Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 61.1%

Time: 29.0s

Alternatives: 10

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = 7.633999776702731e+87
  2. w = -2.430120636094553e-307
  3. h = 1.9715377433787395e-230
  4. D = -4.9999725891039704e-291
  5. d = 1.6038699175221354e+275
  6. M = 7.303165780285422e-56

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.5% 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: 61.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right) \cdot t_1\\ t_3 := t_1 \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + t_1 \cdot \frac{d \cdot \frac{d}{D}}{\frac{D}{c0} \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;t_2 \leq -0.2:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot \left(\frac{D}{\frac{c0}{D}} \cdot \left(\frac{h}{d} \cdot \frac{w \cdot \left(M \cdot M\right)}{d}\right)\right)\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (* (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))) t_1))
        (t_3
         (+
          (* t_1 (* (/ c0 (* w h)) (pow (/ d D) 2.0)))
          (* t_1 (/ (* d (/ d D)) (* (/ D c0) (* w h)))))))
   (if (<= t_2 -0.2)
     t_3
     (if (<= t_2 0.0)
       (* t_1 (* 0.5 (* (/ D (/ c0 D)) (* (/ h d) (/ (* w (* M M)) d)))))
       (if (<= t_2 INFINITY)
         t_3
         (* 0.25 (* (* (/ D d) (/ D d)) (* h (* M M)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (t_0 + sqrt(((t_0 * t_0) - (M * M)))) * t_1;
	double t_3 = (t_1 * ((c0 / (w * h)) * pow((d / D), 2.0))) + (t_1 * ((d * (d / D)) / ((D / c0) * (w * h))));
	double tmp;
	if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = t_1 * (0.5 * ((D / (c0 / D)) * ((h / d) * ((w * (M * M)) / d))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = 0.25 * (((D / d) * (D / d)) * (h * (M * M)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (t_0 + Math.sqrt(((t_0 * t_0) - (M * M)))) * t_1;
	double t_3 = (t_1 * ((c0 / (w * h)) * Math.pow((d / D), 2.0))) + (t_1 * ((d * (d / D)) / ((D / c0) * (w * h))));
	double tmp;
	if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = t_1 * (0.5 * ((D / (c0 / D)) * ((h / d) * ((w * (M * M)) / d))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = 0.25 * (((D / d) * (D / d)) * (h * (M * M)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_1 = c0 / (2.0 * w)
	t_2 = (t_0 + math.sqrt(((t_0 * t_0) - (M * M)))) * t_1
	t_3 = (t_1 * ((c0 / (w * h)) * math.pow((d / D), 2.0))) + (t_1 * ((d * (d / D)) / ((D / c0) * (w * h))))
	tmp = 0
	if t_2 <= -0.2:
		tmp = t_3
	elif t_2 <= 0.0:
		tmp = t_1 * (0.5 * ((D / (c0 / D)) * ((h / d) * ((w * (M * M)) / d))))
	elif t_2 <= math.inf:
		tmp = t_3
	else:
		tmp = 0.25 * (((D / d) * (D / d)) * (h * (M * M)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))) * t_1)
	t_3 = Float64(Float64(t_1 * Float64(Float64(c0 / Float64(w * h)) * (Float64(d / D) ^ 2.0))) + Float64(t_1 * Float64(Float64(d * Float64(d / D)) / Float64(Float64(D / c0) * Float64(w * h)))))
	tmp = 0.0
	if (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(t_1 * Float64(0.5 * Float64(Float64(D / Float64(c0 / D)) * Float64(Float64(h / d) * Float64(Float64(w * Float64(M * M)) / d)))));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(M * M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	t_1 = c0 / (2.0 * w);
	t_2 = (t_0 + sqrt(((t_0 * t_0) - (M * M)))) * t_1;
	t_3 = (t_1 * ((c0 / (w * h)) * ((d / D) ^ 2.0))) + (t_1 * ((d * (d / D)) / ((D / c0) * (w * h))));
	tmp = 0.0;
	if (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = t_1 * (0.5 * ((D / (c0 / D)) * ((h / d) * ((w * (M * M)) / d))));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = 0.25 * (((D / d) * (D / d)) * (h * (M * M)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(N[(D / c0), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.2], t$95$3, If[LessEqual[t$95$2, 0.0], N[(t$95$1 * N[(0.5 * N[(N[(D / N[(c0 / D), $MachinePrecision]), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(N[(w * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 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))))) < -0.20000000000000001 or 0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 81.3%

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

   3. Simplified 81.4%

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

      1. distribute-lft-in 81.4%

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

   5. Applied egg-rr 83.7%

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

   6. Taylor expanded in c0 around inf 81.4%

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

      1. unpow2 81.4%

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

      2. unpow2 81.4%

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

      3. *-commutative 81.4%

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

      4. associate-*r* 83.8%

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

      5. times-frac 83.6%

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

      6. times-frac 83.7%

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

      7. *-commutative 83.7%

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

   8. Simplified 83.7%

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

      1. clear-num 83.7%

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

      2. associate-*r/ 84.7%

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

      3. frac-times 88.3%

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

      4. *-un-lft-identity 88.3%

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

   10. Applied egg-rr 88.3%

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

   if -0.20000000000000001 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 0.0

   1. Initial program 52.3%

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

   3. Simplified 24.3%

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

   4. Taylor expanded in c0 around -inf 22.2%

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

      1. +-commutative 22.2%

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

      2. mul-1-neg 22.2%

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

      3. unsub-neg 22.2%

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

      4. unpow2 22.2%

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

      5. *-commutative 22.2%

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

      6. unpow2 22.2%

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

      7. unpow2 22.2%

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

      8. associate-*r* 15.2%

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

      9. *-commutative 15.2%

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

      10. unpow2 15.2%

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

      11. *-commutative 15.2%

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

   6. Simplified 15.2%

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

   7. Taylor expanded in c0 around 0 50.6%

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

      1. unpow2 50.6%

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

      2. times-frac 64.6%

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

      3. unpow2 64.6%

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

      4. *-commutative 64.6%

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

      5. *-commutative 64.6%

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

      6. *-commutative 64.6%

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

      7. associate-*r* 64.6%

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

      8. unpow2 64.6%

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

      9. associate-/l* 64.9%

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

      10. times-frac 80.0%

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

   9. Simplified 80.0%

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

   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%

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

   3. Simplified 4.7%

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

   4. Taylor expanded in c0 around -inf 0.1%

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

      1. +-commutative 0.1%

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

      2. mul-1-neg 0.1%

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

      3. unsub-neg 0.1%

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

      4. unpow2 0.1%

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

      5. *-commutative 0.1%

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

      6. unpow2 0.1%

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

      7. unpow2 0.1%

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

      8. associate-*r* 0.1%

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

      9. *-commutative 0.1%

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

      10. unpow2 0.1%

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

      11. *-commutative 0.1%

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

   6. Simplified 0.9%

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

   7. Taylor expanded in c0 around 0 43.0%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 43.0%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 42.9%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 42.2%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 42.2%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 44.5%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 44.5%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 44.5%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 44.5%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]

   10. Taylor expanded in d around 0 44.5%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{{d}^{2}}{D}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 44.5%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{\color{blue}{d \cdot d}}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       2. associate-*l/ 53.5%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{D} \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       3. associate-/r/ 53.4%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   12. Simplified 53.4%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   13. Step-by-step derivation

       1. associate-/r/ 57.4%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   14. Applied egg-rr 57.4%

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \cdot \frac{c0}{2 \cdot w} \leq -0.2:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + \frac{c0}{2 \cdot w} \cdot \frac{d \cdot \frac{d}{D}}{\frac{D}{c0} \cdot \left(w \cdot h\right)}\\ \mathbf{elif}\;\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \cdot \frac{c0}{2 \cdot w} \leq 0:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(0.5 \cdot \left(\frac{D}{\frac{c0}{D}} \cdot \left(\frac{h}{d} \cdot \frac{w \cdot \left(M \cdot M\right)}{d}\right)\right)\right)\\ \mathbf{elif}\;\left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \cdot \frac{c0}{2 \cdot w} \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + \frac{c0}{2 \cdot w} \cdot \frac{d \cdot \frac{d}{D}}{\frac{D}{c0} \cdot \left(w \cdot h\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\\ \end{array} \]

