Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 60.8%

Time: 28.1s

Alternatives: 8

Speedup: 30.2×

• could not determine a ground truth

  1. c0 = 3.0971041498318137e+307
  2. w = -4.601696006834109e-239
  3. h = 8.246156584955236e-224
  4. D = 2.3499460353720535e-150
  5. d = 1.2039632980168139e+269
  6. M = -1.4333332286764162e+267

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: 60.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 82.2%

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

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

   1. Initial program 0.0%

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

      1. associate-*l/ 0.6%

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

      2. *-commutative 0.6%

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

      3. fma-def 0.6%

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

      4. associate-*l* 0.6%

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

      5. *-commutative 0.6%

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

      6. associate-*r* 0.6%

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

      7. associate-*l* 0.7%

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

      8. *-commutative 0.7%

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

   3. Simplified 5.3%

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

   4. Taylor expanded in c0 around -inf 1.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   5. Step-by-step derivation

      1. fma-def 1.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)} \]

      2. *-commutative 1.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{c0}^{2} \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      3. unpow2 1.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(c0 \cdot c0\right)} \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      4. distribute-rgt1-in 1.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      5. metadata-eval 1.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      6. mul0-lft 30.9%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      7. associate-/l* 31.5%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      8. unpow2 31.5%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}\right) \]

      9. unpow2 31.5%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}}\right) \]

      10. *-commutative 31.5%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      11. unpow2 31.5%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}}\right) \]

   6. Simplified 31.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]
   7. Step-by-step derivation

      1. associate-/r/ 30.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right) \]

   8. Applied egg-rr 30.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right) \]

   9. Taylor expanded in D around 0 30.9%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 30.9%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}\right) \]

       2. unpow2 30.9%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}}\right) \]

       3. unpow2 30.9%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}}\right) \]

       4. times-frac 34.6%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

       5. unpow2 34.6%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

       6. associate-/l* 39.2%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

   11. Simplified 39.2%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

   12. Taylor expanded in c0 around 0 57.3%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{0}}{w}, 0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

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

Alternative 2: 44.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \frac{0}{w}, 0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{if}\;c0 \leq -2.1 \cdot 10^{+214}:\\ \;\;\;\;\frac{c0}{\frac{w}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{h}}{w}}}\\ \mathbf{elif}\;c0 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 2.2 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{0 \cdot \left(c0 \cdot c0\right)}{w}, 0.25 \cdot \frac{D \cdot \frac{h}{\frac{d}{M \cdot M}}}{\frac{d}{D}}\right)\\ \mathbf{elif}\;c0 \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{c0 \cdot \left(\left(2 \cdot \frac{c0}{h}\right) \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)}{2 \cdot w}\\ \mathbf{elif}\;c0 \leq 1.75 \cdot 10^{+226}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0
         (fma -0.5 (/ 0.0 w) (* 0.25 (* (/ D (/ d D)) (/ (* h (* M M)) d))))))
   (if (<= c0 -2.1e+214)
     (/ c0 (/ w (* (pow (/ d D) 2.0) (/ (/ c0 h) w))))
     (if (<= c0 -1.35e+154)
       t_0
       (if (<= c0 2.2e+44)
         (fma
          -0.5
          (/ (* 0.0 (* c0 c0)) w)
          (* 0.25 (/ (* D (/ h (/ d (* M M)))) (/ d D))))
         (if (<= c0 7.2e+154)
           (/ (* c0 (* (* 2.0 (/ c0 h)) (/ (* (/ d D) (/ d D)) w))) (* 2.0 w))
           (if (<= c0 1.75e+226)
             t_0
             (/
              (* 2.0 (/ (* d (* d (* c0 c0))) (* D (* (* w h) D))))
              (* 2.0 w)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = fma(-0.5, (0.0 / w), (0.25 * ((D / (d / D)) * ((h * (M * M)) / d))));
	double tmp;
	if (c0 <= -2.1e+214) {
		tmp = c0 / (w / (pow((d / D), 2.0) * ((c0 / h) / w)));
	} else if (c0 <= -1.35e+154) {
		tmp = t_0;
	} else if (c0 <= 2.2e+44) {
		tmp = fma(-0.5, ((0.0 * (c0 * c0)) / w), (0.25 * ((D * (h / (d / (M * M)))) / (d / D))));
	} else if (c0 <= 7.2e+154) {
		tmp = (c0 * ((2.0 * (c0 / h)) * (((d / D) * (d / D)) / w))) / (2.0 * w);
	} else if (c0 <= 1.75e+226) {
		tmp = t_0;
	} else {
		tmp = (2.0 * ((d * (d * (c0 * c0))) / (D * ((w * h) * D)))) / (2.0 * w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = fma(-0.5, Float64(0.0 / w), Float64(0.25 * Float64(Float64(D / Float64(d / D)) * Float64(Float64(h * Float64(M * M)) / d))))
	tmp = 0.0
	if (c0 <= -2.1e+214)
		tmp = Float64(c0 / Float64(w / Float64((Float64(d / D) ^ 2.0) * Float64(Float64(c0 / h) / w))));
	elseif (c0 <= -1.35e+154)
		tmp = t_0;
	elseif (c0 <= 2.2e+44)
		tmp = fma(-0.5, Float64(Float64(0.0 * Float64(c0 * c0)) / w), Float64(0.25 * Float64(Float64(D * Float64(h / Float64(d / Float64(M * M)))) / Float64(d / D))));
	elseif (c0 <= 7.2e+154)
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 * Float64(c0 / h)) * Float64(Float64(Float64(d / D) * Float64(d / D)) / w))) / Float64(2.0 * w));
	elseif (c0 <= 1.75e+226)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(d * Float64(d * Float64(c0 * c0))) / Float64(D * Float64(Float64(w * h) * D)))) / Float64(2.0 * w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(-0.5 * N[(0.0 / w), $MachinePrecision] + N[(0.25 * N[(N[(D / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -2.1e+214], N[(c0 / N[(w / N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, -1.35e+154], t$95$0, If[LessEqual[c0, 2.2e+44], N[(-0.5 * N[(N[(0.0 * N[(c0 * c0), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] + N[(0.25 * N[(N[(D * N[(h / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 7.2e+154], N[(N[(c0 * N[(N[(2.0 * N[(c0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 1.75e+226], t$95$0, N[(N[(2.0 * N[(N[(d * N[(d * N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c0 < -2.1000000000000001e214

