Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.1% → 55.3%

Time: 28.6s

Alternatives: 6

Speedup: 30.2×

• could not determine a ground truth

  1. c0 = 3.5077146241479414e+211
  2. w = 5.481449434721425e-291
  3. h = -4.797416469692599e-254
  4. D = -5.271529079666494e-152
  5. d = -7.507822258369487e+300
  6. M = -3.544394125361695e+21

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (w * h) * (D * D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / t_0;
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * ((2.0 * (d * (c0 * d))) / t_0);
	} else {
		tmp = -0.5 * (0.0 / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (w * h) * (D * D)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / t_0
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * ((2.0 * (d * (c0 * d))) / t_0)
	else:
		tmp = -0.5 * (0.0 / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(w * h) * Float64(D * D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / t_0)
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(2.0 * Float64(d * Float64(c0 * d))) / t_0));
	else
		tmp = Float64(-0.5 * Float64(0.0 / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (w * h) * (D * D);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / t_0;
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * ((2.0 * (d * (c0 * d))) / t_0);
	else
		tmp = -0.5 * (0.0 / w);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.8%

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

      1. times-frac 72.4%

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

      2. fma-def 70.3%

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

      3. associate-/r* 70.5%

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

      4. difference-of-squares 70.5%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in c0 around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. unpow2 74.0%

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

      3. associate-*r/ 73.0%

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

      4. unpow2 73.0%

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

      5. unpow2 73.0%

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

      6. *-commutative 73.0%

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

      7. unpow2 73.0%

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

   6. Simplified 73.0%

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

   7. Taylor expanded in c0 around 0 79.2%

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

      1. associate-*r/ 79.2%

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

      2. unpow2 79.2%

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

      3. associate-*l* 80.8%

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

      4. unpow2 80.8%

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

   9. Simplified 80.8%

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

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

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. difference-of-squares 7.3%

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

      3. associate-*l* 7.3%

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

      4. associate-*l* 7.9%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in c0 around -inf 0.7%

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

      1. *-commutative 0.7%

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

      2. unpow2 0.7%

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

      3. distribute-rgt1-in 0.7%

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

      4. metadata-eval 0.7%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w} \]

      5. mul0-lft 34.9%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w} \]