Alternative 2: 38.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := h \cdot \left(M \cdot M\right)\\ t_2 := 0.25 \cdot \frac{D \cdot t_1}{\frac{d}{\frac{D}{d}}}\\ t_3 := t_0 \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + t_0 \cdot \left(\frac{c0}{D} \cdot \left(\frac{d}{D} \cdot \frac{d}{w \cdot h}\right)\right)\\ t_4 := t_0 \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \mathbf{if}\;M \leq 2.7 \cdot 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.05 \cdot 10^{-71}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;M \leq 7.5 \cdot 10^{-42}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot t_1\right)\\ \mathbf{elif}\;M \leq 2.35 \cdot 10^{-6}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;M \leq 5.5 \cdot 10^{+36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;M \leq 1.35 \cdot 10^{+61}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;M \leq 8.8 \cdot 10^{+145}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* h (* M M)))
        (t_2 (* 0.25 (/ (* D t_1) (/ d (/ D d)))))
        (t_3
         (+
          (* t_0 (* (/ c0 (* w h)) (pow (/ d D) 2.0)))
          (* t_0 (* (/ c0 D) (* (/ d D) (/ d (* w h)))))))
        (t_4 (* t_0 (* 2.0 (* (/ c0 (* D D)) (/ (* d d) (* w h)))))))
   (if (<= M 2.7e-227)
     0.0
     (if (<= M 1.05e-71)
       t_3
       (if (<= M 7.5e-42)
         (* 0.25 (* (* (/ D d) (/ D d)) t_1))
         (if (<= M 2.35e-6)
           t_4
           (if (<= M 5.5e+36)
             t_2
             (if (<= M 1.35e+61) t_4 (if (<= M 8.8e+145) t_2 t_3)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = h * (M * M);
	double t_2 = 0.25 * ((D * t_1) / (d / (D / d)));
	double t_3 = (t_0 * ((c0 / (w * h)) * pow((d / D), 2.0))) + (t_0 * ((c0 / D) * ((d / D) * (d / (w * h)))));
	double t_4 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	double tmp;
	if (M <= 2.7e-227) {
		tmp = 0.0;
	} else if (M <= 1.05e-71) {
		tmp = t_3;
	} else if (M <= 7.5e-42) {
		tmp = 0.25 * (((D / d) * (D / d)) * t_1);
	} else if (M <= 2.35e-6) {
		tmp = t_4;
	} else if (M <= 5.5e+36) {
		tmp = t_2;
	} else if (M <= 1.35e+61) {
		tmp = t_4;
	} else if (M <= 8.8e+145) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    t_1 = h * (m * m)
    t_2 = 0.25d0 * ((d * t_1) / (d_1 / (d / d_1)))
    t_3 = (t_0 * ((c0 / (w * h)) * ((d_1 / d) ** 2.0d0))) + (t_0 * ((c0 / d) * ((d_1 / d) * (d_1 / (w * h)))))
    t_4 = t_0 * (2.0d0 * ((c0 / (d * d)) * ((d_1 * d_1) / (w * h))))
    if (m <= 2.7d-227) then
        tmp = 0.0d0
    else if (m <= 1.05d-71) then
        tmp = t_3
    else if (m <= 7.5d-42) then
        tmp = 0.25d0 * (((d / d_1) * (d / d_1)) * t_1)
    else if (m <= 2.35d-6) then
        tmp = t_4
    else if (m <= 5.5d+36) then
        tmp = t_2
    else if (m <= 1.35d+61) then
        tmp = t_4
    else if (m <= 8.8d+145) then
        tmp = t_2
    else
        tmp = t_3
    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);
	double t_1 = h * (M * M);
	double t_2 = 0.25 * ((D * t_1) / (d / (D / d)));
	double t_3 = (t_0 * ((c0 / (w * h)) * Math.pow((d / D), 2.0))) + (t_0 * ((c0 / D) * ((d / D) * (d / (w * h)))));
	double t_4 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	double tmp;
	if (M <= 2.7e-227) {
		tmp = 0.0;
	} else if (M <= 1.05e-71) {
		tmp = t_3;
	} else if (M <= 7.5e-42) {
		tmp = 0.25 * (((D / d) * (D / d)) * t_1);
	} else if (M <= 2.35e-6) {
		tmp = t_4;
	} else if (M <= 5.5e+36) {
		tmp = t_2;
	} else if (M <= 1.35e+61) {
		tmp = t_4;
	} else if (M <= 8.8e+145) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = h * (M * M)
	t_2 = 0.25 * ((D * t_1) / (d / (D / d)))
	t_3 = (t_0 * ((c0 / (w * h)) * math.pow((d / D), 2.0))) + (t_0 * ((c0 / D) * ((d / D) * (d / (w * h)))))
	t_4 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))))
	tmp = 0
	if M <= 2.7e-227:
		tmp = 0.0
	elif M <= 1.05e-71:
		tmp = t_3
	elif M <= 7.5e-42:
		tmp = 0.25 * (((D / d) * (D / d)) * t_1)
	elif M <= 2.35e-6:
		tmp = t_4
	elif M <= 5.5e+36:
		tmp = t_2
	elif M <= 1.35e+61:
		tmp = t_4
	elif M <= 8.8e+145:
		tmp = t_2
	else:
		tmp = t_3
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(h * Float64(M * M))
	t_2 = Float64(0.25 * Float64(Float64(D * t_1) / Float64(d / Float64(D / d))))
	t_3 = Float64(Float64(t_0 * Float64(Float64(c0 / Float64(w * h)) * (Float64(d / D) ^ 2.0))) + Float64(t_0 * Float64(Float64(c0 / D) * Float64(Float64(d / D) * Float64(d / Float64(w * h))))))
	t_4 = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 / Float64(D * D)) * Float64(Float64(d * d) / Float64(w * h)))))
	tmp = 0.0
	if (M <= 2.7e-227)
		tmp = 0.0;
	elseif (M <= 1.05e-71)
		tmp = t_3;
	elseif (M <= 7.5e-42)
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * t_1));
	elseif (M <= 2.35e-6)
		tmp = t_4;
	elseif (M <= 5.5e+36)
		tmp = t_2;
	elseif (M <= 1.35e+61)
		tmp = t_4;
	elseif (M <= 8.8e+145)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = h * (M * M);
	t_2 = 0.25 * ((D * t_1) / (d / (D / d)));
	t_3 = (t_0 * ((c0 / (w * h)) * ((d / D) ^ 2.0))) + (t_0 * ((c0 / D) * ((d / D) * (d / (w * h)))));
	t_4 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	tmp = 0.0;
	if (M <= 2.7e-227)
		tmp = 0.0;
	elseif (M <= 1.05e-71)
		tmp = t_3;
	elseif (M <= 7.5e-42)
		tmp = 0.25 * (((D / d) * (D / d)) * t_1);
	elseif (M <= 2.35e-6)
		tmp = t_4;
	elseif (M <= 5.5e+36)
		tmp = t_2;
	elseif (M <= 1.35e+61)
		tmp = t_4;
	elseif (M <= 8.8e+145)
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.25 * N[(N[(D * t$95$1), $MachinePrecision] / N[(d / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(c0 / D), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[(2.0 * N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 2.7e-227], 0.0, If[LessEqual[M, 1.05e-71], t$95$3, If[LessEqual[M, 7.5e-42], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 2.35e-6], t$95$4, If[LessEqual[M, 5.5e+36], t$95$2, If[LessEqual[M, 1.35e+61], t$95$4, If[LessEqual[M, 8.8e+145], t$95$2, t$95$3]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if M < 2.7e-227

   1. Initial program 28.2%

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

   3. Simplified 28.3%

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

   4. Taylor expanded in c0 around -inf 4.3%

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

      1. mul-1-neg 4.3%

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

      2. distribute-lft-in 4.3%

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

   6. Simplified 34.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   7. Taylor expanded in c0 around 0 39.6%

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

   if 2.7e-227 < M < 1.0500000000000001e-71 or 8.80000000000000035e145 < M

   1. Initial program 18.6%

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

   3. Simplified 23.8%

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

      1. distribute-lft-in 23.8%

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

   5. Applied egg-rr 37.5%

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

   6. Taylor expanded in c0 around inf 34.0%

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

      1. unpow2 34.0%

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

      2. unpow2 34.0%

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

      3. *-commutative 34.0%

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

      4. associate-*r* 40.9%

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

      5. times-frac 45.0%

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

      6. times-frac 51.4%

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

      7. *-commutative 51.4%

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

   8. Simplified 51.4%

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

   if 1.0500000000000001e-71 < M < 7.49999999999999972e-42

   1. Initial program 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. unsub-neg 0.0%

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

      4. unpow2 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. unpow2 0.0%

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

      8. associate-*r* 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. *-commutative 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in c0 around 0 34.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 34.3%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 34.3%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 34.3%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 34.3%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 35.8%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 35.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 35.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 35.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]

   10. Taylor expanded in d around 0 35.8%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{{d}^{2}}{D}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 35.8%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{\color{blue}{d \cdot d}}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       2. associate-*l/ 35.6%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{D} \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       3. associate-/r/ 35.6%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   12. Simplified 35.6%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   13. Step-by-step derivation

       1. associate-/r/ 51.1%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   14. Applied egg-rr 51.1%

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   if 7.49999999999999972e-42 < M < 2.34999999999999995e-6 or 5.5000000000000002e36 < M < 1.3500000000000001e61

   1. Initial program 77.0%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in c0 around inf 77.1%

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

      1. times-frac 83.7%

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

      2. unpow2 83.7%

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

      3. unpow2 83.7%

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

   6. Simplified 83.7%

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

   if 2.34999999999999995e-6 < M < 5.5000000000000002e36 or 1.3500000000000001e61 < M < 8.80000000000000035e145

   1. Initial program 22.3%

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

   3. Simplified 28.0%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. unsub-neg 0.0%

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

      4. unpow2 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. unpow2 0.0%

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

      8. associate-*r* 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. *-commutative 0.0%

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

   6. Simplified 1.1%

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

   7. Taylor expanded in c0 around 0 36.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 36.4%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 36.2%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 35.6%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 35.6%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 35.9%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 35.9%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 35.9%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 35.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 39.7%

          \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d \cdot d}{D}}} \]

       2. associate-/l* 45.9%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{\frac{d}{\frac{D}{d}}}} \]

   11. Applied egg-rr 45.9%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.7 \cdot 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.05 \cdot 10^{-71}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{D} \cdot \left(\frac{d}{D} \cdot \frac{d}{w \cdot h}\right)\right)\\ \mathbf{elif}\;M \leq 7.5 \cdot 10^{-42}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\\ \mathbf{elif}\;M \leq 2.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \mathbf{elif}\;M \leq 5.5 \cdot 10^{+36}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}\\ \mathbf{elif}\;M \leq 1.35 \cdot 10^{+61}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \mathbf{elif}\;M \leq 8.8 \cdot 10^{+145}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{D} \cdot \left(\frac{d}{D} \cdot \frac{d}{w \cdot h}\right)\right)\\ \end{array} \]

