%

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

       2. frac-times 40.5%

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

       3. associate-*l/ 39.9%

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

       4. associate-/r* 42.3%

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

       5. associate-*r/ 41.1%

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

       6. frac-times 41.7%

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

   15. Applied egg-rr 41.7%

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

   if 2.50000000000000021e-218 < D < 1.0000000000000001e-123 or 1.35000000000000009e-74 < D < 7.5000000000000003e-62

   1. Initial program 10.0%

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

      1. +-commutative 10.0%

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

      2. +-commutative 10.0%

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

      3. times-frac 10.0%

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

      4. fma-neg 10.0%

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

   3. Simplified 10.1%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. distribute-lft1-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. mul0-lft 53.1%

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

      6. distribute-lft-neg-in 53.1%

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

      7. distribute-rgt-neg-in 53.1%

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

      8. metadata-eval 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in c0 around 0 59.9%

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

   if 1.0000000000000001e-123 < D < 1.35000000000000009e-74

   1. Initial program 54.6%

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

      1. +-commutative 54.6%

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

      2. +-commutative 54.6%

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

      3. times-frac 54.6%

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

      4. fma-neg 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in c0 around inf 62.6%

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

      1. expm1-log1p-u 31.8%

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

      2. expm1-udef 31.8%

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

   7. Applied egg-rr 31.8%

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

      1. expm1-def 31.8%

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

      2. expm1-log1p 62.6%

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

      3. associate-*l/ 69.5%

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

      4. associate-/l* 69.5%

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

      5. *-commutative 69.5%

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

      6. times-frac 69.5%

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

      7. metadata-eval 69.5%

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

      8. associate-*l/ 69.5%

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

      9. unpow2 69.5%

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

      10. unpow2 69.5%

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

      11. times-frac 69.7%

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

      12. unpow2 69.7%

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

      13. times-frac 61.9%

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

   9. Simplified 61.9%

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

       1. pow2 61.9%

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

       2. frac-times 69.7%

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

       3. associate-*l/ 69.7%

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

       4. *-commutative 69.7%

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

       5. associate-/r* 62.6%

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

       6. associate-*r/ 62.6%

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

       7. pow2 62.6%

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

   11. Applied egg-rr 62.6%

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

   if 7.5000000000000003e-62 < D

   1. Initial program 22.9%

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

      1. +-commutative 22.9%

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

      2. +-commutative 22.9%

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

      3. times-frac 22.9%

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

      4. fma-neg 22.9%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in c0 around inf 40.2%