   1. Initial program 26.7%

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

      1. associate-*l* 23.5%

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

      2. difference-of-squares 30.2%

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

      3. associate-*l* 30.2%

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

      4. associate-*l* 46.7%

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

   3. Simplified 46.7%

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

      1. flip-+ 3.3%

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

   5. Applied egg-rr 3.3%

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

      1. unpow2 3.3%

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

      2. associate--r- 3.3%

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

      3. +-inverses 13.5%

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

      4. unpow2 13.5%

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

      5. associate-/l/ 10.2%

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

      6. associate-/l/ 6.8%

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

   7. Simplified 6.8%

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

   8. Taylor expanded in M around 0 34.2%

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

      1. associate-/r* 34.3%

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

      2. *-commutative 34.3%

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

      3. associate-*r/ 34.3%

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

      4. unpow2 34.3%

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

      5. unpow2 34.3%

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

      6. times-frac 37.7%

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

      7. unpow2 37.7%

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

      8. *-commutative 37.7%

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

      9. times-frac 47.9%

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

   10. Simplified 47.9%

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

       1. associate-*l/ 47.9%

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

       2. associate-*r* 47.9%

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

   12. Applied egg-rr 47.9%

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

       1. associate-/l* 51.1%

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

       2. associate-*l* 51.1%

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

       3. times-frac 37.7%

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

       4. *-commutative 37.7%

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

       5. times-frac 51.1%

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

       6. unpow2 51.1%

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

       7. times-frac 47.7%

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

       8. unpow2 47.7%

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

       9. unpow2 47.7%

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

       10. associate-/r* 47.7%

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

       11. unpow2 47.7%

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

       12. *-commutative 47.7%

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

       13. unpow2 47.7%

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

       14. times-frac 47.7%

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

       15. metadata-eval 47.7%

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

   14. Simplified 54.5%

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

   if -2.1000000000000001e214 < c0 < -1.35000000000000003e154 or 7.2000000000000001e154 < c0 < 1.7499999999999999e226