Alternative 3: 38.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{d}{w \cdot h}\\ t_2 := h \cdot \left(M \cdot M\right)\\ t_3 := 0.25 \cdot \frac{D \cdot t_2}{\frac{d}{\frac{D}{d}}}\\ t_4 := t_0 \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\\ t_5 := t_0 \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \mathbf{if}\;M \leq 2.7 \cdot 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.2 \cdot 10^{-71}:\\ \;\;\;\;t_4 + t_0 \cdot \left(\frac{\frac{d}{D}}{\frac{D}{c0}} \cdot t_1\right)\\ \mathbf{elif}\;M \leq 6 \cdot 10^{-42}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot t_2\right)\\ \mathbf{elif}\;M \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;M \leq 5.5 \cdot 10^{+41}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;M \leq 3.1 \cdot 10^{+63}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;M \leq 1.4 \cdot 10^{+145}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4 + t_0 \cdot \left(\frac{c0}{D} \cdot \left(\frac{d}{D} \cdot t_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ d (* w h)))
        (t_2 (* h (* M M)))
        (t_3 (* 0.25 (/ (* D t_2) (/ d (/ D d)))))
        (t_4 (* t_0 (* (/ c0 (* w h)) (pow (/ d D) 2.0))))
        (t_5 (* t_0 (* 2.0 (* (/ c0 (* D D)) (/ (* d d) (* w h)))))))
   (if (<= M 2.7e-227)
     0.0
     (if (<= M 1.2e-71)
       (+ t_4 (* t_0 (* (/ (/ d D) (/ D c0)) t_1)))
       (if (<= M 6e-42)
         (* 0.25 (* (* (/ D d) (/ D d)) t_2))
         (if (<= M 2.7e-6)
           t_5
           (if (<= M 5.5e+41)
             t_3
             (if (<= M 3.1e+63)
               t_5
               (if (<= M 1.4e+145)
                 t_3
                 (+ t_4 (* t_0 (* (/ c0 D) (* (/ d D) t_1)))))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = d / (w * h);
	double t_2 = h * (M * M);
	double t_3 = 0.25 * ((D * t_2) / (d / (D / d)));
	double t_4 = t_0 * ((c0 / (w * h)) * pow((d / D), 2.0));
	double t_5 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	double tmp;
	if (M <= 2.7e-227) {
		tmp = 0.0;
	} else if (M <= 1.2e-71) {
		tmp = t_4 + (t_0 * (((d / D) / (D / c0)) * t_1));
	} else if (M <= 6e-42) {
		tmp = 0.25 * (((D / d) * (D / d)) * t_2);
	} else if (M <= 2.7e-6) {
		tmp = t_5;
	} else if (M <= 5.5e+41) {
		tmp = t_3;
	} else if (M <= 3.1e+63) {
		tmp = t_5;
	} else if (M <= 1.4e+145) {
		tmp = t_3;
	} else {
		tmp = t_4 + (t_0 * ((c0 / D) * ((d / D) * t_1)));
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    t_1 = d_1 / (w * h)
    t_2 = h * (m * m)
    t_3 = 0.25d0 * ((d * t_2) / (d_1 / (d / d_1)))
    t_4 = t_0 * ((c0 / (w * h)) * ((d_1 / d) ** 2.0d0))
    t_5 = t_0 * (2.0d0 * ((c0 / (d * d)) * ((d_1 * d_1) / (w * h))))
    if (m <= 2.7d-227) then
        tmp = 0.0d0
    else if (m <= 1.2d-71) then
        tmp = t_4 + (t_0 * (((d_1 / d) / (d / c0)) * t_1))
    else if (m <= 6d-42) then
        tmp = 0.25d0 * (((d / d_1) * (d / d_1)) * t_2)
    else if (m <= 2.7d-6) then
        tmp = t_5
    else if (m <= 5.5d+41) then
        tmp = t_3
    else if (m <= 3.1d+63) then
        tmp = t_5
    else if (m <= 1.4d+145) then
        tmp = t_3
    else
        tmp = t_4 + (t_0 * ((c0 / d) * ((d_1 / d) * t_1)))
    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);
	double t_1 = d / (w * h);
	double t_2 = h * (M * M);
	double t_3 = 0.25 * ((D * t_2) / (d / (D / d)));
	double t_4 = t_0 * ((c0 / (w * h)) * Math.pow((d / D), 2.0));
	double t_5 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	double tmp;
	if (M <= 2.7e-227) {
		tmp = 0.0;
	} else if (M <= 1.2e-71) {
		tmp = t_4 + (t_0 * (((d / D) / (D / c0)) * t_1));
	} else if (M <= 6e-42) {
		tmp = 0.25 * (((D / d) * (D / d)) * t_2);
	} else if (M <= 2.7e-6) {
		tmp = t_5;
	} else if (M <= 5.5e+41) {
		tmp = t_3;
	} else if (M <= 3.1e+63) {
		tmp = t_5;
	} else if (M <= 1.4e+145) {
		tmp = t_3;
	} else {
		tmp = t_4 + (t_0 * ((c0 / D) * ((d / D) * t_1)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = d / (w * h)
	t_2 = h * (M * M)
	t_3 = 0.25 * ((D * t_2) / (d / (D / d)))
	t_4 = t_0 * ((c0 / (w * h)) * math.pow((d / D), 2.0))
	t_5 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))))
	tmp = 0
	if M <= 2.7e-227:
		tmp = 0.0
	elif M <= 1.2e-71:
		tmp = t_4 + (t_0 * (((d / D) / (D / c0)) * t_1))
	elif M <= 6e-42:
		tmp = 0.25 * (((D / d) * (D / d)) * t_2)
	elif M <= 2.7e-6:
		tmp = t_5
	elif M <= 5.5e+41:
		tmp = t_3
	elif M <= 3.1e+63:
		tmp = t_5
	elif M <= 1.4e+145:
		tmp = t_3
	else:
		tmp = t_4 + (t_0 * ((c0 / D) * ((d / D) * t_1)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(d / Float64(w * h))
	t_2 = Float64(h * Float64(M * M))
	t_3 = Float64(0.25 * Float64(Float64(D * t_2) / Float64(d / Float64(D / d))))
	t_4 = Float64(t_0 * Float64(Float64(c0 / Float64(w * h)) * (Float64(d / D) ^ 2.0)))
	t_5 = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 / Float64(D * D)) * Float64(Float64(d * d) / Float64(w * h)))))
	tmp = 0.0
	if (M <= 2.7e-227)
		tmp = 0.0;
	elseif (M <= 1.2e-71)
		tmp = Float64(t_4 + Float64(t_0 * Float64(Float64(Float64(d / D) / Float64(D / c0)) * t_1)));
	elseif (M <= 6e-42)
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * t_2));
	elseif (M <= 2.7e-6)
		tmp = t_5;
	elseif (M <= 5.5e+41)
		tmp = t_3;
	elseif (M <= 3.1e+63)
		tmp = t_5;
	elseif (M <= 1.4e+145)
		tmp = t_3;
	else
		tmp = Float64(t_4 + Float64(t_0 * Float64(Float64(c0 / D) * Float64(Float64(d / D) * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = d / (w * h);
	t_2 = h * (M * M);
	t_3 = 0.25 * ((D * t_2) / (d / (D / d)));
	t_4 = t_0 * ((c0 / (w * h)) * ((d / D) ^ 2.0));
	t_5 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	tmp = 0.0;
	if (M <= 2.7e-227)
		tmp = 0.0;
	elseif (M <= 1.2e-71)
		tmp = t_4 + (t_0 * (((d / D) / (D / c0)) * t_1));
	elseif (M <= 6e-42)
		tmp = 0.25 * (((D / d) * (D / d)) * t_2);
	elseif (M <= 2.7e-6)
		tmp = t_5;
	elseif (M <= 5.5e+41)
		tmp = t_3;
	elseif (M <= 3.1e+63)
		tmp = t_5;
	elseif (M <= 1.4e+145)
		tmp = t_3;
	else
		tmp = t_4 + (t_0 * ((c0 / D) * ((d / D) * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.25 * N[(N[(D * t$95$2), $MachinePrecision] / N[(d / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * N[(2.0 * N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 2.7e-227], 0.0, If[LessEqual[M, 1.2e-71], N[(t$95$4 + N[(t$95$0 * N[(N[(N[(d / D), $MachinePrecision] / N[(D / c0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 6e-42], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 2.7e-6], t$95$5, If[LessEqual[M, 5.5e+41], t$95$3, If[LessEqual[M, 3.1e+63], t$95$5, If[LessEqual[M, 1.4e+145], t$95$3, N[(t$95$4 + N[(t$95$0 * N[(N[(c0 / D), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if M < 2.7e-227

   1. Initial program 28.2%

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

   3. Simplified 28.3%

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

   4. Taylor expanded in c0 around -inf 4.3%

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

      1. mul-1-neg 4.3%

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

      2. distribute-lft-in 4.3%

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

   6. Simplified 34.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   7. Taylor expanded in c0 around 0 39.6%

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

   if 2.7e-227 < M < 1.2e-71

   1. Initial program 27.2%

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

   3. Simplified 35.6%

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

      1. distribute-lft-in 35.6%

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

   5. Applied egg-rr 57.6%

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

   6. Taylor expanded in c0 around inf 28.9%

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

      1. unpow2 28.9%

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

      2. unpow2 28.9%

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

      3. *-commutative 28.9%

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

      4. associate-*r* 37.2%

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

      5. times-frac 43.9%

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

      6. times-frac 51.4%

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

      7. *-commutative 51.4%

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

   8. Simplified 51.4%

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

      1. clear-num 51.4%

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

      2. associate-*r/ 53.7%

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

      3. frac-times 56.5%

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

      4. *-un-lft-identity 56.5%

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

   10. Applied egg-rr 56.5%

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

       1. times-frac 53.1%

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

   12. Applied egg-rr 53.1%

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

   if 1.2e-71 < M < 6.00000000000000054e-42

   1. Initial program 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. unsub-neg 0.0%

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

      4. unpow2 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. unpow2 0.0%

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

      8. associate-*r* 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. *-commutative 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in c0 around 0 34.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 34.3%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 34.3%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 34.3%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 34.3%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 35.8%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 35.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 35.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 35.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]

   10. Taylor expanded in d around 0 35.8%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{{d}^{2}}{D}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 35.8%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{\color{blue}{d \cdot d}}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       2. associate-*l/ 35.6%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{D} \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       3. associate-/r/ 35.6%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   12. Simplified 35.6%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   13. Step-by-step derivation

       1. associate-/r/ 51.1%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   14. Applied egg-rr 51.1%

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   if 6.00000000000000054e-42 < M < 2.69999999999999998e-6 or 5.5000000000000003e41 < M < 3.1000000000000001e63