   6. Simplified 34.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]

   7. Taylor expanded in c0 around 0 43.2%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

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

Alternative 2: 37.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{0}{w}\\ t_1 := \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{if}\;c0 \leq -8.2 \cdot 10^{+196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c0 \leq -3.55 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq -2.2 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c0 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq -5.2 \cdot 10^{-120} \lor \neg \left(c0 \leq 32500000000\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{0 \cdot \left(c0 \cdot c0\right)}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ 0.0 w)))
        (t_1 (* (* (/ d D) (/ d D)) (/ (* c0 c0) (* h (* w w))))))
   (if (<= c0 -8.2e+196)
     t_1
     (if (<= c0 -3.55e+90)
       t_0
       (if (<= c0 -2.2e+39)
         t_1
         (if (<= c0 -5e-50)
           t_0
           (if (or (<= c0 -5.2e-120) (not (<= c0 32500000000.0)))
             t_1
             (* -0.5 (/ (* 0.0 (* c0 c0)) w)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -0.5 * (0.0 / w);
	double t_1 = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	double tmp;
	if (c0 <= -8.2e+196) {
		tmp = t_1;
	} else if (c0 <= -3.55e+90) {
		tmp = t_0;
	} else if (c0 <= -2.2e+39) {
		tmp = t_1;
	} else if (c0 <= -5e-50) {
		tmp = t_0;
	} else if ((c0 <= -5.2e-120) || !(c0 <= 32500000000.0)) {
		tmp = t_1;
	} else {
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) * (0.0d0 / w)
    t_1 = ((d_1 / d) * (d_1 / d)) * ((c0 * c0) / (h * (w * w)))
    if (c0 <= (-8.2d+196)) then
        tmp = t_1
    else if (c0 <= (-3.55d+90)) then
        tmp = t_0
    else if (c0 <= (-2.2d+39)) then
        tmp = t_1
    else if (c0 <= (-5d-50)) then
        tmp = t_0
    else if ((c0 <= (-5.2d-120)) .or. (.not. (c0 <= 32500000000.0d0))) then
        tmp = t_1
    else
        tmp = (-0.5d0) * ((0.0d0 * (c0 * c0)) / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -0.5 * (0.0 / w);
	double t_1 = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	double tmp;
	if (c0 <= -8.2e+196) {
		tmp = t_1;
	} else if (c0 <= -3.55e+90) {
		tmp = t_0;
	} else if (c0 <= -2.2e+39) {
		tmp = t_1;
	} else if (c0 <= -5e-50) {
		tmp = t_0;
	} else if ((c0 <= -5.2e-120) || !(c0 <= 32500000000.0)) {
		tmp = t_1;
	} else {
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = -0.5 * (0.0 / w)
	t_1 = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)))
	tmp = 0
	if c0 <= -8.2e+196:
		tmp = t_1
	elif c0 <= -3.55e+90:
		tmp = t_0
	elif c0 <= -2.2e+39:
		tmp = t_1
	elif c0 <= -5e-50:
		tmp = t_0
	elif (c0 <= -5.2e-120) or not (c0 <= 32500000000.0):
		tmp = t_1
	else:
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(-0.5 * Float64(0.0 / w))
	t_1 = Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(Float64(c0 * c0) / Float64(h * Float64(w * w))))
	tmp = 0.0
	if (c0 <= -8.2e+196)
		tmp = t_1;
	elseif (c0 <= -3.55e+90)
		tmp = t_0;
	elseif (c0 <= -2.2e+39)
		tmp = t_1;
	elseif (c0 <= -5e-50)
		tmp = t_0;
	elseif ((c0 <= -5.2e-120) || !(c0 <= 32500000000.0))
		tmp = t_1;
	else
		tmp = Float64(-0.5 * Float64(Float64(0.0 * Float64(c0 * c0)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = -0.5 * (0.0 / w);
	t_1 = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	tmp = 0.0;
	if (c0 <= -8.2e+196)
		tmp = t_1;
	elseif (c0 <= -3.55e+90)
		tmp = t_0;
	elseif (c0 <= -2.2e+39)
		tmp = t_1;
	elseif (c0 <= -5e-50)
		tmp = t_0;
	elseif ((c0 <= -5.2e-120) || ~((c0 <= 32500000000.0)))
		tmp = t_1;
	else
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(-0.5 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * c0), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -8.2e+196], t$95$1, If[LessEqual[c0, -3.55e+90], t$95$0, If[LessEqual[c0, -2.2e+39], t$95$1, If[LessEqual[c0, -5e-50], t$95$0, If[Or[LessEqual[c0, -5.2e-120], N[Not[LessEqual[c0, 32500000000.0]], $MachinePrecision]], t$95$1, N[(-0.5 * N[(N[(0.0 * N[(c0 * c0), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -8.1999999999999993e196 or -3.54999999999999988e90 < c0 < -2.2000000000000001e39 or -4.99999999999999968e-50 < c0 < -5.2000000000000002e-120 or 3.25e10 < c0

   1. Initial program 35.2%

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

      1. times-frac 32.9%

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

      2. fma-def 32.2%

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

      3. associate-/r* 32.2%

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

      4. difference-of-squares 39.1%

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

   3. Simplified 44.7%

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

      1. fma-udef 45.4%

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

      2. associate-/l/ 40.8%

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

      3. frac-times 44.7%

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

      4. pow2 44.7%

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

      5. fma-udef 44.7%

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

      6. associate-/l/ 39.9%

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

      7. times-frac 39.2%

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

      8. associate-/l/ 39.1%

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

      9. times-frac 39.1%

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

   5. Applied egg-rr 42.9%

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

   6. Taylor expanded in c0 around inf 40.0%

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

      1. times-frac 37.1%

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

      2. unpow2 37.1%

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

      3. associate-/r* 42.5%

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

      4. unpow2 42.5%

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

      5. associate-*r/ 49.3%

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

      6. associate-*l/ 49.3%

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

      7. unpow2 49.3%

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

      8. unpow2 49.3%

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

      9. *-commutative 49.3%

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

      10. unpow2 49.3%

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

   8. Simplified 49.3%

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

      1. unpow2 49.3%

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

   10. Applied egg-rr 49.3%

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

   if -8.1999999999999993e196 < c0 < -3.54999999999999988e90 or -2.2000000000000001e39 < c0 < -4.99999999999999968e-50