   1. Initial program 28.7%

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

      1. associate-*l/ 25.9%

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

      2. *-commutative 25.9%

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

      3. fma-def 23.1%

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

      4. associate-*l* 23.1%

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

      5. *-commutative 23.1%

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

      6. associate-*r* 23.1%

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

      7. associate-*l* 23.1%

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

      8. *-commutative 23.1%

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

   3. Simplified 23.2%

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

   4. Taylor expanded in c0 around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   5. Step-by-step derivation

      1. fma-def 0.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{c0}^{2} \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      3. unpow2 0.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(c0 \cdot c0\right)} \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      4. distribute-rgt1-in 0.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      5. metadata-eval 0.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      6. mul0-lft 0.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      7. associate-/l* 0.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      8. unpow2 0.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}\right) \]

      9. unpow2 0.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}}\right) \]

      10. *-commutative 0.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      11. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}}\right) \]

   6. Simplified 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]
   7. Step-by-step derivation

      1. associate-/r/ 0.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right) \]

   9. Taylor expanded in D around 0 0.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}\right) \]

       2. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}}\right) \]

       3. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}}\right) \]

       4. times-frac 0.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

       5. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

       6. associate-/l* 0.0%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

   11. Simplified 0.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

   12. Taylor expanded in c0 around 0 69.3%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{0}}{w}, 0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

   if -1.35000000000000003e154 < c0 < 2.19999999999999996e44

   1. Initial program 20.0%

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

      1. associate-*l/ 20.0%

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

      2. *-commutative 20.0%

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

      3. fma-def 18.6%

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

      4. associate-*l* 18.0%

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

      5. *-commutative 18.0%

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

      6. associate-*r* 18.0%

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

      7. associate-*l* 18.1%

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

      8. *-commutative 18.1%

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

   3. Simplified 19.3%

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

   4. Taylor expanded in c0 around -inf 7.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   5. Step-by-step derivation

      1. fma-def 7.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)} \]

      2. *-commutative 7.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{c0}^{2} \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      3. unpow2 7.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(c0 \cdot c0\right)} \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      4. distribute-rgt1-in 7.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      5. metadata-eval 7.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      6. mul0-lft 41.1%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      7. associate-/l* 41.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      8. unpow2 41.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}\right) \]

      9. unpow2 41.8%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}}\right) \]

      10. *-commutative 41.8%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      11. unpow2 41.8%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}}\right) \]

   6. Simplified 41.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]
   7. Step-by-step derivation

      1. associate-/r/ 41.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right) \]

   8. Applied egg-rr 41.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right) \]

   9. Taylor expanded in D around 0 41.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 41.1%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}\right) \]

       2. unpow2 41.1%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}}\right) \]

       3. unpow2 41.1%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}}\right) \]

       4. times-frac 45.5%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

       5. unpow2 45.5%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

       6. associate-/l* 50.8%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

   11. Simplified 50.8%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]
   12. Step-by-step derivation

       1. associate-*l/ 53.6%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\frac{D \cdot \frac{h \cdot \left(M \cdot M\right)}{d}}{\frac{d}{D}}}\right) \]

       2. associate-/l* 54.9%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot \color{blue}{\frac{h}{\frac{d}{M \cdot M}}}}{\frac{d}{D}}\right) \]

   13. Applied egg-rr 54.9%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\frac{D \cdot \frac{h}{\frac{d}{M \cdot M}}}{\frac{d}{D}}}\right) \]