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

      1. expm1-log1p-u 19.9%

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

      2. expm1-udef 16.3%

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

   7. Applied egg-rr 18.1%

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

      1. expm1-def 21.8%

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

      2. expm1-log1p 42.3%

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

      3. associate-*l/ 36.7%

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

      4. associate-/l* 42.3%

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

      5. *-commutative 42.3%

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

      6. times-frac 42.3%

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

      7. metadata-eval 42.3%

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

      8. associate-*l/ 38.6%

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

      9. unpow2 38.6%

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

      10. unpow2 38.6%

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

      11. times-frac 56.1%

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

      12. unpow2 56.1%

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

      13. times-frac 59.9%

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

   9. Simplified 59.9%

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

       1. expm1-log1p-u 30.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 24.4%

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

       3. *-un-lft-identity 24.4%

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

   11. Applied egg-rr 24.4%

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

       1. expm1-def 30.0%

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

       2. expm1-log1p 59.9%

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

       3. associate-/r/ 59.9%

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

   13. Simplified 59.9%

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

       1. pow2 59.9%

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

       2. clear-num 59.9%

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

       3. frac-times 60.0%

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

       4. *-un-lft-identity 60.0%

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

   15. Applied egg-rr 60.0%

       \[\leadsto \frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{\frac{d}{\frac{D}{d} \cdot D}}}{h}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.5 \cdot 10^{-218}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0}{w} \cdot \left(d \cdot \frac{d}{D}\right)}{h \cdot D}\\ \mathbf{elif}\;D \leq 10^{-123}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.35 \cdot 10^{-74}:\\ \;\;\;\;\frac{c0}{\frac{w}{\frac{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}{h}}}\\ \mathbf{elif}\;D \leq 7.5 \cdot 10^{-62}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 4.8 \cdot 10^{-218}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{c0 \cdot \left(d \cdot \frac{d}{D}\right)}{\left(w \cdot h\right) \cdot D}\\ \mathbf{elif}\;D \leq 1.8 \cdot 10^{-124}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)\\ \mathbf{elif}\;D \leq 7.5 \cdot 10^{-62}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 4.8e-218)
   (* (/ c0 w) (/ (* c0 (* d (/ d D))) (* (* w h) D)))
   (if (<= D 1.8e-124)
     0.0
     (if (<= D 1.8e-74)
       (* (/ c0 w) (* (/ c0 w) (/ (* (/ d D) (/ d D)) h)))
       (if (<= D 7.5e-62)
         0.0
         (* (/ c0 w) (* (/ c0 w) (/ (/ d (* D (/ D d))) h))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 4.8e-218) {
		tmp = (c0 / w) * ((c0 * (d * (d / D))) / ((w * h) * D));
	} else if (D <= 1.8e-124) {
		tmp = 0.0;
	} else if (D <= 1.8e-74) {
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	} else if (D <= 7.5e-62) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	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 (d <= 4.8d-218) then
        tmp = (c0 / w) * ((c0 * (d_1 * (d_1 / d))) / ((w * h) * d))
    else if (d <= 1.8d-124) then
        tmp = 0.0d0
    else if (d <= 1.8d-74) then
        tmp = (c0 / w) * ((c0 / w) * (((d_1 / d) * (d_1 / d)) / h))
    else if (d <= 7.5d-62) then
        tmp = 0.0d0
    else
        tmp = (c0 / w) * ((c0 / w) * ((d_1 / (d * (d / d_1))) / h))
    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 (D <= 4.8e-218) {
		tmp = (c0 / w) * ((c0 * (d * (d / D))) / ((w * h) * D));
	} else if (D <= 1.8e-124) {
		tmp = 0.0;
	} else if (D <= 1.8e-74) {
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	} else if (D <= 7.5e-62) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 4.8e-218:
		tmp = (c0 / w) * ((c0 * (d * (d / D))) / ((w * h) * D))
	elif D <= 1.8e-124:
		tmp = 0.0
	elif D <= 1.8e-74:
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h))
	elif D <= 7.5e-62:
		tmp = 0.0
	else:
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 4.8e-218)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 * Float64(d * Float64(d / D))) / Float64(Float64(w * h) * D)));
	elseif (D <= 1.8e-124)
		tmp = 0.0;
	elseif (D <= 1.8e-74)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 / w) * Float64(Float64(Float64(d / D) * Float64(d / D)) / h)));
	elseif (D <= 7.5e-62)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 / w) * Float64(Float64(d / Float64(D * Float64(D / d))) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 4.8e-218)
		tmp = (c0 / w) * ((c0 * (d * (d / D))) / ((w * h) * D));
	elseif (D <= 1.8e-124)
		tmp = 0.0;
	elseif (D <= 1.8e-74)
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	elseif (D <= 7.5e-62)
		tmp = 0.0;
	else
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 4.8e-218], N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 * N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 1.8e-124], 0.0, If[LessEqual[D, 1.8e-74], N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 7.5e-62], 0.0, N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] * N[(N[(d / N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if D < 4.8000000000000002e-218

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

      2. +-commutative 23.7%

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

      3. times-frac 23.0%

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

      4. fma-neg 23.0%

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

   3. Simplified 23.8%

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

   5. Taylor expanded in c0 around inf 33.0%

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

      1. expm1-log1p-u 14.6%

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

      2. expm1-udef 14.6%

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

   7. Applied egg-rr 15.3%

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

      1. expm1-def 15.3%

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

      2. expm1-log1p 32.6%

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

      3. associate-*l/ 32.6%

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

      4. associate-/l* 32.6%

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

      5. *-commutative 32.6%

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

      6. times-frac 32.6%

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

      7. metadata-eval 32.6%

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

      8. associate-*l/ 34.0%

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

      9. unpow2 34.0%

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

      10. unpow2 34.0%

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

      11. times-frac 40.7%

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

      12. unpow2 40.7%

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

      13. times-frac 41.7%

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

   9. Simplified 41.7%

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

       1. expm1-log1p-u 20.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 20.9%

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

       3. *-un-lft-identity 20.9%

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

   11. Applied egg-rr 20.9%

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

       1. expm1-def 20.5%

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

       2. expm1-log1p 41.7%

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

       3. associate-/r/ 42.8%

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

   13. Simplified 42.8%

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

       1. pow2 42.8%

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

       2. frac-times 40.5%

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

       3. associate-*l/ 39.9%

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

       4. *-commutative 39.9%

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

       5. associate-*r/ 38.7%

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

       6. frac-times 39.7%

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

   15. Applied egg-rr 39.7%

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

   if 4.8000000000000002e-218 < D < 1.80000000000000005e-124 or 1.8000000000000001e-74 < D < 7.5000000000000003e-62