   1. Initial program 18.8%

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

      1. associate-*l* 14.6%

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

      2. difference-of-squares 16.6%

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

      3. associate-*l* 16.6%

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

      4. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in c0 around -inf 2.2%

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

      1. *-commutative 2.2%

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

      2. unpow2 2.2%

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

      3. distribute-rgt1-in 2.2%

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

      4. metadata-eval 2.2%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w} \]

      5. mul0-lft 33.5%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w} \]

   6. Simplified 33.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]

   7. Taylor expanded in c0 around 0 44.4%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{w} \]

   if -5.2000000000000002e-120 < c0 < 3.25e10

   1. Initial program 19.0%

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

      1. associate-*l* 17.6%

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

      2. difference-of-squares 20.2%

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

      3. associate-*l* 20.3%

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

      4. associate-*l* 20.5%

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

   3. Simplified 20.5%

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

   4. Taylor expanded in c0 around -inf 7.0%

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

      1. *-commutative 7.0%

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

      2. unpow2 7.0%

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

      3. distribute-rgt1-in 7.0%

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

      4. metadata-eval 7.0%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w} \]

      5. mul0-lft 49.7%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w} \]

   6. Simplified 49.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -8.2 \cdot 10^{+196}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;c0 \leq -3.55 \cdot 10^{+90}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{elif}\;c0 \leq -2.2 \cdot 10^{+39}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;c0 \leq -5 \cdot 10^{-50}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{elif}\;c0 \leq -5.2 \cdot 10^{-120} \lor \neg \left(c0 \leq 32500000000\right):\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{0 \cdot \left(c0 \cdot c0\right)}{w}\\ \end{array} \]