   if 2.19999999999999996e44 < c0 < 7.2000000000000001e154

   1. Initial program 43.5%

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

      1. associate-*l* 43.5%

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

      2. difference-of-squares 52.2%

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

      3. associate-*l* 52.2%

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

      4. associate-*l* 56.5%

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

   3. Simplified 56.5%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.3%

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

      1. unpow2 0.3%

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

      2. associate--r- 0.8%

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

      3. +-inverses 13.8%

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

      4. unpow2 13.8%

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

      5. associate-/l/ 13.5%

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

      6. associate-/l/ 13.8%

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

   7. Simplified 13.8%

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

   8. Taylor expanded in M around 0 52.8%

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

      1. associate-/r* 52.8%

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

      2. *-commutative 52.8%

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

      3. associate-*r/ 52.7%

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

      4. unpow2 52.7%

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

      5. unpow2 52.7%

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

      6. times-frac 61.4%

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

      7. unpow2 61.4%

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

      8. *-commutative 61.4%

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

      9. times-frac 65.8%

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

   10. Simplified 65.8%

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

       1. associate-*l/ 65.9%

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

       2. associate-*r* 65.9%

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

   12. Applied egg-rr 65.9%

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

       1. unpow2 65.8%

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

   14. Applied egg-rr 65.9%

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

   if 1.7499999999999999e226 < c0

   1. Initial program 47.5%

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

      1. associate-*l* 47.5%

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

      2. difference-of-squares 52.3%

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

      3. associate-*l* 52.3%

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

      4. associate-*l* 52.3%

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

   3. Simplified 52.3%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. unpow2 0.0%

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

      2. associate--r- 0.0%

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

      3. +-inverses 14.5%

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

      4. unpow2 14.5%

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

      5. associate-/l/ 14.8%

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

      6. associate-/l/ 14.9%

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

   7. Simplified 14.9%

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

   8. Taylor expanded in M around 0 57.7%

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

      1. associate-/r* 57.7%

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

      2. *-commutative 57.7%

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

      3. associate-*r/ 52.9%

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

      4. unpow2 52.9%

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

      5. unpow2 52.9%

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

      6. times-frac 62.5%

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

      7. unpow2 62.5%

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

      8. *-commutative 62.5%

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

      9. times-frac 57.7%

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

   10. Simplified 57.7%

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

       1. associate-*l/ 57.8%

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

       2. associate-*r* 57.8%

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

   12. Applied egg-rr 57.8%

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

   13. Taylor expanded in c0 around 0 53.0%

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

       1. unpow2 53.0%

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

       2. associate-*l* 58.0%

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

       3. unpow2 58.0%

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

       4. unpow2 58.0%

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

       5. associate-*r* 62.8%

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

   15. Simplified 62.8%

       \[\leadsto \frac{\color{blue}{2 \cdot \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}}}{2 \cdot w} \]
3. Recombined 5 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -2.1 \cdot 10^{+214}:\\ \;\;\;\;\frac{c0}{\frac{w}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{h}}{w}}}\\ \mathbf{elif}\;c0 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{0}{w}, 0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;c0 \leq 2.2 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{0 \cdot \left(c0 \cdot c0\right)}{w}, 0.25 \cdot \frac{D \cdot \frac{h}{\frac{d}{M \cdot M}}}{\frac{d}{D}}\right)\\ \mathbf{elif}\;c0 \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{c0 \cdot \left(\left(2 \cdot \frac{c0}{h}\right) \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)}{2 \cdot w}\\ \mathbf{elif}\;c0 \leq 1.75 \cdot 10^{+226}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{0}{w}, 0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}}{2 \cdot w}\\ \end{array} \]

Alternative 3: 42.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \frac{0}{w}, 0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{if}\;c0 \leq -4.2 \cdot 10^{+217}:\\ \;\;\;\;\frac{c0}{\frac{w}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{h}}{w}}}\\ \mathbf{elif}\;c0 \leq 1.35 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 1.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{c0 \cdot \left(\left(2 \cdot \frac{c0}{h}\right) \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)}{2 \cdot w}\\ \mathbf{elif}\;c0 \leq 1.35 \cdot 10^{+228}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0
         (fma -0.5 (/ 0.0 w) (* 0.25 (* (/ D (/ d D)) (/ (* h (* M M)) d))))))
   (if (<= c0 -4.2e+217)
     (/ c0 (/ w (* (pow (/ d D) 2.0) (/ (/ c0 h) w))))
     (if (<= c0 1.35e+46)
       t_0
       (if (<= c0 1.9e+155)
         (/ (* c0 (* (* 2.0 (/ c0 h)) (/ (* (/ d D) (/ d D)) w))) (* 2.0 w))
         (if (<= c0 1.35e+228)
           t_0
           (/
            (* 2.0 (/ (* d (* d (* c0 c0))) (* D (* (* w h) D))))
            (* 2.0 w))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = fma(-0.5, (0.0 / w), (0.25 * ((D / (d / D)) * ((h * (M * M)) / d))));
	double tmp;
	if (c0 <= -4.2e+217) {
		tmp = c0 / (w / (pow((d / D), 2.0) * ((c0 / h) / w)));
	} else if (c0 <= 1.35e+46) {
		tmp = t_0;
	} else if (c0 <= 1.9e+155) {
		tmp = (c0 * ((2.0 * (c0 / h)) * (((d / D) * (d / D)) / w))) / (2.0 * w);
	} else if (c0 <= 1.35e+228) {
		tmp = t_0;
	} else {
		tmp = (2.0 * ((d * (d * (c0 * c0))) / (D * ((w * h) * D)))) / (2.0 * w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = fma(-0.5, Float64(0.0 / w), Float64(0.25 * Float64(Float64(D / Float64(d / D)) * Float64(Float64(h * Float64(M * M)) / d))))
	tmp = 0.0
	if (c0 <= -4.2e+217)
		tmp = Float64(c0 / Float64(w / Float64((Float64(d / D) ^ 2.0) * Float64(Float64(c0 / h) / w))));
	elseif (c0 <= 1.35e+46)
		tmp = t_0;
	elseif (c0 <= 1.9e+155)
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 * Float64(c0 / h)) * Float64(Float64(Float64(d / D) * Float64(d / D)) / w))) / Float64(2.0 * w));
	elseif (c0 <= 1.35e+228)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(d * Float64(d * Float64(c0 * c0))) / Float64(D * Float64(Float64(w * h) * D)))) / Float64(2.0 * w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(-0.5 * N[(0.0 / w), $MachinePrecision] + N[(0.25 * N[(N[(D / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -4.2e+217], N[(c0 / N[(w / N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 1.35e+46], t$95$0, If[LessEqual[c0, 1.9e+155], N[(N[(c0 * N[(N[(2.0 * N[(c0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 1.35e+228], t$95$0, N[(N[(2.0 * N[(N[(d * N[(d * N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c0 < -4.2000000000000002e217