   1. Initial program 10.0%

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

      1. +-commutative 10.0%

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

      2. +-commutative 10.0%

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

      3. times-frac 10.0%

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

      4. fma-neg 10.0%

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

   3. Simplified 10.1%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. distribute-lft1-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. mul0-lft 53.1%

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

      6. distribute-lft-neg-in 53.1%

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

      7. distribute-rgt-neg-in 53.1%

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

      8. metadata-eval 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in c0 around 0 59.9%

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

   if 1.80000000000000005e-124 < D < 1.8000000000000001e-74

   1. Initial program 54.6%

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

      1. +-commutative 54.6%

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

      2. +-commutative 54.6%

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

      3. times-frac 54.6%

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

      4. fma-neg 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in c0 around inf 62.6%

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

      1. expm1-log1p-u 31.8%

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

      2. expm1-udef 31.8%

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

   7. Applied egg-rr 31.8%

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

      1. expm1-def 31.8%

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

      2. expm1-log1p 62.6%

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

      3. associate-*l/ 69.5%

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

      4. associate-/l* 69.5%

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

      5. *-commutative 69.5%

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

      6. times-frac 69.5%

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

      7. metadata-eval 69.5%

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

      8. associate-*l/ 69.5%

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

      9. unpow2 69.5%

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

      10. unpow2 69.5%

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

      11. times-frac 69.7%

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

      12. unpow2 69.7%

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

      13. times-frac 61.9%

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

   9. Simplified 61.9%

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

       1. expm1-log1p-u 31.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 31.0%

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

       3. *-un-lft-identity 31.0%

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

   11. Applied egg-rr 31.0%

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

       1. expm1-def 31.0%

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

       2. expm1-log1p 61.9%

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

       3. associate-/r/ 61.9%

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

   13. Simplified 61.9%

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

       1. pow2 61.9%

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

   15. Applied egg-rr 61.9%

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

   if 7.5000000000000003e-62 < D

   1. Initial program 22.9%

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

      1. +-commutative 22.9%

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

      2. +-commutative 22.9%

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

      3. times-frac 22.9%

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

      4. fma-neg 22.9%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in c0 around inf 40.2%