Alternative 3: 37.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(w \cdot w\right)\\ t_1 := -0.5 \cdot \frac{0}{w}\\ t_2 := \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{t_0}\\ \mathbf{if}\;c0 \leq -3.3 \cdot 10^{+201}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq -1.15 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c0 \leq -3.7 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c0 \leq -1.75 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c0 \leq -1.3 \cdot 10^{-118}:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot t_0}\\ \mathbf{elif}\;c0 \leq 440000000000:\\ \;\;\;\;-0.5 \cdot \frac{0 \cdot \left(c0 \cdot c0\right)}{w}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* w w)))
        (t_1 (* -0.5 (/ 0.0 w)))
        (t_2 (* (* (/ d D) (/ d D)) (/ (* c0 c0) t_0))))
   (if (<= c0 -3.3e+201)
     t_2
     (if (<= c0 -1.15e+90)
       t_1
       (if (<= c0 -3.7e+33)
         t_2
         (if (<= c0 -1.75e-51)
           t_1
           (if (<= c0 -1.3e-118)
             (/ (* (* d d) (* c0 c0)) (* (* D D) t_0))
             (if (<= c0 440000000000.0)
               (* -0.5 (/ (* 0.0 (* c0 c0)) w))
               t_2))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (w * w);
	double t_1 = -0.5 * (0.0 / w);
	double t_2 = ((d / D) * (d / D)) * ((c0 * c0) / t_0);
	double tmp;
	if (c0 <= -3.3e+201) {
		tmp = t_2;
	} else if (c0 <= -1.15e+90) {
		tmp = t_1;
	} else if (c0 <= -3.7e+33) {
		tmp = t_2;
	} else if (c0 <= -1.75e-51) {
		tmp = t_1;
	} else if (c0 <= -1.3e-118) {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * t_0);
	} else if (c0 <= 440000000000.0) {
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = h * (w * w)
    t_1 = (-0.5d0) * (0.0d0 / w)
    t_2 = ((d_1 / d) * (d_1 / d)) * ((c0 * c0) / t_0)
    if (c0 <= (-3.3d+201)) then
        tmp = t_2
    else if (c0 <= (-1.15d+90)) then
        tmp = t_1
    else if (c0 <= (-3.7d+33)) then
        tmp = t_2
    else if (c0 <= (-1.75d-51)) then
        tmp = t_1
    else if (c0 <= (-1.3d-118)) then
        tmp = ((d_1 * d_1) * (c0 * c0)) / ((d * d) * t_0)
    else if (c0 <= 440000000000.0d0) then
        tmp = (-0.5d0) * ((0.0d0 * (c0 * c0)) / w)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (w * w);
	double t_1 = -0.5 * (0.0 / w);
	double t_2 = ((d / D) * (d / D)) * ((c0 * c0) / t_0);
	double tmp;
	if (c0 <= -3.3e+201) {
		tmp = t_2;
	} else if (c0 <= -1.15e+90) {
		tmp = t_1;
	} else if (c0 <= -3.7e+33) {
		tmp = t_2;
	} else if (c0 <= -1.75e-51) {
		tmp = t_1;
	} else if (c0 <= -1.3e-118) {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * t_0);
	} else if (c0 <= 440000000000.0) {
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (w * w)
	t_1 = -0.5 * (0.0 / w)
	t_2 = ((d / D) * (d / D)) * ((c0 * c0) / t_0)
	tmp = 0
	if c0 <= -3.3e+201:
		tmp = t_2
	elif c0 <= -1.15e+90:
		tmp = t_1
	elif c0 <= -3.7e+33:
		tmp = t_2
	elif c0 <= -1.75e-51:
		tmp = t_1
	elif c0 <= -1.3e-118:
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * t_0)
	elif c0 <= 440000000000.0:
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w)
	else:
		tmp = t_2
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(w * w))
	t_1 = Float64(-0.5 * Float64(0.0 / w))
	t_2 = Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(Float64(c0 * c0) / t_0))
	tmp = 0.0
	if (c0 <= -3.3e+201)
		tmp = t_2;
	elseif (c0 <= -1.15e+90)
		tmp = t_1;
	elseif (c0 <= -3.7e+33)
		tmp = t_2;
	elseif (c0 <= -1.75e-51)
		tmp = t_1;
	elseif (c0 <= -1.3e-118)
		tmp = Float64(Float64(Float64(d * d) * Float64(c0 * c0)) / Float64(Float64(D * D) * t_0));
	elseif (c0 <= 440000000000.0)
		tmp = Float64(-0.5 * Float64(Float64(0.0 * Float64(c0 * c0)) / w));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (w * w);
	t_1 = -0.5 * (0.0 / w);
	t_2 = ((d / D) * (d / D)) * ((c0 * c0) / t_0);
	tmp = 0.0;
	if (c0 <= -3.3e+201)
		tmp = t_2;
	elseif (c0 <= -1.15e+90)
		tmp = t_1;
	elseif (c0 <= -3.7e+33)
		tmp = t_2;
	elseif (c0 <= -1.75e-51)
		tmp = t_1;
	elseif (c0 <= -1.3e-118)
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * t_0);
	elseif (c0 <= 440000000000.0)
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * c0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -3.3e+201], t$95$2, If[LessEqual[c0, -1.15e+90], t$95$1, If[LessEqual[c0, -3.7e+33], t$95$2, If[LessEqual[c0, -1.75e-51], t$95$1, If[LessEqual[c0, -1.3e-118], N[(N[(N[(d * d), $MachinePrecision] * N[(c0 * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 440000000000.0], N[(-0.5 * N[(N[(0.0 * N[(c0 * c0), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c0 < -3.3e201 or -1.15e90 < c0 < -3.6999999999999999e33 or 4.4e11 < c0