   1. Initial program 26.7%

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

      1. associate-*l* 23.5%

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

      2. difference-of-squares 30.2%

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

      3. associate-*l* 30.2%

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

      4. associate-*l* 46.7%

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

   3. Simplified 46.7%

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

      1. flip-+ 3.3%

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

   5. Applied egg-rr 3.3%

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

      1. unpow2 3.3%

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

      2. associate--r- 3.3%

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

      3. +-inverses 13.5%

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

      4. unpow2 13.5%

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

      5. associate-/l/ 10.2%

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

      6. associate-/l/ 6.8%

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

   7. Simplified 6.8%

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

   8. Taylor expanded in M around 0 34.2%

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

      1. associate-/r* 34.3%

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

      2. *-commutative 34.3%

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

      3. associate-*r/ 34.3%

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

      4. unpow2 34.3%

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

      5. unpow2 34.3%

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

      6. times-frac 37.7%

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

      7. unpow2 37.7%

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

      8. *-commutative 37.7%

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

      9. times-frac 47.9%

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

   10. Simplified 47.9%

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

       1. associate-*l/ 47.9%

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

       2. associate-*r* 47.9%

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

   12. Applied egg-rr 47.9%

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

       1. associate-/l* 51.1%

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

       2. associate-*l* 51.1%

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

       3. times-frac 37.7%

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

       4. *-commutative 37.7%

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

       5. times-frac 51.1%

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

       6. unpow2 51.1%

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

       7. times-frac 47.7%

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

       8. unpow2 47.7%

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

       9. unpow2 47.7%

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

       10. associate-/r* 47.7%

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

       11. unpow2 47.7%

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

       12. *-commutative 47.7%

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

       13. unpow2 47.7%

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

       14. times-frac 47.7%

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

       15. metadata-eval 47.7%

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

   14. Simplified 54.5%

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

   if -4.2000000000000002e217 < c0 < 1.3500000000000001e46 or 1.9e155 < c0 < 1.3500000000000001e228

   1. Initial program 21.7%

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

      1. associate-*l/ 21.1%

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

      2. *-commutative 21.1%

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

      3. fma-def 19.5%

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

      4. associate-*l* 19.0%

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

      5. *-commutative 19.0%

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

      6. associate-*r* 19.0%

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

      7. associate-*l* 19.0%

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

      8. *-commutative 19.0%

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

   3. Simplified 20.0%

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

   4. Taylor expanded in c0 around -inf 6.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   5. Step-by-step derivation

      1. fma-def 6.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right)} \]

      2. *-commutative 6.2%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{{c0}^{2} \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      3. unpow2 6.2%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(c0 \cdot c0\right)} \cdot \left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      4. distribute-rgt1-in 6.2%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      5. metadata-eval 6.2%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      6. mul0-lft 33.2%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot \color{blue}{0}}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      7. associate-/l* 33.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\frac{{D}^{2}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}}\right) \]

      8. unpow2 33.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{\color{blue}{D \cdot D}}{\frac{{d}^{2}}{{M}^{2} \cdot h}}\right) \]