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

      1. expm1-log1p-u 19.9%

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

      2. expm1-udef 16.3%

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

   7. Applied egg-rr 18.1%

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

      1. expm1-def 21.8%

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

      2. expm1-log1p 42.3%

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

      3. associate-*l/ 36.7%

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

      4. associate-/l* 42.3%

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

      5. *-commutative 42.3%

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

      6. times-frac 42.3%

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

      7. metadata-eval 42.3%

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

      8. associate-*l/ 38.6%

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

      9. unpow2 38.6%

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

      10. unpow2 38.6%

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

      11. times-frac 56.1%

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

      12. unpow2 56.1%

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

      13. times-frac 59.9%

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

   9. Simplified 59.9%

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

       1. expm1-log1p-u 30.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 24.4%

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

       3. *-un-lft-identity 24.4%

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

   11. Applied egg-rr 24.4%

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

       1. expm1-def 30.0%

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

       2. expm1-log1p 59.9%

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

       3. associate-/r/ 59.9%

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

   13. Simplified 59.9%

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

       1. pow2 59.9%

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

       2. clear-num 59.9%

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

       3. frac-times 60.0%

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

       4. *-un-lft-identity 60.0%

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

   15. Applied egg-rr 60.0%

       \[\leadsto \frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{\frac{d}{\frac{D}{d} \cdot D}}}{h}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.8 \cdot 10^{-218}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{c0 \cdot \left(d \cdot \frac{d}{D}\right)}{\left(w \cdot h\right) \cdot D}\\ \mathbf{elif}\;D \leq 1.8 \cdot 10^{-124}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)\\ \mathbf{elif}\;D \leq 7.5 \cdot 10^{-62}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 7.2 \cdot 10^{-219}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0}{w} \cdot \left(d \cdot \frac{d}{D}\right)}{h \cdot D}\\ \mathbf{elif}\;D \leq 2.05 \cdot 10^{-122}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)\\ \mathbf{elif}\;D \leq 4.3 \cdot 10^{-61}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 7.2e-219)
   (* (/ c0 w) (/ (* (/ c0 w) (* d (/ d D))) (* h D)))
   (if (<= D 2.05e-122)
     0.0
     (if (<= D 1.45e-74)
       (* (/ c0 w) (* (/ c0 w) (/ (* (/ d D) (/ d D)) h)))
       (if (<= D 4.3e-61)
         0.0
         (* (/ c0 w) (* (/ c0 w) (/ (/ d (* D (/ D d))) h))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 7.2e-219) {
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	} else if (D <= 2.05e-122) {
		tmp = 0.0;
	} else if (D <= 1.45e-74) {
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	} else if (D <= 4.3e-61) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	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 (d <= 7.2d-219) then
        tmp = (c0 / w) * (((c0 / w) * (d_1 * (d_1 / d))) / (h * d))
    else if (d <= 2.05d-122) then
        tmp = 0.0d0
    else if (d <= 1.45d-74) then
        tmp = (c0 / w) * ((c0 / w) * (((d_1 / d) * (d_1 / d)) / h))
    else if (d <= 4.3d-61) then
        tmp = 0.0d0
    else
        tmp = (c0 / w) * ((c0 / w) * ((d_1 / (d * (d / d_1))) / h))
    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 (D <= 7.2e-219) {
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	} else if (D <= 2.05e-122) {
		tmp = 0.0;
	} else if (D <= 1.45e-74) {
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	} else if (D <= 4.3e-61) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 7.2e-219:
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D))
	elif D <= 2.05e-122:
		tmp = 0.0
	elif D <= 1.45e-74:
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h))
	elif D <= 4.3e-61:
		tmp = 0.0
	else:
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 7.2e-219)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(Float64(c0 / w) * Float64(d * Float64(d / D))) / Float64(h * D)));
	elseif (D <= 2.05e-122)
		tmp = 0.0;
	elseif (D <= 1.45e-74)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 / w) * Float64(Float64(Float64(d / D) * Float64(d / D)) / h)));
	elseif (D <= 4.3e-61)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 / w) * Float64(Float64(d / Float64(D * Float64(D / d))) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 7.2e-219)
		tmp = (c0 / w) * (((c0 / w) * (d * (d / D))) / (h * D));
	elseif (D <= 2.05e-122)
		tmp = 0.0;
	elseif (D <= 1.45e-74)
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	elseif (D <= 4.3e-61)
		tmp = 0.0;
	else
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 7.2e-219], N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(c0 / w), $MachinePrecision] * N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 2.05e-122], 0.0, If[LessEqual[D, 1.45e-74], N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 4.3e-61], 0.0, N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] * N[(N[(d / N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if D < 7.19999999999999947e-219

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

      2. +-commutative 23.7%

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

      3. times-frac 23.0%

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

      4. fma-neg 23.0%

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

   3. Simplified 23.8%

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

   5. Taylor expanded in c0 around inf 33.0%

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

      1. expm1-log1p-u 14.6%

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

      2. expm1-udef 14.6%

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

   7. Applied egg-rr 15.3%

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

      1. expm1-def 15.3%

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

      2. expm1-log1p 32.6%

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

      3. associate-*l/ 32.6%

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

      4. associate-/l* 32.6%

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

      5. *-commutative 32.6%

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

      6. times-frac 32.6%

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

      7. metadata-eval 32.6%

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

      8. associate-*l/ 34.0%

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

      9. unpow2 34.0%

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

      10. unpow2 34.0%

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

      11. times-frac 40.7%

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

      12. unpow2 40.7%

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

      13. times-frac 41.7%

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

   9. Simplified 41.7%

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

       1. expm1-log1p-u 20.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 20.9%

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

       3. *-un-lft-identity 20.9%

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

   11. Applied egg-rr 20.9%

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

       1. expm1-def 20.5%

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

       2. expm1-log1p 41.7%

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

       3. associate-/r/ 42.8%

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

   13. Simplified 42.8%

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

       1. pow2 42.8%

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

       2. frac-times 40.5%

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

       3. associate-*l/ 39.9%

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

       4. associate-/r* 42.3%

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

       5. associate-*r/ 41.1%

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

       6. frac-times 41.7%

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

   15. Applied egg-rr 41.7%

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

   if 7.19999999999999947e-219 < D < 2.05e-122 or 1.45e-74 < D < 4.3000000000000003e-61