   1. Initial program 77.0%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in c0 around inf 77.1%

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

      1. times-frac 83.7%

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

      2. unpow2 83.7%

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

      3. unpow2 83.7%

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

   6. Simplified 83.7%

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

   if 2.69999999999999998e-6 < M < 5.5000000000000003e41 or 3.1000000000000001e63 < M < 1.3999999999999999e145

   1. Initial program 22.3%

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

   3. Simplified 28.0%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. unsub-neg 0.0%

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

      4. unpow2 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. unpow2 0.0%

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

      8. associate-*r* 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. *-commutative 0.0%

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

   6. Simplified 1.1%

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

   7. Taylor expanded in c0 around 0 36.4%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 36.4%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 36.2%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 35.6%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 35.6%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 35.9%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 35.9%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 35.9%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 35.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 39.7%

          \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d \cdot d}{D}}} \]

       2. associate-/l* 45.9%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{\frac{d}{\frac{D}{d}}}} \]

   11. Applied egg-rr 45.9%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}} \]

   if 1.3999999999999999e145 < M

   1. Initial program 4.5%

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

   3. Simplified 4.5%

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

      1. distribute-lft-in 4.5%

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

   5. Applied egg-rr 4.5%

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

   6. Taylor expanded in c0 around inf 42.4%

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

      1. unpow2 42.4%

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

      2. unpow2 42.4%

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

      3. *-commutative 42.4%

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

      4. associate-*r* 46.9%

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

      5. times-frac 46.9%

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

      6. times-frac 51.5%

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

      7. *-commutative 51.5%

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

   8. Simplified 51.5%

      \[\leadsto \frac{c0}{w \cdot 2} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + \frac{c0}{w \cdot 2} \cdot \color{blue}{\left(\frac{c0}{D} \cdot \left(\frac{d}{D} \cdot \frac{d}{h \cdot w}\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.7 \cdot 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d}{D}}{\frac{D}{c0}} \cdot \frac{d}{w \cdot h}\right)\\ \mathbf{elif}\;M \leq 6 \cdot 10^{-42}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\\ \mathbf{elif}\;M \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \mathbf{elif}\;M \leq 5.5 \cdot 10^{+41}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}\\ \mathbf{elif}\;M \leq 3.1 \cdot 10^{+63}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \mathbf{elif}\;M \leq 1.4 \cdot 10^{+145}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{D} \cdot \left(\frac{d}{D} \cdot \frac{d}{w \cdot h}\right)\right)\\ \end{array} \]

Alternative 4: 38.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := h \cdot \left(M \cdot M\right)\\ t_2 := 0.25 \cdot \frac{D \cdot t_1}{\frac{d}{\frac{D}{d}}}\\ t_3 := t_0 \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)\\ t_4 := t_0 \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \mathbf{if}\;M \leq 2.7 \cdot 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.25 \cdot 10^{-71}:\\ \;\;\;\;t_3 + t_0 \cdot \left(\frac{\frac{d}{D}}{\frac{D}{c0}} \cdot \frac{d}{w \cdot h}\right)\\ \mathbf{elif}\;M \leq 7.5 \cdot 10^{-42}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot t_1\right)\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;M \leq 2.9 \cdot 10^{+40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;M \leq 4 \cdot 10^{+60}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;M \leq 2.5 \cdot 10^{+140}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3 + t_0 \cdot \frac{c0 \cdot \left(d \cdot \frac{d}{D}\right)}{D \cdot \left(w \cdot h\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* h (* M M)))
        (t_2 (* 0.25 (/ (* D t_1) (/ d (/ D d)))))
        (t_3 (* t_0 (* (/ c0 (* w h)) (pow (/ d D) 2.0))))
        (t_4 (* t_0 (* 2.0 (* (/ c0 (* D D)) (/ (* d d) (* w h)))))))
   (if (<= M 2.7e-227)
     0.0
     (if (<= M 1.25e-71)
       (+ t_3 (* t_0 (* (/ (/ d D) (/ D c0)) (/ d (* w h)))))
       (if (<= M 7.5e-42)
         (* 0.25 (* (* (/ D d) (/ D d)) t_1))
         (if (<= M 2.2e-6)
           t_4
           (if (<= M 2.9e+40)
             t_2
             (if (<= M 4e+60)
               t_4
               (if (<= M 2.5e+140)
                 t_2
                 (+
                  t_3
                  (* t_0 (/ (* c0 (* d (/ d D))) (* D (* w h))))))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = h * (M * M);
	double t_2 = 0.25 * ((D * t_1) / (d / (D / d)));
	double t_3 = t_0 * ((c0 / (w * h)) * pow((d / D), 2.0));
	double t_4 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	double tmp;
	if (M <= 2.7e-227) {
		tmp = 0.0;
	} else if (M <= 1.25e-71) {
		tmp = t_3 + (t_0 * (((d / D) / (D / c0)) * (d / (w * h))));
	} else if (M <= 7.5e-42) {
		tmp = 0.25 * (((D / d) * (D / d)) * t_1);
	} else if (M <= 2.2e-6) {
		tmp = t_4;
	} else if (M <= 2.9e+40) {
		tmp = t_2;
	} else if (M <= 4e+60) {
		tmp = t_4;
	} else if (M <= 2.5e+140) {
		tmp = t_2;
	} else {
		tmp = t_3 + (t_0 * ((c0 * (d * (d / D))) / (D * (w * h))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    t_1 = h * (m * m)
    t_2 = 0.25d0 * ((d * t_1) / (d_1 / (d / d_1)))
    t_3 = t_0 * ((c0 / (w * h)) * ((d_1 / d) ** 2.0d0))
    t_4 = t_0 * (2.0d0 * ((c0 / (d * d)) * ((d_1 * d_1) / (w * h))))
    if (m <= 2.7d-227) then
        tmp = 0.0d0
    else if (m <= 1.25d-71) then
        tmp = t_3 + (t_0 * (((d_1 / d) / (d / c0)) * (d_1 / (w * h))))
    else if (m <= 7.5d-42) then
        tmp = 0.25d0 * (((d / d_1) * (d / d_1)) * t_1)
    else if (m <= 2.2d-6) then
        tmp = t_4
    else if (m <= 2.9d+40) then
        tmp = t_2
    else if (m <= 4d+60) then
        tmp = t_4
    else if (m <= 2.5d+140) then
        tmp = t_2
    else
        tmp = t_3 + (t_0 * ((c0 * (d_1 * (d_1 / d))) / (d * (w * h))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = h * (M * M);
	double t_2 = 0.25 * ((D * t_1) / (d / (D / d)));
	double t_3 = t_0 * ((c0 / (w * h)) * Math.pow((d / D), 2.0));
	double t_4 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	double tmp;
	if (M <= 2.7e-227) {
		tmp = 0.0;
	} else if (M <= 1.25e-71) {
		tmp = t_3 + (t_0 * (((d / D) / (D / c0)) * (d / (w * h))));
	} else if (M <= 7.5e-42) {
		tmp = 0.25 * (((D / d) * (D / d)) * t_1);
	} else if (M <= 2.2e-6) {
		tmp = t_4;
	} else if (M <= 2.9e+40) {
		tmp = t_2;
	} else if (M <= 4e+60) {
		tmp = t_4;
	} else if (M <= 2.5e+140) {
		tmp = t_2;
	} else {
		tmp = t_3 + (t_0 * ((c0 * (d * (d / D))) / (D * (w * h))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = h * (M * M)
	t_2 = 0.25 * ((D * t_1) / (d / (D / d)))
	t_3 = t_0 * ((c0 / (w * h)) * math.pow((d / D), 2.0))
	t_4 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))))
	tmp = 0
	if M <= 2.7e-227:
		tmp = 0.0
	elif M <= 1.25e-71:
		tmp = t_3 + (t_0 * (((d / D) / (D / c0)) * (d / (w * h))))
	elif M <= 7.5e-42:
		tmp = 0.25 * (((D / d) * (D / d)) * t_1)
	elif M <= 2.2e-6:
		tmp = t_4
	elif M <= 2.9e+40:
		tmp = t_2
	elif M <= 4e+60:
		tmp = t_4
	elif M <= 2.5e+140:
		tmp = t_2
	else:
		tmp = t_3 + (t_0 * ((c0 * (d * (d / D))) / (D * (w * h))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(h * Float64(M * M))
	t_2 = Float64(0.25 * Float64(Float64(D * t_1) / Float64(d / Float64(D / d))))
	t_3 = Float64(t_0 * Float64(Float64(c0 / Float64(w * h)) * (Float64(d / D) ^ 2.0)))
	t_4 = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 / Float64(D * D)) * Float64(Float64(d * d) / Float64(w * h)))))
	tmp = 0.0
	if (M <= 2.7e-227)
		tmp = 0.0;
	elseif (M <= 1.25e-71)
		tmp = Float64(t_3 + Float64(t_0 * Float64(Float64(Float64(d / D) / Float64(D / c0)) * Float64(d / Float64(w * h)))));
	elseif (M <= 7.5e-42)
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * t_1));
	elseif (M <= 2.2e-6)
		tmp = t_4;
	elseif (M <= 2.9e+40)
		tmp = t_2;
	elseif (M <= 4e+60)
		tmp = t_4;
	elseif (M <= 2.5e+140)
		tmp = t_2;
	else
		tmp = Float64(t_3 + Float64(t_0 * Float64(Float64(c0 * Float64(d * Float64(d / D))) / Float64(D * Float64(w * h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = h * (M * M);
	t_2 = 0.25 * ((D * t_1) / (d / (D / d)));
	t_3 = t_0 * ((c0 / (w * h)) * ((d / D) ^ 2.0));
	t_4 = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	tmp = 0.0;
	if (M <= 2.7e-227)
		tmp = 0.0;
	elseif (M <= 1.25e-71)
		tmp = t_3 + (t_0 * (((d / D) / (D / c0)) * (d / (w * h))));
	elseif (M <= 7.5e-42)
		tmp = 0.25 * (((D / d) * (D / d)) * t_1);
	elseif (M <= 2.2e-6)
		tmp = t_4;
	elseif (M <= 2.9e+40)
		tmp = t_2;
	elseif (M <= 4e+60)
		tmp = t_4;
	elseif (M <= 2.5e+140)
		tmp = t_2;
	else
		tmp = t_3 + (t_0 * ((c0 * (d * (d / D))) / (D * (w * h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.25 * N[(N[(D * t$95$1), $MachinePrecision] / N[(d / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[(2.0 * N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 2.7e-227], 0.0, If[LessEqual[M, 1.25e-71], N[(t$95$3 + N[(t$95$0 * N[(N[(N[(d / D), $MachinePrecision] / N[(D / c0), $MachinePrecision]), $MachinePrecision] * N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 7.5e-42], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 2.2e-6], t$95$4, If[LessEqual[M, 2.9e+40], t$95$2, If[LessEqual[M, 4e+60], t$95$4, If[LessEqual[M, 2.5e+140], t$95$2, N[(t$95$3 + N[(t$95$0 * N[(N[(c0 * N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if M < 2.7e-227