      9. unpow2 33.7%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{\color{blue}{d \cdot d}}{{M}^{2} \cdot h}}\right) \]

      10. *-commutative 33.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{\color{blue}{h \cdot {M}^{2}}}}\right) \]

      11. unpow2 33.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \color{blue}{\left(M \cdot M\right)}}}\right) \]

   6. Simplified 33.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]
   7. Step-by-step derivation

      1. associate-/r/ 33.1%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right) \]

   8. Applied egg-rr 33.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D \cdot D}{d \cdot d} \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}\right) \]

   9. Taylor expanded in D around 0 33.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 33.2%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2}}\right) \]

       2. unpow2 33.2%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)}{{d}^{2}}\right) \]

       3. unpow2 33.2%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}}\right) \]

       4. times-frac 36.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

       5. unpow2 36.7%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{d} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

       6. associate-/l* 41.1%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \left(\color{blue}{\frac{D}{\frac{d}{D}}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

   11. Simplified 41.1%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(c0 \cdot c0\right) \cdot 0}{w}, 0.25 \cdot \color{blue}{\left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}\right) \]

   12. Taylor expanded in c0 around 0 54.4%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{0}}{w}, 0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right) \]

   if 1.3500000000000001e46 < c0 < 1.9e155

   1. Initial program 43.5%

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

      1. associate-*l* 43.5%

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

      2. difference-of-squares 52.2%

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

      3. associate-*l* 52.2%

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

      4. associate-*l* 56.5%

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

   3. Simplified 56.5%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.3%

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

      1. unpow2 0.3%

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

      2. associate--r- 0.8%

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

      3. +-inverses 13.8%

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

      4. unpow2 13.8%

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

      5. associate-/l/ 13.5%

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

      6. associate-/l/ 13.8%

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

   7. Simplified 13.8%

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

   8. Taylor expanded in M around 0 52.8%

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

      1. associate-/r* 52.8%

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

      2. *-commutative 52.8%

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

      3. associate-*r/ 52.7%

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

      4. unpow2 52.7%

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

      5. unpow2 52.7%

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

      6. times-frac 61.4%

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

      7. unpow2 61.4%

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

      8. *-commutative 61.4%

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

      9. times-frac 65.8%

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

   10. Simplified 65.8%

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

       1. associate-*l/ 65.9%

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

       2. associate-*r* 65.9%

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

   12. Applied egg-rr 65.9%

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

       1. unpow2 65.8%

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

   14. Applied egg-rr 65.9%

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

   if 1.3500000000000001e228 < c0

   1. Initial program 47.5%

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

      1. associate-*l* 47.5%

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

      2. difference-of-squares 52.3%

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

      3. associate-*l* 52.3%

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

      4. associate-*l* 52.3%

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

   3. Simplified 52.3%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. unpow2 0.0%

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

      2. associate--r- 0.0%

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

      3. +-inverses 14.5%

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

      4. unpow2 14.5%

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

      5. associate-/l/ 14.8%

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

      6. associate-/l/ 14.9%

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

   7. Simplified 14.9%

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

   8. Taylor expanded in M around 0 57.7%

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

      1. associate-/r* 57.7%

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

      2. *-commutative 57.7%

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

      3. associate-*r/ 52.9%

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

      4. unpow2 52.9%

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

      5. unpow2 52.9%

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

      6. times-frac 62.5%

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

      7. unpow2 62.5%

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

      8. *-commutative 62.5%

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

      9. times-frac 57.7%

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

   10. Simplified 57.7%

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

       1. associate-*l/ 57.8%

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

       2. associate-*r* 57.8%

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

   12. Applied egg-rr 57.8%

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

   13. Taylor expanded in c0 around 0 53.0%

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

       1. unpow2 53.0%

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

       2. associate-*l* 58.0%

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

       3. unpow2 58.0%

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

       4. unpow2 58.0%

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

       5. associate-*r* 62.8%

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

   15. Simplified 62.8%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -4.2 \cdot 10^{+217}:\\ \;\;\;\;\frac{c0}{\frac{w}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{\frac{c0}{h}}{w}}}\\ \mathbf{elif}\;c0 \leq 1.35 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{0}{w}, 0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{elif}\;c0 \leq 1.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{c0 \cdot \left(\left(2 \cdot \frac{c0}{h}\right) \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)}{2 \cdot w}\\ \mathbf{elif}\;c0 \leq 1.35 \cdot 10^{+228}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{0}{w}, 0.25 \cdot \left(\frac{D}{\frac{d}{D}} \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}}{2 \cdot w}\\ \end{array} \]