   1. Initial program 10.0%

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

      1. +-commutative 10.0%

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

      2. +-commutative 10.0%

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

      3. times-frac 10.0%

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

      4. fma-neg 10.0%

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

   3. Simplified 10.1%

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

   5. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. neg-mul-1 0.0%

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

      3. distribute-lft1-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. mul0-lft 53.1%

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

      6. distribute-lft-neg-in 53.1%

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

      7. distribute-rgt-neg-in 53.1%

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

      8. metadata-eval 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in c0 around 0 59.9%

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

   if 2.05e-122 < D < 1.45e-74

   1. Initial program 54.6%

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

      1. +-commutative 54.6%

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

      2. +-commutative 54.6%

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

      3. times-frac 54.6%

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

      4. fma-neg 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in c0 around inf 62.6%

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

      1. expm1-log1p-u 31.8%

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

      2. expm1-udef 31.8%

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

   7. Applied egg-rr 31.8%

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

      1. expm1-def 31.8%

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

      2. expm1-log1p 62.6%

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

      3. associate-*l/ 69.5%

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

      4. associate-/l* 69.5%

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

      5. *-commutative 69.5%

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

      6. times-frac 69.5%

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

      7. metadata-eval 69.5%

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

      8. associate-*l/ 69.5%

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

      9. unpow2 69.5%

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

      10. unpow2 69.5%

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

      11. times-frac 69.7%

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

      12. unpow2 69.7%

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

      13. times-frac 61.9%

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

   9. Simplified 61.9%

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

       1. expm1-log1p-u 31.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 31.0%

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

       3. *-un-lft-identity 31.0%

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

   11. Applied egg-rr 31.0%

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

       1. expm1-def 31.0%

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

       2. expm1-log1p 61.9%

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

       3. associate-/r/ 61.9%

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

   13. Simplified 61.9%

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

       1. pow2 61.9%

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

   15. Applied egg-rr 61.9%

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

   if 4.3000000000000003e-61 < D

   1. Initial program 22.9%

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

      1. +-commutative 22.9%

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

      2. +-commutative 22.9%

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

      3. times-frac 22.9%

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

      4. fma-neg 22.9%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in c0 around inf 40.2%