   1. Initial program 28.2%

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

   3. Simplified 28.3%

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

   4. Taylor expanded in c0 around -inf 4.3%

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

      1. mul-1-neg 4.3%

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

      2. distribute-lft-in 4.3%

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

   6. Simplified 34.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

   7. Taylor expanded in c0 around 0 39.6%

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

   if 2.7e-227 < M < 1.24999999999999999e-71

   1. Initial program 27.2%

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

   3. Simplified 35.6%

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

      1. distribute-lft-in 35.6%

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

   5. Applied egg-rr 57.6%

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

   6. Taylor expanded in c0 around inf 28.9%

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

      1. unpow2 28.9%

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

      2. unpow2 28.9%

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

      3. *-commutative 28.9%

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

      4. associate-*r* 37.2%

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

      5. times-frac 43.9%

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

      6. times-frac 51.4%

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

      7. *-commutative 51.4%

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

   8. Simplified 51.4%

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

      1. clear-num 51.4%

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

      2. associate-*r/ 53.7%

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

      3. frac-times 56.5%

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

      4. *-un-lft-identity 56.5%

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

   10. Applied egg-rr 56.5%

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

       1. times-frac 53.1%

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

   12. Applied egg-rr 53.1%

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

   if 1.24999999999999999e-71 < M < 7.49999999999999972e-42

   1. Initial program 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. unsub-neg 0.0%

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

      4. unpow2 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. unpow2 0.0%

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

      8. associate-*r* 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. *-commutative 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in c0 around 0 34.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 34.3%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 34.3%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 34.3%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 34.3%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 35.8%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 35.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 35.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 35.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]

   10. Taylor expanded in d around 0 35.8%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{{d}^{2}}{D}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 35.8%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{\color{blue}{d \cdot d}}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       2. associate-*l/ 35.6%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{D} \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       3. associate-/r/ 35.6%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   12. Simplified 35.6%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   13. Step-by-step derivation

       1. associate-/r/ 51.1%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   14. Applied egg-rr 51.1%

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   if 7.49999999999999972e-42 < M < 2.2000000000000001e-6 or 2.90000000000000017e40 < M < 3.9999999999999998e60

   1. Initial program 77.0%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in c0 around inf 77.1%

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

      1. times-frac 83.7%

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

      2. unpow2 83.7%

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

      3. unpow2 83.7%

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

   6. Simplified 83.7%

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

   if 2.2000000000000001e-6 < M < 2.90000000000000017e40 or 3.9999999999999998e60 < M < 2.50000000000000004e140

   1. Initial program 24.5%

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

   3. Simplified 30.8%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. unsub-neg 0.0%

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

      4. unpow2 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. unpow2 0.0%

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

      8. associate-*r* 0.0%

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

      9. *-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. *-commutative 0.0%

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

   6. Simplified 1.2%

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

   7. Taylor expanded in c0 around 0 36.5%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 36.5%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 36.2%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 35.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 35.5%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 35.9%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 35.9%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 35.9%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 35.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 40.1%

          \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d \cdot d}{D}}} \]

       2. associate-/l* 43.6%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{\frac{d}{\frac{D}{d}}}} \]

   11. Applied egg-rr 43.6%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}} \]

   if 2.50000000000000004e140 < M

   1. Initial program 4.1%

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

   3. Simplified 4.1%

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

      1. distribute-lft-in 4.1%

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

   5. Applied egg-rr 4.1%

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

   6. Taylor expanded in c0 around inf 37.5%

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

      1. unpow2 37.5%

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

      2. unpow2 37.5%

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

      3. *-commutative 37.5%

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

      4. associate-*r* 41.5%

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

      5. times-frac 41.5%

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

      6. times-frac 45.6%

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

      7. *-commutative 45.6%

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

   8. Simplified 45.6%

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

      1. *-commutative 45.6%

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

      2. associate-*r/ 45.6%

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

      3. frac-times 49.5%

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

   10. Applied egg-rr 49.5%

       \[\leadsto \frac{c0}{w \cdot 2} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + \frac{c0}{w \cdot 2} \cdot \color{blue}{\frac{\left(\frac{d}{D} \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot D}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.7 \cdot 10^{-227}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.25 \cdot 10^{-71}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + \frac{c0}{2 \cdot w} \cdot \left(\frac{\frac{d}{D}}{\frac{D}{c0}} \cdot \frac{d}{w \cdot h}\right)\\ \mathbf{elif}\;M \leq 7.5 \cdot 10^{-42}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \mathbf{elif}\;M \leq 2.9 \cdot 10^{+40}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}\\ \mathbf{elif}\;M \leq 4 \cdot 10^{+60}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \mathbf{elif}\;M \leq 2.5 \cdot 10^{+140}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right) + \frac{c0}{2 \cdot w} \cdot \frac{c0 \cdot \left(d \cdot \frac{d}{D}\right)}{D \cdot \left(w \cdot h\right)}\\ \end{array} \]

Alternative 5: 40.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;c0 \leq -4.4 \cdot 10^{+85}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_1}{\frac{d}{\frac{D}{d}}}\\ \mathbf{elif}\;c0 \leq -8.2 \cdot 10^{-15}:\\ \;\;\;\;t_0 \cdot \left(\frac{c0}{D} \cdot \left(\frac{d}{D} \cdot \frac{d}{w \cdot h}\right)\right) + t_0 \cdot \frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\\ \mathbf{elif}\;c0 \leq 3.4 \cdot 10^{+127} \lor \neg \left(c0 \leq 1.1 \cdot 10^{+243}\right):\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (* h (* M M))))
   (if (<= c0 -4.4e+85)
     (* 0.25 (/ (* D t_1) (/ d (/ D d))))
     (if (<= c0 -8.2e-15)
       (+
        (* t_0 (* (/ c0 D) (* (/ d D) (/ d (* w h)))))
        (* t_0 (/ (* c0 (* d d)) (* D (* D (* w h))))))
       (if (or (<= c0 3.4e+127) (not (<= c0 1.1e+243)))
         (* 0.25 (* (* (/ D d) (/ D d)) t_1))
         (* t_0 (* 2.0 (* (/ c0 (* D D)) (/ (* d d) (* w h))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = h * (M * M);
	double tmp;
	if (c0 <= -4.4e+85) {
		tmp = 0.25 * ((D * t_1) / (d / (D / d)));
	} else if (c0 <= -8.2e-15) {
		tmp = (t_0 * ((c0 / D) * ((d / D) * (d / (w * h))))) + (t_0 * ((c0 * (d * d)) / (D * (D * (w * h)))));
	} else if ((c0 <= 3.4e+127) || !(c0 <= 1.1e+243)) {
		tmp = 0.25 * (((D / d) * (D / d)) * t_1);
	} else {
		tmp = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    t_1 = h * (m * m)
    if (c0 <= (-4.4d+85)) then
        tmp = 0.25d0 * ((d * t_1) / (d_1 / (d / d_1)))
    else if (c0 <= (-8.2d-15)) then
        tmp = (t_0 * ((c0 / d) * ((d_1 / d) * (d_1 / (w * h))))) + (t_0 * ((c0 * (d_1 * d_1)) / (d * (d * (w * h)))))
    else if ((c0 <= 3.4d+127) .or. (.not. (c0 <= 1.1d+243))) then
        tmp = 0.25d0 * (((d / d_1) * (d / d_1)) * t_1)
    else
        tmp = t_0 * (2.0d0 * ((c0 / (d * d)) * ((d_1 * d_1) / (w * h))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = h * (M * M);
	double tmp;
	if (c0 <= -4.4e+85) {
		tmp = 0.25 * ((D * t_1) / (d / (D / d)));
	} else if (c0 <= -8.2e-15) {
		tmp = (t_0 * ((c0 / D) * ((d / D) * (d / (w * h))))) + (t_0 * ((c0 * (d * d)) / (D * (D * (w * h)))));
	} else if ((c0 <= 3.4e+127) || !(c0 <= 1.1e+243)) {
		tmp = 0.25 * (((D / d) * (D / d)) * t_1);
	} else {
		tmp = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = h * (M * M)
	tmp = 0
	if c0 <= -4.4e+85:
		tmp = 0.25 * ((D * t_1) / (d / (D / d)))
	elif c0 <= -8.2e-15:
		tmp = (t_0 * ((c0 / D) * ((d / D) * (d / (w * h))))) + (t_0 * ((c0 * (d * d)) / (D * (D * (w * h)))))
	elif (c0 <= 3.4e+127) or not (c0 <= 1.1e+243):
		tmp = 0.25 * (((D / d) * (D / d)) * t_1)
	else:
		tmp = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (c0 <= -4.4e+85)
		tmp = Float64(0.25 * Float64(Float64(D * t_1) / Float64(d / Float64(D / d))));
	elseif (c0 <= -8.2e-15)
		tmp = Float64(Float64(t_0 * Float64(Float64(c0 / D) * Float64(Float64(d / D) * Float64(d / Float64(w * h))))) + Float64(t_0 * Float64(Float64(c0 * Float64(d * d)) / Float64(D * Float64(D * Float64(w * h))))));
	elseif ((c0 <= 3.4e+127) || !(c0 <= 1.1e+243))
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * t_1));
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(c0 / Float64(D * D)) * Float64(Float64(d * d) / Float64(w * h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = h * (M * M);
	tmp = 0.0;
	if (c0 <= -4.4e+85)
		tmp = 0.25 * ((D * t_1) / (d / (D / d)));
	elseif (c0 <= -8.2e-15)
		tmp = (t_0 * ((c0 / D) * ((d / D) * (d / (w * h))))) + (t_0 * ((c0 * (d * d)) / (D * (D * (w * h)))));
	elseif ((c0 <= 3.4e+127) || ~((c0 <= 1.1e+243)))
		tmp = 0.25 * (((D / d) * (D / d)) * t_1);
	else
		tmp = t_0 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -4.4e+85], N[(0.25 * N[(N[(D * t$95$1), $MachinePrecision] / N[(d / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -8.2e-15], N[(N[(t$95$0 * N[(N[(c0 / D), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(D * N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c0, 3.4e+127], N[Not[LessEqual[c0, 1.1e+243]], $MachinePrecision]], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c0 < -4.4000000000000003e85