   1. Initial program 34.1%

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

      1. times-frac 31.5%

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

      2. fma-def 30.7%

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

      3. associate-/r* 30.7%

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

      4. difference-of-squares 35.9%

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

   3. Simplified 42.0%

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

      1. fma-udef 42.7%

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

      2. associate-/l/ 37.7%

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

      3. frac-times 41.9%

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

      4. pow2 41.9%

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

      5. fma-udef 41.9%

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

      6. associate-/l/ 36.8%

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

      7. times-frac 35.9%

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

      8. associate-/l/ 35.9%

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

      9. times-frac 35.9%

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

   5. Applied egg-rr 41.7%

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

   6. Taylor expanded in c0 around inf 36.8%

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

      1. times-frac 34.4%

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

      2. unpow2 34.4%

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

      3. associate-/r* 40.4%

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

      4. unpow2 40.4%

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

      5. associate-*r/ 47.1%

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

      6. associate-*l/ 47.1%

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

      7. unpow2 47.1%

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

      8. unpow2 47.1%

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

      9. *-commutative 47.1%

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

      10. unpow2 47.1%

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

   8. Simplified 47.1%

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

      1. unpow2 47.1%

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

   10. Applied egg-rr 47.1%

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

   if -3.3e201 < c0 < -1.15e90 or -3.6999999999999999e33 < c0 < -1.7499999999999999e-51

   1. Initial program 18.8%

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

      1. associate-*l* 14.6%

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

      2. difference-of-squares 16.6%

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

      3. associate-*l* 16.6%

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

      4. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in c0 around -inf 2.2%

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

      1. *-commutative 2.2%

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

      2. unpow2 2.2%

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

      3. distribute-rgt1-in 2.2%

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

      4. metadata-eval 2.2%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w} \]

      5. mul0-lft 33.5%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w} \]

   6. Simplified 33.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]

   7. Taylor expanded in c0 around 0 44.4%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{w} \]

   if -1.7499999999999999e-51 < c0 < -1.3e-118

   1. Initial program 45.9%

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

      1. times-frac 45.9%

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

      2. fma-def 46.0%

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

      3. associate-/r* 46.0%

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

      4. difference-of-squares 69.1%

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

   3. Simplified 69.6%

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

      1. fma-udef 69.6%

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

      2. associate-/l/ 69.2%

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

      3. frac-times 70.4%

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

      4. pow2 70.4%

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

      5. fma-udef 70.4%

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

      6. associate-/l/ 69.0%

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

      7. times-frac 69.1%

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

      8. associate-/l/ 69.1%

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

      9. times-frac 69.1%

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

   5. Applied egg-rr 53.5%

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

   6. Taylor expanded in c0 around inf 69.6%

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

      1. times-frac 61.8%

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

      2. unpow2 61.8%

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

      3. unpow2 61.8%

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

      4. unpow2 61.8%

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

      5. *-commutative 61.8%

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

      6. unpow2 61.8%

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

   8. Simplified 61.8%

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

      1. frac-times 69.6%

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

   10. Applied egg-rr 69.6%

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

   if -1.3e-118 < c0 < 4.4e11

   1. Initial program 19.0%

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

      1. associate-*l* 17.6%

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

      2. difference-of-squares 20.2%

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

      3. associate-*l* 20.3%

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

      4. associate-*l* 20.5%

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

   3. Simplified 20.5%

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

   4. Taylor expanded in c0 around -inf 7.0%

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

      1. *-commutative 7.0%

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

      2. unpow2 7.0%

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

      3. distribute-rgt1-in 7.0%

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

      4. metadata-eval 7.0%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w} \]

      5. mul0-lft 49.7%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w} \]

   6. Simplified 49.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -3.3 \cdot 10^{+201}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;c0 \leq -1.15 \cdot 10^{+90}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{elif}\;c0 \leq -3.7 \cdot 10^{+33}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;c0 \leq -1.75 \cdot 10^{-51}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{elif}\;c0 \leq -1.3 \cdot 10^{-118}:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \mathbf{elif}\;c0 \leq 440000000000:\\ \;\;\;\;-0.5 \cdot \frac{0 \cdot \left(c0 \cdot c0\right)}{w}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \end{array} \]