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

      1. expm1-log1p-u 19.9%

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

      2. expm1-udef 16.3%

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

   7. Applied egg-rr 18.1%

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

      1. expm1-def 21.8%

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

      2. expm1-log1p 42.3%

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

      3. associate-*l/ 36.7%

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

      4. associate-/l* 42.3%

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

      5. *-commutative 42.3%

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

      6. times-frac 42.3%

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

      7. metadata-eval 42.3%

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

      8. associate-*l/ 38.6%

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

      9. unpow2 38.6%

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

      10. unpow2 38.6%

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

      11. times-frac 56.1%

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

      12. unpow2 56.1%

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

      13. times-frac 59.9%

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

   9. Simplified 59.9%

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

       1. expm1-log1p-u 30.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 24.4%

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

       3. *-un-lft-identity 24.4%

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

   11. Applied egg-rr 24.4%

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

       1. expm1-def 30.0%

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

       2. expm1-log1p 59.9%

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

       3. associate-/r/ 59.9%

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

   13. Simplified 59.9%

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

       1. pow2 59.9%

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

       2. clear-num 59.9%

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

       3. frac-times 60.0%

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

       4. *-un-lft-identity 60.0%

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

   15. Applied egg-rr 60.0%

       \[\leadsto \frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{\frac{d}{\frac{D}{d} \cdot D}}}{h}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 7.2 \cdot 10^{-219}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\frac{c0}{w} \cdot \left(d \cdot \frac{d}{D}\right)}{h \cdot D}\\ \mathbf{elif}\;D \leq 2.05 \cdot 10^{-122}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)\\ \mathbf{elif}\;D \leq 4.3 \cdot 10^{-61}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -4.5 \cdot 10^{+231}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -5 \cdot 10^{-127} \lor \neg \left(c0 \leq 1.2 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= c0 -4.5e+231)
   0.0
   (if (or (<= c0 -5e-127) (not (<= c0 1.2e-93)))
     (* (/ c0 w) (* (/ c0 w) (/ (* (/ d D) (/ d D)) h)))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (c0 <= -4.5e+231) {
		tmp = 0.0;
	} else if ((c0 <= -5e-127) || !(c0 <= 1.2e-93)) {
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	} 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 (c0 <= (-4.5d+231)) then
        tmp = 0.0d0
    else if ((c0 <= (-5d-127)) .or. (.not. (c0 <= 1.2d-93))) then
        tmp = (c0 / w) * ((c0 / w) * (((d_1 / d) * (d_1 / d)) / h))
    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 (c0 <= -4.5e+231) {
		tmp = 0.0;
	} else if ((c0 <= -5e-127) || !(c0 <= 1.2e-93)) {
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if c0 <= -4.5e+231:
		tmp = 0.0
	elif (c0 <= -5e-127) or not (c0 <= 1.2e-93):
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (c0 <= -4.5e+231)
		tmp = 0.0;
	elseif ((c0 <= -5e-127) || !(c0 <= 1.2e-93))
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 / w) * Float64(Float64(Float64(d / D) * Float64(d / D)) / h)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (c0 <= -4.5e+231)
		tmp = 0.0;
	elseif ((c0 <= -5e-127) || ~((c0 <= 1.2e-93)))
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[c0, -4.5e+231], 0.0, If[Or[LessEqual[c0, -5e-127], N[Not[LessEqual[c0, 1.2e-93]], $MachinePrecision]], N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if c0 < -4.49999999999999991e231 or -4.9999999999999997e-127 < c0 < 1.2000000000000001e-93

   1. Initial program 14.7%

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

      1. +-commutative 14.7%

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

      2. +-commutative 14.7%

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

      3. times-frac 13.5%

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

      4. fma-neg 13.5%

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

   3. Simplified 13.6%

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

   5. Taylor expanded in c0 around -inf 1.5%

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

      1. associate-*r* 1.5%

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

      2. neg-mul-1 1.5%

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

      3. distribute-lft1-in 1.5%

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

      4. metadata-eval 1.5%

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

      5. mul0-lft 48.5%

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

      6. distribute-lft-neg-in 48.5%

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

      7. distribute-rgt-neg-in 48.5%

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

      8. metadata-eval 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in c0 around 0 53.4%

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

   if -4.49999999999999991e231 < c0 < -4.9999999999999997e-127 or 1.2000000000000001e-93 < c0

   1. Initial program 27.9%

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

      1. +-commutative 27.9%

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

      2. +-commutative 27.9%

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

      3. times-frac 27.9%

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

      4. fma-neg 27.9%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in c0 around inf 39.2%