   1. Initial program 34.3%

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

   3. Simplified 30.8%

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

   4. Taylor expanded in c0 around -inf 0.7%

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

      1. +-commutative 0.7%

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

      2. mul-1-neg 0.7%

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

      3. unsub-neg 0.7%

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

      4. unpow2 0.7%

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

      5. *-commutative 0.7%

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

      6. unpow2 0.7%

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

      7. unpow2 0.7%

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

      8. associate-*r* 0.7%

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

      9. *-commutative 0.7%

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

      10. unpow2 0.7%

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

      11. *-commutative 0.7%

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

   6. Simplified 2.2%

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

   7. Taylor expanded in c0 around 0 31.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 31.7%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 29.9%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 29.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 29.5%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 35.0%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 35.0%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 35.0%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 35.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 40.6%

          \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d \cdot d}{D}}} \]

       2. associate-/l* 49.5%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{\frac{d}{\frac{D}{d}}}} \]

   11. Applied egg-rr 49.5%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}} \]

   if -4.4000000000000003e85 < c0 < -8.20000000000000072e-15

   1. Initial program 35.6%

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

   3. Simplified 39.0%

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

      1. distribute-lft-in 39.0%

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

   5. Applied egg-rr 54.7%

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

   6. Taylor expanded in c0 around inf 45.8%

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

      1. unpow2 45.8%

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

      2. unpow2 45.8%

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

      3. *-commutative 45.8%

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

      4. associate-*r* 52.3%

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

      5. times-frac 53.2%

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

      6. times-frac 56.3%

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

      7. *-commutative 56.3%

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

   8. Simplified 56.3%

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

      1. unpow2 56.3%

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

      2. times-frac 49.2%

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

      3. frac-times 45.8%

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

      4. *-commutative 45.8%

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

      5. *-commutative 45.8%

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

      6. associate-*l* 53.2%

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

      7. *-commutative 53.2%

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

   10. Applied egg-rr 53.2%

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

   if -8.20000000000000072e-15 < c0 < 3.39999999999999977e127 or 1.10000000000000004e243 < c0

   1. Initial program 17.8%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in c0 around -inf 0.9%

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

      1. +-commutative 0.9%

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

      2. mul-1-neg 0.9%

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

      3. unsub-neg 0.9%

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

      4. unpow2 0.9%

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

      5. *-commutative 0.9%

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

      6. unpow2 0.9%

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

      7. unpow2 0.9%

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

      8. associate-*r* 0.9%

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

      9. *-commutative 0.9%

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

      10. unpow2 0.9%

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

      11. *-commutative 0.9%

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

   6. Simplified 1.1%

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

   7. Taylor expanded in c0 around 0 40.5%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 40.5%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 40.4%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 40.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 40.5%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 40.8%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 40.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 40.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 40.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]

   10. Taylor expanded in d around 0 40.8%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{{d}^{2}}{D}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 40.8%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{\color{blue}{d \cdot d}}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       2. associate-*l/ 45.9%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{D} \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       3. associate-/r/ 45.8%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   12. Simplified 45.8%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   13. Step-by-step derivation

       1. associate-/r/ 50.0%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   14. Applied egg-rr 50.0%

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   if 3.39999999999999977e127 < c0 < 1.10000000000000004e243

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in c0 around inf 58.5%

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

      1. times-frac 58.6%

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

      2. unpow2 58.6%

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

      3. unpow2 58.6%

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

   6. Simplified 58.6%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -4.4 \cdot 10^{+85}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}\\ \mathbf{elif}\;c0 \leq -8.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{D} \cdot \left(\frac{d}{D} \cdot \frac{d}{w \cdot h}\right)\right) + \frac{c0}{2 \cdot w} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\\ \mathbf{elif}\;c0 \leq 3.4 \cdot 10^{+127} \lor \neg \left(c0 \leq 1.1 \cdot 10^{+243}\right):\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \end{array} \]

Alternative 6: 40.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot d}{D \cdot D}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;c0 \leq -5.2 \cdot 10^{+85}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_2}{\frac{d}{\frac{D}{d}}}\\ \mathbf{elif}\;c0 \leq -4.8 \cdot 10^{-14}:\\ \;\;\;\;t_1 \cdot \left(\frac{c0}{w \cdot h} \cdot t_0 + c0 \cdot \frac{t_0}{w \cdot h}\right)\\ \mathbf{elif}\;c0 \leq 2.6 \cdot 10^{+127} \lor \neg \left(c0 \leq 1.05 \cdot 10^{+243}\right):\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* d d) (* D D))) (t_1 (/ c0 (* 2.0 w))) (t_2 (* h (* M M))))
   (if (<= c0 -5.2e+85)
     (* 0.25 (/ (* D t_2) (/ d (/ D d))))
     (if (<= c0 -4.8e-14)
       (* t_1 (+ (* (/ c0 (* w h)) t_0) (* c0 (/ t_0 (* w h)))))
       (if (or (<= c0 2.6e+127) (not (<= c0 1.05e+243)))
         (* 0.25 (* (* (/ D d) (/ D d)) t_2))
         (* t_1 (* 2.0 (* (/ c0 (* D D)) (/ (* d d) (* w h))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * d) / (D * D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = h * (M * M);
	double tmp;
	if (c0 <= -5.2e+85) {
		tmp = 0.25 * ((D * t_2) / (d / (D / d)));
	} else if (c0 <= -4.8e-14) {
		tmp = t_1 * (((c0 / (w * h)) * t_0) + (c0 * (t_0 / (w * h))));
	} else if ((c0 <= 2.6e+127) || !(c0 <= 1.05e+243)) {
		tmp = 0.25 * (((D / d) * (D / d)) * t_2);
	} else {
		tmp = t_1 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (d_1 * d_1) / (d * d)
    t_1 = c0 / (2.0d0 * w)
    t_2 = h * (m * m)
    if (c0 <= (-5.2d+85)) then
        tmp = 0.25d0 * ((d * t_2) / (d_1 / (d / d_1)))
    else if (c0 <= (-4.8d-14)) then
        tmp = t_1 * (((c0 / (w * h)) * t_0) + (c0 * (t_0 / (w * h))))
    else if ((c0 <= 2.6d+127) .or. (.not. (c0 <= 1.05d+243))) then
        tmp = 0.25d0 * (((d / d_1) * (d / d_1)) * t_2)
    else
        tmp = t_1 * (2.0d0 * ((c0 / (d * d)) * ((d_1 * d_1) / (w * h))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * d) / (D * D);
	double t_1 = c0 / (2.0 * w);
	double t_2 = h * (M * M);
	double tmp;
	if (c0 <= -5.2e+85) {
		tmp = 0.25 * ((D * t_2) / (d / (D / d)));
	} else if (c0 <= -4.8e-14) {
		tmp = t_1 * (((c0 / (w * h)) * t_0) + (c0 * (t_0 / (w * h))));
	} else if ((c0 <= 2.6e+127) || !(c0 <= 1.05e+243)) {
		tmp = 0.25 * (((D / d) * (D / d)) * t_2);
	} else {
		tmp = t_1 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d * d) / (D * D)
	t_1 = c0 / (2.0 * w)
	t_2 = h * (M * M)
	tmp = 0
	if c0 <= -5.2e+85:
		tmp = 0.25 * ((D * t_2) / (d / (D / d)))
	elif c0 <= -4.8e-14:
		tmp = t_1 * (((c0 / (w * h)) * t_0) + (c0 * (t_0 / (w * h))))
	elif (c0 <= 2.6e+127) or not (c0 <= 1.05e+243):
		tmp = 0.25 * (((D / d) * (D / d)) * t_2)
	else:
		tmp = t_1 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * d) / Float64(D * D))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (c0 <= -5.2e+85)
		tmp = Float64(0.25 * Float64(Float64(D * t_2) / Float64(d / Float64(D / d))));
	elseif (c0 <= -4.8e-14)
		tmp = Float64(t_1 * Float64(Float64(Float64(c0 / Float64(w * h)) * t_0) + Float64(c0 * Float64(t_0 / Float64(w * h)))));
	elseif ((c0 <= 2.6e+127) || !(c0 <= 1.05e+243))
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * t_2));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(Float64(c0 / Float64(D * D)) * Float64(Float64(d * d) / Float64(w * h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d * d) / (D * D);
	t_1 = c0 / (2.0 * w);
	t_2 = h * (M * M);
	tmp = 0.0;
	if (c0 <= -5.2e+85)
		tmp = 0.25 * ((D * t_2) / (d / (D / d)));
	elseif (c0 <= -4.8e-14)
		tmp = t_1 * (((c0 / (w * h)) * t_0) + (c0 * (t_0 / (w * h))));
	elseif ((c0 <= 2.6e+127) || ~((c0 <= 1.05e+243)))
		tmp = 0.25 * (((D / d) * (D / d)) * t_2);
	else
		tmp = t_1 * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -5.2e+85], N[(0.25 * N[(N[(D * t$95$2), $MachinePrecision] / N[(d / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -4.8e-14], N[(t$95$1 * N[(N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(c0 * N[(t$95$0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c0, 2.6e+127], N[Not[LessEqual[c0, 1.05e+243]], $MachinePrecision]], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c0 < -5.20000000000000021e85