Alternative 4: 43.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{if}\;c0 \leq -5.4 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq -3.5 \cdot 10^{-51}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{elif}\;c0 \leq -4.5 \cdot 10^{-128} \lor \neg \left(c0 \leq 5.5 \cdot 10^{+31}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{0 \cdot \left(c0 \cdot c0\right)}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0
         (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))))))
   (if (<= c0 -5.4e+29)
     t_0
     (if (<= c0 -3.5e-51)
       (* -0.5 (/ 0.0 w))
       (if (or (<= c0 -4.5e-128) (not (<= c0 5.5e+31)))
         t_0
         (* -0.5 (/ (* 0.0 (* c0 c0)) w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) / h) * ((d / D) * (d / D))));
	double tmp;
	if (c0 <= -5.4e+29) {
		tmp = t_0;
	} else if (c0 <= -3.5e-51) {
		tmp = -0.5 * (0.0 / w);
	} else if ((c0 <= -4.5e-128) || !(c0 <= 5.5e+31)) {
		tmp = t_0;
	} else {
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c0 / (2.0d0 * w)) * (2.0d0 * (((c0 / w) / h) * ((d_1 / d) * (d_1 / d))))
    if (c0 <= (-5.4d+29)) then
        tmp = t_0
    else if (c0 <= (-3.5d-51)) then
        tmp = (-0.5d0) * (0.0d0 / w)
    else if ((c0 <= (-4.5d-128)) .or. (.not. (c0 <= 5.5d+31))) then
        tmp = t_0
    else
        tmp = (-0.5d0) * ((0.0d0 * (c0 * c0)) / w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) / h) * ((d / D) * (d / D))));
	double tmp;
	if (c0 <= -5.4e+29) {
		tmp = t_0;
	} else if (c0 <= -3.5e-51) {
		tmp = -0.5 * (0.0 / w);
	} else if ((c0 <= -4.5e-128) || !(c0 <= 5.5e+31)) {
		tmp = t_0;
	} else {
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) / h) * ((d / D) * (d / D))))
	tmp = 0
	if c0 <= -5.4e+29:
		tmp = t_0
	elif c0 <= -3.5e-51:
		tmp = -0.5 * (0.0 / w)
	elif (c0 <= -4.5e-128) or not (c0 <= 5.5e+31):
		tmp = t_0
	else:
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(Float64(c0 / w) / h) * Float64(Float64(d / D) * Float64(d / D)))))
	tmp = 0.0
	if (c0 <= -5.4e+29)
		tmp = t_0;
	elseif (c0 <= -3.5e-51)
		tmp = Float64(-0.5 * Float64(0.0 / w));
	elseif ((c0 <= -4.5e-128) || !(c0 <= 5.5e+31))
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(Float64(0.0 * Float64(c0 * c0)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) / h) * ((d / D) * (d / D))));
	tmp = 0.0;
	if (c0 <= -5.4e+29)
		tmp = t_0;
	elseif (c0 <= -3.5e-51)
		tmp = -0.5 * (0.0 / w);
	elseif ((c0 <= -4.5e-128) || ~((c0 <= 5.5e+31)))
		tmp = t_0;
	else
		tmp = -0.5 * ((0.0 * (c0 * c0)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -5.4e+29], t$95$0, If[LessEqual[c0, -3.5e-51], N[(-0.5 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c0, -4.5e-128], N[Not[LessEqual[c0, 5.5e+31]], $MachinePrecision]], t$95$0, N[(-0.5 * N[(N[(0.0 * N[(c0 * c0), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -5.4e29 or -3.4999999999999997e-51 < c0 < -4.4999999999999999e-128 or 5.50000000000000002e31 < c0