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

      1. expm1-log1p-u 18.0%

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

      2. expm1-udef 18.0%

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

   7. Applied egg-rr 19.2%

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

      1. expm1-def 19.2%

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

      2. expm1-log1p 40.5%

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

      3. associate-*l/ 40.5%

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

      4. associate-/l* 41.0%

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

      5. *-commutative 41.0%

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

      6. times-frac 41.0%

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

      7. metadata-eval 41.0%

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

      8. associate-*l/ 41.7%

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

      9. unpow2 41.7%

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

      10. unpow2 41.7%

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

      11. times-frac 50.2%

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

      12. unpow2 50.2%

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

      13. times-frac 50.8%

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

   9. Simplified 50.8%

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

       1. expm1-log1p-u 24.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 24.1%

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

       3. *-un-lft-identity 24.1%

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

   11. Applied egg-rr 24.1%

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

       1. expm1-def 24.0%

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

       2. expm1-log1p 50.8%

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

       3. associate-/r/ 51.3%

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

   13. Simplified 51.3%

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

       1. pow2 51.3%

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

   15. Applied egg-rr 51.3%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -4.5 \cdot 10^{+231}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -5 \cdot 10^{-127} \lor \neg \left(c0 \leq 1.2 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq -3.6 \cdot 10^{+232}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -3.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)\\ \mathbf{elif}\;c0 \leq 7.5 \cdot 10^{-95}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= c0 -3.6e+232)
   0.0
   (if (<= c0 -3.4e-128)
     (* (/ c0 w) (* (/ c0 w) (/ (* (/ d D) (/ d D)) h)))
     (if (<= c0 7.5e-95)
       0.0
       (* (/ c0 w) (* (/ c0 w) (/ (/ d (* D (/ D d))) h)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (c0 <= -3.6e+232) {
		tmp = 0.0;
	} else if (c0 <= -3.4e-128) {
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	} else if (c0 <= 7.5e-95) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (c0 <= (-3.6d+232)) then
        tmp = 0.0d0
    else if (c0 <= (-3.4d-128)) then
        tmp = (c0 / w) * ((c0 / w) * (((d_1 / d) * (d_1 / d)) / h))
    else if (c0 <= 7.5d-95) then
        tmp = 0.0d0
    else
        tmp = (c0 / w) * ((c0 / w) * ((d_1 / (d * (d / d_1))) / h))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (c0 <= -3.6e+232) {
		tmp = 0.0;
	} else if (c0 <= -3.4e-128) {
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	} else if (c0 <= 7.5e-95) {
		tmp = 0.0;
	} else {
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if c0 <= -3.6e+232:
		tmp = 0.0
	elif c0 <= -3.4e-128:
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h))
	elif c0 <= 7.5e-95:
		tmp = 0.0
	else:
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (c0 <= -3.6e+232)
		tmp = 0.0;
	elseif (c0 <= -3.4e-128)
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 / w) * Float64(Float64(Float64(d / D) * Float64(d / D)) / h)));
	elseif (c0 <= 7.5e-95)
		tmp = 0.0;
	else
		tmp = Float64(Float64(c0 / w) * Float64(Float64(c0 / w) * Float64(Float64(d / Float64(D * Float64(D / d))) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (c0 <= -3.6e+232)
		tmp = 0.0;
	elseif (c0 <= -3.4e-128)
		tmp = (c0 / w) * ((c0 / w) * (((d / D) * (d / D)) / h));
	elseif (c0 <= 7.5e-95)
		tmp = 0.0;
	else
		tmp = (c0 / w) * ((c0 / w) * ((d / (D * (D / d))) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[c0, -3.6e+232], 0.0, If[LessEqual[c0, -3.4e-128], N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 7.5e-95], 0.0, N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / w), $MachinePrecision] * N[(N[(d / N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -3.59999999999999993e232 or -3.39999999999999975e-128 < c0 < 7.5000000000000003e-95

   1. Initial program 14.7%

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

      1. +-commutative 14.7%

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

      2. +-commutative 14.7%

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

      3. times-frac 13.5%

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

      4. fma-neg 13.5%

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

   3. Simplified 13.6%

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

   5. Taylor expanded in c0 around -inf 1.5%

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

      1. associate-*r* 1.5%

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

      2. neg-mul-1 1.5%

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

      3. distribute-lft1-in 1.5%

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

      4. metadata-eval 1.5%

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

      5. mul0-lft 48.5%

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

      6. distribute-lft-neg-in 48.5%

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

      7. distribute-rgt-neg-in 48.5%

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

      8. metadata-eval 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in c0 around 0 53.4%