   1. Initial program 34.3%

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

   3. Simplified 30.8%

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

   4. Taylor expanded in c0 around -inf 0.7%

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

      1. +-commutative 0.7%

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

      2. mul-1-neg 0.7%

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

      3. unsub-neg 0.7%

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

      4. unpow2 0.7%

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

      5. *-commutative 0.7%

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

      6. unpow2 0.7%

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

      7. unpow2 0.7%

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

      8. associate-*r* 0.7%

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

      9. *-commutative 0.7%

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

      10. unpow2 0.7%

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

      11. *-commutative 0.7%

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

   6. Simplified 2.2%

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

   7. Taylor expanded in c0 around 0 31.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 31.7%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 29.9%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 29.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 29.5%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 35.0%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 35.0%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 35.0%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 35.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 40.6%

          \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d \cdot d}{D}}} \]

       2. associate-/l* 49.5%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{\frac{d}{\frac{D}{d}}}} \]

   11. Applied egg-rr 49.5%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}} \]

   if -5.20000000000000021e85 < c0 < -4.8e-14

   1. Initial program 35.6%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in c0 around inf 45.8%

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

      1. associate-*r* 45.8%

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

      2. *-commutative 45.8%

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

      3. unpow2 45.8%

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

      4. *-commutative 45.8%

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

      5. associate-*r/ 46.4%

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

      6. *-commutative 46.4%

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

      7. unpow2 46.4%

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

      8. *-commutative 46.4%

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

      9. associate-*r* 46.4%

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

      10. associate-/r* 49.2%

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

      11. unpow2 49.2%

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

      12. unpow2 49.2%

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

   6. Simplified 49.2%

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

   if -4.8e-14 < c0 < 2.6000000000000002e127 or 1.05e243 < c0

   1. Initial program 17.8%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in c0 around -inf 0.9%

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

      1. +-commutative 0.9%

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

      2. mul-1-neg 0.9%

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

      3. unsub-neg 0.9%

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

      4. unpow2 0.9%

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

      5. *-commutative 0.9%

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

      6. unpow2 0.9%

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

      7. unpow2 0.9%

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

      8. associate-*r* 0.9%

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

      9. *-commutative 0.9%

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

      10. unpow2 0.9%

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

      11. *-commutative 0.9%

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

   6. Simplified 1.1%

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

   7. Taylor expanded in c0 around 0 40.5%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 40.5%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 40.4%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 40.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 40.5%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 40.8%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 40.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 40.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 40.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]

   10. Taylor expanded in d around 0 40.8%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{{d}^{2}}{D}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 40.8%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{\color{blue}{d \cdot d}}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       2. associate-*l/ 45.9%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{D} \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       3. associate-/r/ 45.8%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   12. Simplified 45.8%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   13. Step-by-step derivation

       1. associate-/r/ 50.0%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   14. Applied egg-rr 50.0%

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   if 2.6000000000000002e127 < c0 < 1.05e243

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in c0 around inf 58.5%

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

      1. times-frac 58.6%

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

      2. unpow2 58.6%

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

      3. unpow2 58.6%

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

   6. Simplified 58.6%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -5.2 \cdot 10^{+85}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}\\ \mathbf{elif}\;c0 \leq -4.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + c0 \cdot \frac{\frac{d \cdot d}{D \cdot D}}{w \cdot h}\right)\\ \mathbf{elif}\;c0 \leq 2.6 \cdot 10^{+127} \lor \neg \left(c0 \leq 1.05 \cdot 10^{+243}\right):\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \end{array} \]

Alternative 7: 40.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;c0 \leq -3.8 \cdot 10^{+85}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_0}{\frac{d}{\frac{D}{d}}}\\ \mathbf{elif}\;c0 \leq -6.8 \cdot 10^{-14} \lor \neg \left(c0 \leq 2.8 \cdot 10^{+127}\right) \land c0 \leq 5.5 \cdot 10^{+242}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M))))
   (if (<= c0 -3.8e+85)
     (* 0.25 (/ (* D t_0) (/ d (/ D d))))
     (if (or (<= c0 -6.8e-14) (and (not (<= c0 2.8e+127)) (<= c0 5.5e+242)))
       (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ c0 (* D D)) (/ (* d d) (* w h)))))
       (* 0.25 (* (* (/ D d) (/ D d)) t_0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double tmp;
	if (c0 <= -3.8e+85) {
		tmp = 0.25 * ((D * t_0) / (d / (D / d)));
	} else if ((c0 <= -6.8e-14) || (!(c0 <= 2.8e+127) && (c0 <= 5.5e+242))) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	} else {
		tmp = 0.25 * (((D / d) * (D / d)) * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = h * (m * m)
    if (c0 <= (-3.8d+85)) then
        tmp = 0.25d0 * ((d * t_0) / (d_1 / (d / d_1)))
    else if ((c0 <= (-6.8d-14)) .or. (.not. (c0 <= 2.8d+127)) .and. (c0 <= 5.5d+242)) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * ((c0 / (d * d)) * ((d_1 * d_1) / (w * h))))
    else
        tmp = 0.25d0 * (((d / d_1) * (d / d_1)) * t_0)
    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 * (M * M);
	double tmp;
	if (c0 <= -3.8e+85) {
		tmp = 0.25 * ((D * t_0) / (d / (D / d)));
	} else if ((c0 <= -6.8e-14) || (!(c0 <= 2.8e+127) && (c0 <= 5.5e+242))) {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	} else {
		tmp = 0.25 * (((D / d) * (D / d)) * t_0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	tmp = 0
	if c0 <= -3.8e+85:
		tmp = 0.25 * ((D * t_0) / (d / (D / d)))
	elif (c0 <= -6.8e-14) or (not (c0 <= 2.8e+127) and (c0 <= 5.5e+242)):
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))))
	else:
		tmp = 0.25 * (((D / d) * (D / d)) * t_0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (c0 <= -3.8e+85)
		tmp = Float64(0.25 * Float64(Float64(D * t_0) / Float64(d / Float64(D / d))));
	elseif ((c0 <= -6.8e-14) || (!(c0 <= 2.8e+127) && (c0 <= 5.5e+242)))
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(c0 / Float64(D * D)) * Float64(Float64(d * d) / Float64(w * h)))));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	tmp = 0.0;
	if (c0 <= -3.8e+85)
		tmp = 0.25 * ((D * t_0) / (d / (D / d)));
	elseif ((c0 <= -6.8e-14) || (~((c0 <= 2.8e+127)) && (c0 <= 5.5e+242)))
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / (D * D)) * ((d * d) / (w * h))));
	else
		tmp = 0.25 * (((D / d) * (D / d)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -3.8e+85], N[(0.25 * N[(N[(D * t$95$0), $MachinePrecision] / N[(d / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c0, -6.8e-14], And[N[Not[LessEqual[c0, 2.8e+127]], $MachinePrecision], LessEqual[c0, 5.5e+242]]], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -3.79999999999999992e85

   1. Initial program 34.3%

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

   3. Simplified 30.8%

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

   4. Taylor expanded in c0 around -inf 0.7%

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

      1. +-commutative 0.7%

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

      2. mul-1-neg 0.7%

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

      3. unsub-neg 0.7%

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

      4. unpow2 0.7%

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

      5. *-commutative 0.7%

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

      6. unpow2 0.7%

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

      7. unpow2 0.7%

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

      8. associate-*r* 0.7%

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

      9. *-commutative 0.7%

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

      10. unpow2 0.7%

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

      11. *-commutative 0.7%

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

   6. Simplified 2.2%

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

   7. Taylor expanded in c0 around 0 31.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 31.7%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 29.9%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 29.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 29.5%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 35.0%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 35.0%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 35.0%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 35.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 40.6%

          \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d \cdot d}{D}}} \]

       2. associate-/l* 49.5%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{\frac{d}{\frac{D}{d}}}} \]

   11. Applied egg-rr 49.5%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}} \]

   if -3.79999999999999992e85 < c0 < -6.80000000000000006e-14 or 2.8000000000000002e127 < c0 < 5.50000000000000022e242

   1. Initial program 42.8%

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

   3. Simplified 44.5%

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

   4. Taylor expanded in c0 around inf 52.2%

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

      1. times-frac 53.8%

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

      2. unpow2 53.8%

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

      3. unpow2 53.8%

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

   6. Simplified 53.8%

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

   if -6.80000000000000006e-14 < c0 < 2.8000000000000002e127 or 5.50000000000000022e242 < c0

   1. Initial program 17.8%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in c0 around -inf 0.9%

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

      1. +-commutative 0.9%

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

      2. mul-1-neg 0.9%

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

      3. unsub-neg 0.9%

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

      4. unpow2 0.9%

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

      5. *-commutative 0.9%

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

      6. unpow2 0.9%

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

      7. unpow2 0.9%

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

      8. associate-*r* 0.9%

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

      9. *-commutative 0.9%

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

      10. unpow2 0.9%

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

      11. *-commutative 0.9%

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

   6. Simplified 1.1%

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

   7. Taylor expanded in c0 around 0 40.5%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 40.5%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 40.4%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 40.5%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 40.5%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 40.8%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 40.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 40.8%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 40.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]

   10. Taylor expanded in d around 0 40.8%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{{d}^{2}}{D}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 40.8%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{\color{blue}{d \cdot d}}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       2. associate-*l/ 45.9%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{D} \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       3. associate-/r/ 45.8%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   12. Simplified 45.8%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   13. Step-by-step derivation

       1. associate-/r/ 50.0%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   14. Applied egg-rr 50.0%

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -3.8 \cdot 10^{+85}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}\\ \mathbf{elif}\;c0 \leq -6.8 \cdot 10^{-14} \lor \neg \left(c0 \leq 2.8 \cdot 10^{+127}\right) \land c0 \leq 5.5 \cdot 10^{+242}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{w \cdot h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\\ \end{array} \]