   1. Initial program 32.1%

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

      1. times-frac 29.6%

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

      2. fma-def 29.6%

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

      3. associate-/r* 29.6%

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

      4. difference-of-squares 36.0%

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

   3. Simplified 40.1%

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

   4. Taylor expanded in c0 around inf 38.2%

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

      1. *-commutative 38.2%

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

      2. unpow2 38.2%

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

      3. associate-*r/ 37.8%

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

      4. unpow2 37.8%

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

      5. unpow2 37.8%

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

      6. *-commutative 37.8%

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

      7. unpow2 37.8%

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

   6. Simplified 37.8%

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

   7. Taylor expanded in c0 around 0 40.7%

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

      1. times-frac 38.9%

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

      2. unpow2 38.9%

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

      3. associate-/r* 44.1%

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

      4. unpow2 44.1%

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

      5. associate-*l/ 50.9%

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

      6. associate-/r/ 50.9%

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

      7. *-commutative 50.9%

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

      8. associate-/r* 51.2%

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

      9. associate-/r/ 51.2%

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

      10. associate-*r/ 51.8%

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

      11. unpow2 51.8%

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

   9. Simplified 51.8%

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

       1. unpow2 44.6%

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

   11. Applied egg-rr 51.8%

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

   if -5.4e29 < c0 < -3.4999999999999997e-51

   1. Initial program 17.7%

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

      1. associate-*l* 6.0%

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

      2. difference-of-squares 6.0%

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

      3. associate-*l* 6.0%

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

      4. associate-*l* 6.0%

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

   3. Simplified 6.0%

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

   4. Taylor expanded in c0 around -inf 5.9%

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

      1. *-commutative 5.9%

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

      2. unpow2 5.9%

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

      3. distribute-rgt1-in 5.9%

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

      4. metadata-eval 5.9%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w} \]

      5. mul0-lft 54.4%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w} \]

   6. Simplified 54.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]

   7. Taylor expanded in c0 around 0 54.4%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{w} \]

   if -4.4999999999999999e-128 < c0 < 5.50000000000000002e31

   1. Initial program 20.1%

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

      1. associate-*l* 18.8%

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

      2. difference-of-squares 21.2%

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

      3. associate-*l* 21.2%

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

      4. associate-*l* 21.5%

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

   3. Simplified 21.5%

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

   4. Taylor expanded in c0 around -inf 7.8%

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

      1. *-commutative 7.8%

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

      2. unpow2 7.8%

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

      3. distribute-rgt1-in 7.8%

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

      4. metadata-eval 7.8%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w} \]

      5. mul0-lft 48.6%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w} \]

   6. Simplified 48.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -5.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{elif}\;c0 \leq -3.5 \cdot 10^{-51}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{elif}\;c0 \leq -4.5 \cdot 10^{-128} \lor \neg \left(c0 \leq 5.5 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{0 \cdot \left(c0 \cdot c0\right)}{w}\\ \end{array} \]

Alternative 5: 38.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{-107}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* D D) 1e-107)
   (* -0.5 (/ 0.0 w))
   (if (<= (* D D) 5e+301)
     (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ (* d d) (* D D)) (/ c0 (* w h)))))
     (* (* (/ d D) (/ d D)) (/ (* c0 c0) (* h (* w w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 1e-107) {
		tmp = -0.5 * (0.0 / w);
	} else if ((D * D) <= 5e+301) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d * d) / (D * D)) * (c0 / (w * h))));
	} else {
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	}
	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 * d) <= 1d-107) then
        tmp = (-0.5d0) * (0.0d0 / w)
    else if ((d * d) <= 5d+301) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (((d_1 * d_1) / (d * d)) * (c0 / (w * h))))
    else
        tmp = ((d_1 / d) * (d_1 / d)) * ((c0 * c0) / (h * (w * w)))
    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 * D) <= 1e-107) {
		tmp = -0.5 * (0.0 / w);
	} else if ((D * D) <= 5e+301) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d * d) / (D * D)) * (c0 / (w * h))));
	} else {
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (D * D) <= 1e-107:
		tmp = -0.5 * (0.0 / w)
	elif (D * D) <= 5e+301:
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d * d) / (D * D)) * (c0 / (w * h))))
	else:
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(D * D) <= 1e-107)
		tmp = Float64(-0.5 * Float64(0.0 / w));
	elseif (Float64(D * D) <= 5e+301)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(Float64(d * d) / Float64(D * D)) * Float64(c0 / Float64(w * h)))));
	else
		tmp = Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(Float64(c0 * c0) / Float64(h * Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((D * D) <= 1e-107)
		tmp = -0.5 * (0.0 / w);
	elseif ((D * D) <= 5e+301)
		tmp = (c0 / (2.0 * w)) * (2.0 * (((d * d) / (D * D)) * (c0 / (w * h))));
	else
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(D * D), $MachinePrecision], 1e-107], N[(-0.5 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * D), $MachinePrecision], 5e+301], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * c0), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 D D) < 1e-107

   1. Initial program 24.0%

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

      1. associate-*l* 23.3%

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

      2. difference-of-squares 29.5%

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

      3. associate-*l* 29.5%

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

      4. associate-*l* 30.3%

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

   3. Simplified 30.3%

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

   4. Taylor expanded in c0 around -inf 4.2%

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

      1. *-commutative 4.2%

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

      2. unpow2 4.2%

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

      3. distribute-rgt1-in 4.2%

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

      4. metadata-eval 4.2%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w} \]