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

   if -3.59999999999999993e232 < c0 < -3.39999999999999975e-128

   1. Initial program 32.3%

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

      1. +-commutative 32.3%

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

      2. +-commutative 32.3%

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

      3. times-frac 32.3%

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

      4. fma-neg 32.3%

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

   3. Simplified 32.4%

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

   5. Taylor expanded in c0 around inf 46.1%

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

      1. expm1-log1p-u 21.7%

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

      2. expm1-udef 21.7%

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

   7. Applied egg-rr 21.8%

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

      1. expm1-def 21.8%

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

      2. expm1-log1p 46.3%

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

      3. associate-*l/ 47.5%

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

      4. associate-/l* 47.6%

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

      5. *-commutative 47.6%

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

      6. times-frac 47.6%

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

      7. metadata-eval 47.6%

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

      8. associate-*l/ 47.5%

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

      9. unpow2 47.5%

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

      10. unpow2 47.5%

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

      11. times-frac 58.2%

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

      12. unpow2 58.2%

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

      13. times-frac 57.0%

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

   9. Simplified 57.0%

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

       1. expm1-log1p-u 28.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 27.2%

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

       3. *-un-lft-identity 27.2%

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

   11. Applied egg-rr 27.2%

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

       1. expm1-def 28.4%

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

       2. expm1-log1p 57.0%

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

       3. associate-/r/ 58.2%

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

   13. Simplified 58.2%

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

       1. pow2 58.2%

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

   15. Applied egg-rr 58.2%

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

   if 7.5000000000000003e-95 < c0

   1. Initial program 24.5%

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

      1. +-commutative 24.5%

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

      2. +-commutative 24.5%

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

      3. times-frac 24.5%

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

      4. fma-neg 24.5%

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

   3. Simplified 26.7%

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

   5. Taylor expanded in c0 around inf 33.8%

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

      1. expm1-log1p-u 15.1%

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

      2. expm1-udef 15.1%

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

   7. Applied egg-rr 17.2%

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

      1. expm1-def 17.2%

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

      2. expm1-log1p 35.9%

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

      3. associate-*l/ 34.9%

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

      4. associate-/l* 35.9%

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

      5. *-commutative 35.9%

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

      6. times-frac 35.9%

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

      7. metadata-eval 35.9%

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

      8. associate-*l/ 37.1%

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

      9. unpow2 37.1%

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

      10. unpow2 37.1%

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

      11. times-frac 44.0%

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

      12. unpow2 44.0%

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

      13. times-frac 45.8%

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

   9. Simplified 45.8%

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

       1. expm1-log1p-u 20.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{1 \cdot \frac{w}{\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}}}\right)\right)} \]

       2. expm1-udef 21.6%

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

       3. *-un-lft-identity 21.6%

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

   11. Applied egg-rr 21.6%

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

       1. expm1-def 20.6%

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

       2. expm1-log1p 45.8%

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

       3. associate-/r/ 45.8%

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

   13. Simplified 45.8%

       \[\leadsto \color{blue}{\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{h}\right)} \]
   14. Step-by-step derivation

       1. pow2 45.8%

          \[\leadsto \frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{\frac{d}{D} \cdot \frac{d}{D}}}{h}\right) \]

       2. clear-num 45.8%

          \[\leadsto \frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{\frac{1}{\frac{D}{d}}} \cdot \frac{d}{D}}{h}\right) \]

       3. frac-times 46.9%

          \[\leadsto \frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{\frac{1 \cdot d}{\frac{D}{d} \cdot D}}}{h}\right) \]

       4. *-un-lft-identity 46.9%

          \[\leadsto \frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{\color{blue}{d}}{\frac{D}{d} \cdot D}}{h}\right) \]

   15. Applied egg-rr 46.9%

       \[\leadsto \frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\color{blue}{\frac{d}{\frac{D}{d} \cdot D}}}{h}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -3.6 \cdot 10^{+232}:\\ \;\;\;\;0\\ \mathbf{elif}\;c0 \leq -3.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right)\\ \mathbf{elif}\;c0 \leq 7.5 \cdot 10^{-95}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w} \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D \cdot \frac{D}{d}}}{h}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.3% 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 23.5%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Step-by-step derivation

   1. +-commutative 23.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\right)} \]

   2. +-commutative 23.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)} \]

   3. times-frac 23.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   4. fma-neg 23.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\color{blue}{\mathsf{fma}\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, -M \cdot M\right)}}\right) \]

3. Simplified 23.9%

   \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) - M \cdot M}\right)} \]
4. Add Preprocessing

5. Taylor expanded in c0 around -inf 3.0%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* 3.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(-1 \cdot c0\right) \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)} \]

   2. neg-mul-1 3.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(-c0\right)} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

   3. distribute-lft1-in 3.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}\right) \]

   4. metadata-eval 3.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right) \]

   5. mul0-lft 30.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-c0\right) \cdot \color{blue}{0}\right) \]

   6. distribute-lft-neg-in 30.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-c0 \cdot 0\right)} \]

   7. distribute-rgt-neg-in 30.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot \left(-0\right)\right)} \]

   8. metadata-eval 30.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \color{blue}{0}\right) \]

7. Simplified 30.7%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(c0 \cdot 0\right)} \]

8. Taylor expanded in c0 around 0 34.9%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 34.9%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))))