Alternative 8: 39.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;c0 \leq -4.6 \cdot 10^{+27}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot t_0}{\frac{d}{\frac{D}{d}}}\\ \mathbf{elif}\;c0 \leq -3.5 \cdot 10^{-14} \lor \neg \left(c0 \leq 3.8 \cdot 10^{+127}\right) \land c0 \leq 5.2 \cdot 10^{+234}:\\ \;\;\;\;\frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* h (* M M))))
   (if (<= c0 -4.6e+27)
     (* 0.25 (/ (* D t_0) (/ d (/ D d))))
     (if (or (<= c0 -3.5e-14) (and (not (<= c0 3.8e+127)) (<= c0 5.2e+234)))
       (* (/ (* c0 c0) (* D D)) (/ (* d d) (* h (* w w))))
       (* 0.25 (* (* (/ D d) (/ D d)) t_0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = h * (M * M);
	double tmp;
	if (c0 <= -4.6e+27) {
		tmp = 0.25 * ((D * t_0) / (d / (D / d)));
	} else if ((c0 <= -3.5e-14) || (!(c0 <= 3.8e+127) && (c0 <= 5.2e+234))) {
		tmp = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	} else {
		tmp = 0.25 * (((D / d) * (D / d)) * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = h * (m * m)
    if (c0 <= (-4.6d+27)) then
        tmp = 0.25d0 * ((d * t_0) / (d_1 / (d / d_1)))
    else if ((c0 <= (-3.5d-14)) .or. (.not. (c0 <= 3.8d+127)) .and. (c0 <= 5.2d+234)) then
        tmp = ((c0 * c0) / (d * d)) * ((d_1 * d_1) / (h * (w * w)))
    else
        tmp = 0.25d0 * (((d / d_1) * (d / d_1)) * t_0)
    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 * (M * M);
	double tmp;
	if (c0 <= -4.6e+27) {
		tmp = 0.25 * ((D * t_0) / (d / (D / d)));
	} else if ((c0 <= -3.5e-14) || (!(c0 <= 3.8e+127) && (c0 <= 5.2e+234))) {
		tmp = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	} else {
		tmp = 0.25 * (((D / d) * (D / d)) * t_0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = h * (M * M)
	tmp = 0
	if c0 <= -4.6e+27:
		tmp = 0.25 * ((D * t_0) / (d / (D / d)))
	elif (c0 <= -3.5e-14) or (not (c0 <= 3.8e+127) and (c0 <= 5.2e+234)):
		tmp = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)))
	else:
		tmp = 0.25 * (((D / d) * (D / d)) * t_0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (c0 <= -4.6e+27)
		tmp = Float64(0.25 * Float64(Float64(D * t_0) / Float64(d / Float64(D / d))));
	elseif ((c0 <= -3.5e-14) || (!(c0 <= 3.8e+127) && (c0 <= 5.2e+234)))
		tmp = Float64(Float64(Float64(c0 * c0) / Float64(D * D)) * Float64(Float64(d * d) / Float64(h * Float64(w * w))));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = h * (M * M);
	tmp = 0.0;
	if (c0 <= -4.6e+27)
		tmp = 0.25 * ((D * t_0) / (d / (D / d)));
	elseif ((c0 <= -3.5e-14) || (~((c0 <= 3.8e+127)) && (c0 <= 5.2e+234)))
		tmp = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	else
		tmp = 0.25 * (((D / d) * (D / d)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -4.6e+27], N[(0.25 * N[(N[(D * t$95$0), $MachinePrecision] / N[(d / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c0, -3.5e-14], And[N[Not[LessEqual[c0, 3.8e+127]], $MachinePrecision], LessEqual[c0, 5.2e+234]]], N[(N[(N[(c0 * c0), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -4.6000000000000001e27

   1. Initial program 31.0%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in c0 around -inf 0.5%

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

      1. +-commutative 0.5%

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

      2. mul-1-neg 0.5%

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

      3. unsub-neg 0.5%

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

      4. unpow2 0.5%

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

      5. *-commutative 0.5%

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

      6. unpow2 0.5%

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

      7. unpow2 0.5%

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

      8. associate-*r* 0.5%

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

      9. *-commutative 0.5%

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

      10. unpow2 0.5%

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

      11. *-commutative 0.5%

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

   6. Simplified 3.0%

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

   7. Taylor expanded in c0 around 0 26.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 26.6%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 25.2%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 24.9%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 24.9%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 30.4%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 30.4%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 30.4%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 30.4%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 34.6%

          \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d \cdot d}{D}}} \]

       2. associate-/l* 44.0%

          \[\leadsto 0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{\frac{d}{\frac{D}{d}}}} \]

   11. Applied egg-rr 44.0%

       \[\leadsto 0.25 \cdot \color{blue}{\frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}} \]

   if -4.6000000000000001e27 < c0 < -3.5000000000000002e-14 or 3.7999999999999998e127 < c0 < 5.2000000000000003e234

   1. Initial program 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in c0 around inf 60.9%

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

      1. times-frac 60.9%

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

      2. unpow2 60.9%

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

      3. unpow2 60.9%

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

      4. unpow2 60.9%

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

      5. unpow2 60.9%

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

   6. Simplified 60.9%

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

   if -3.5000000000000002e-14 < c0 < 3.7999999999999998e127 or 5.2000000000000003e234 < c0

   1. Initial program 18.1%

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

   3. Simplified 23.0%

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

   4. Taylor expanded in c0 around -inf 0.8%

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

      1. +-commutative 0.8%

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

      2. mul-1-neg 0.8%

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

      3. unsub-neg 0.8%

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

      4. unpow2 0.8%

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

      5. *-commutative 0.8%

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

      6. unpow2 0.8%

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

      7. unpow2 0.8%

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

      8. associate-*r* 0.9%

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

      9. *-commutative 0.9%

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

      10. unpow2 0.9%

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

      11. *-commutative 0.9%

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

   6. Simplified 1.0%

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

   7. Taylor expanded in c0 around 0 39.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 39.6%

         \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

      2. associate-/l* 39.6%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

      3. associate-/r/ 39.6%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

      4. unpow2 39.6%

         \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

      5. associate-/l* 40.0%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

      6. unpow2 40.0%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

      7. *-commutative 40.0%

         \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

   9. Simplified 40.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]

   10. Taylor expanded in d around 0 40.0%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{{d}^{2}}{D}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 40.0%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{\color{blue}{d \cdot d}}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       2. associate-*l/ 44.9%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{D} \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

       3. associate-/r/ 44.9%

          \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   12. Simplified 44.9%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
   13. Step-by-step derivation

       1. associate-/r/ 49.0%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

   14. Applied egg-rr 49.0%

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -4.6 \cdot 10^{+27}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\frac{d}{\frac{D}{d}}}\\ \mathbf{elif}\;c0 \leq -3.5 \cdot 10^{-14} \lor \neg \left(c0 \leq 3.8 \cdot 10^{+127}\right) \land c0 \leq 5.2 \cdot 10^{+234}:\\ \;\;\;\;\frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\\ \end{array} \]

Alternative 9: 41.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (((D / d) * (D / d)) * (h * (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
    code = 0.25d0 * (((d / d_1) * (d / d_1)) * (h * (m * m)))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.25 * (((D / d) * (D / d)) * (h * (M * M)));
}

Python

def code(c0, w, h, D, d, M):
	return 0.25 * (((D / d) * (D / d)) * (h * (M * M)))

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(M * M))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.25 * (((D / d) * (D / d)) * (h * (M * M)));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.5%

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

3. Simplified 29.8%

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

4. Taylor expanded in c0 around -inf 2.0%

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

   1. +-commutative 2.0%

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

   2. mul-1-neg 2.0%

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

   3. unsub-neg 2.0%

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

   4. unpow2 2.0%

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

   5. *-commutative 2.0%

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

   6. unpow2 2.0%

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

   7. unpow2 2.0%

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

   8. associate-*r* 1.9%

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

   9. *-commutative 1.9%

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

   10. unpow2 1.9%

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

   11. *-commutative 1.9%

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

6. Simplified 1.9%

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

7. Taylor expanded in c0 around 0 32.0%

   \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 32.0%

      \[\leadsto 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]

   2. associate-/l* 31.6%

      \[\leadsto 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{d \cdot d}{{M}^{2} \cdot h}}} \]

   3. associate-/r/ 31.1%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right)} \]

   4. unpow2 31.1%

      \[\leadsto 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d \cdot d} \cdot \left({M}^{2} \cdot h\right)\right) \]

   5. associate-/l* 33.0%

      \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d \cdot d}{D}}} \cdot \left({M}^{2} \cdot h\right)\right) \]

   6. unpow2 33.0%

      \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)\right) \]

   7. *-commutative 33.0%

      \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \color{blue}{\left(h \cdot \left(M \cdot M\right)\right)}\right) \]

9. Simplified 33.0%

   \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D}{\frac{d \cdot d}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)} \]

10. Taylor expanded in d around 0 33.0%

    \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{{d}^{2}}{D}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
11. Step-by-step derivation

    1. unpow2 33.0%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\frac{\color{blue}{d \cdot d}}{D}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

    2. associate-*l/ 38.9%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{D} \cdot d}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

    3. associate-/r/ 38.8%

       \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

12. Simplified 38.8%

    \[\leadsto 0.25 \cdot \left(\frac{D}{\color{blue}{\frac{d}{\frac{D}{d}}}} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]
13. Step-by-step derivation

    1. associate-/r/ 41.4%

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

14. Applied egg-rr 41.4%

    \[\leadsto 0.25 \cdot \left(\color{blue}{\left(\frac{D}{d} \cdot \frac{D}{d}\right)} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

15. Final simplification 41.4%

    \[\leadsto 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \]

Alternative 10: 33.6% accurate, 151.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 27.5%

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

3. Simplified 29.8%

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

4. Taylor expanded in c0 around -inf 3.9%

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

   1. mul-1-neg 3.9%

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

   2. distribute-lft-in 3.9%

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

6. Simplified 28.8%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]

7. Taylor expanded in c0 around 0 32.8%

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

8. Final simplification 32.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023292 
(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))))))