      5. mul0-lft 34.4%

         \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w} \]

   6. Simplified 34.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]

   7. Taylor expanded in c0 around 0 39.5%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{w} \]

   if 1e-107 < (*.f64 D D) < 5.0000000000000004e301

   1. Initial program 41.7%

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

      1. times-frac 37.1%

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

      2. fma-def 37.2%

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

      3. times-frac 37.2%

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

      4. difference-of-squares 40.8%

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

   3. Simplified 40.8%

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

      1. sqrt-prod 42.0%

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

      2. associate-*l* 40.8%

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

      3. div-inv 40.8%

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

      4. clear-num 40.8%

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

      5. associate-*r/ 40.8%

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

      6. *-commutative 40.8%

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

      7. times-frac 42.3%

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

      8. associate-/l/ 42.3%

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

      9. fma-neg 42.3%

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

      10. associate-/l/ 42.3%

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

      11. frac-times 42.2%

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

      12. pow2 42.2%

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

   5. Applied egg-rr 42.2%

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

   6. Taylor expanded in c0 around inf 47.9%

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

      1. times-frac 46.7%

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

      2. unpow2 46.7%

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

      3. unpow2 46.7%

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

   8. Simplified 46.7%

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

   if 5.0000000000000004e301 < (*.f64 D D)

   1. Initial program 0.2%

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

      1. times-frac 0.2%

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

      2. fma-def 0.2%

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

      3. associate-/r* 0.7%

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

      4. difference-of-squares 0.7%

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

   3. Simplified 28.3%

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

      1. fma-udef 28.2%

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

      2. associate-/l/ 1.5%

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

      3. frac-times 28.4%

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

      4. pow2 28.4%

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

      5. fma-udef 28.4%

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

      6. associate-/l/ 0.1%

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

      7. times-frac 0.3%

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

      8. associate-/l/ 0.7%

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

      9. times-frac 0.7%

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

   5. Applied egg-rr 39.5%

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

   6. Taylor expanded in c0 around inf 0.2%

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

      1. times-frac 0.4%

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

      2. unpow2 0.4%

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

      3. associate-/r* 32.7%

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

      4. unpow2 32.7%

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

      5. associate-*r/ 43.8%

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

      6. associate-*l/ 48.1%

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

      7. unpow2 48.1%

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

      8. unpow2 48.1%

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

      9. *-commutative 48.1%

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

      10. unpow2 48.1%

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

   8. Simplified 48.1%

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

      1. unpow2 48.1%

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

   10. Applied egg-rr 48.1%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 10^{-107}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{elif}\;D \cdot D \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{d \cdot d}{D \cdot D} \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \end{array} \]

Alternative 6: 33.7% accurate, 30.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{0}{w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* -0.5 (/ 0.0 w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return -0.5 * (0.0 / w);
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (-0.5d0) * (0.0d0 / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return -0.5 * (0.0 / w);
}

Python

def code(c0, w, h, D, d, M):
	return -0.5 * (0.0 / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(-0.5 * Float64(0.0 / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = -0.5 * (0.0 / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(-0.5 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.3%

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

   1. associate-*l* 26.1%

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

   2. difference-of-squares 30.8%

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

   3. associate-*l* 30.8%

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

   4. associate-*l* 31.3%

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

3. Simplified 31.3%

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

4. Taylor expanded in c0 around -inf 3.8%

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

   1. *-commutative 3.8%

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

   2. unpow2 3.8%

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

   3. distribute-rgt1-in 3.8%

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

   4. metadata-eval 3.8%

      \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w} \]

   5. mul0-lft 26.2%

      \[\leadsto -0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w} \]

6. Simplified 26.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot 0}{w}} \]

7. Taylor expanded in c0 around 0 32.2%

   \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{w} \]

8. Final simplification 32.2%

   \[\leadsto -0.5 \cdot \frac{0}{w} \]

Reproduce

herbie shell --seed 2023178 
(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))))))