Alternative 4: 42.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{0}{w}\\ \mathbf{if}\;M \cdot M \leq 2.7 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \cdot M \leq 1.65 \cdot 10^{+50}:\\ \;\;\;\;\frac{2 \cdot \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}}{2 \cdot w}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{+130}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(\left(2 \cdot \frac{c0}{h}\right) \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ 0.0 w))))
   (if (<= (* M M) 2.7e-155)
     t_0
     (if (<= (* M M) 1.65e+50)
       (/ (* 2.0 (/ (* d (* d (* c0 c0))) (* D (* (* w h) D)))) (* 2.0 w))
       (if (<= (* M M) 5e+130)
         t_0
         (/
          (* c0 (* (* 2.0 (/ c0 h)) (/ (* (/ d D) (/ d D)) w)))
          (* 2.0 w)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -0.5 * (0.0 / w);
	double tmp;
	if ((M * M) <= 2.7e-155) {
		tmp = t_0;
	} else if ((M * M) <= 1.65e+50) {
		tmp = (2.0 * ((d * (d * (c0 * c0))) / (D * ((w * h) * D)))) / (2.0 * w);
	} else if ((M * M) <= 5e+130) {
		tmp = t_0;
	} else {
		tmp = (c0 * ((2.0 * (c0 / h)) * (((d / D) * (d / D)) / w))) / (2.0 * w);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (0.0d0 / w)
    if ((m * m) <= 2.7d-155) then
        tmp = t_0
    else if ((m * m) <= 1.65d+50) then
        tmp = (2.0d0 * ((d_1 * (d_1 * (c0 * c0))) / (d * ((w * h) * d)))) / (2.0d0 * w)
    else if ((m * m) <= 5d+130) then
        tmp = t_0
    else
        tmp = (c0 * ((2.0d0 * (c0 / h)) * (((d_1 / d) * (d_1 / d)) / w))) / (2.0d0 * w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -0.5 * (0.0 / w);
	double tmp;
	if ((M * M) <= 2.7e-155) {
		tmp = t_0;
	} else if ((M * M) <= 1.65e+50) {
		tmp = (2.0 * ((d * (d * (c0 * c0))) / (D * ((w * h) * D)))) / (2.0 * w);
	} else if ((M * M) <= 5e+130) {
		tmp = t_0;
	} else {
		tmp = (c0 * ((2.0 * (c0 / h)) * (((d / D) * (d / D)) / w))) / (2.0 * w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = -0.5 * (0.0 / w)
	tmp = 0
	if (M * M) <= 2.7e-155:
		tmp = t_0
	elif (M * M) <= 1.65e+50:
		tmp = (2.0 * ((d * (d * (c0 * c0))) / (D * ((w * h) * D)))) / (2.0 * w)
	elif (M * M) <= 5e+130:
		tmp = t_0
	else:
		tmp = (c0 * ((2.0 * (c0 / h)) * (((d / D) * (d / D)) / w))) / (2.0 * w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(-0.5 * Float64(0.0 / w))
	tmp = 0.0
	if (Float64(M * M) <= 2.7e-155)
		tmp = t_0;
	elseif (Float64(M * M) <= 1.65e+50)
		tmp = Float64(Float64(2.0 * Float64(Float64(d * Float64(d * Float64(c0 * c0))) / Float64(D * Float64(Float64(w * h) * D)))) / Float64(2.0 * w));
	elseif (Float64(M * M) <= 5e+130)
		tmp = t_0;
	else
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 * Float64(c0 / h)) * Float64(Float64(Float64(d / D) * Float64(d / D)) / w))) / Float64(2.0 * w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = -0.5 * (0.0 / w);
	tmp = 0.0;
	if ((M * M) <= 2.7e-155)
		tmp = t_0;
	elseif ((M * M) <= 1.65e+50)
		tmp = (2.0 * ((d * (d * (c0 * c0))) / (D * ((w * h) * D)))) / (2.0 * w);
	elseif ((M * M) <= 5e+130)
		tmp = t_0;
	else
		tmp = (c0 * ((2.0 * (c0 / h)) * (((d / D) * (d / D)) / w))) / (2.0 * w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(-0.5 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * M), $MachinePrecision], 2.7e-155], t$95$0, If[LessEqual[N[(M * M), $MachinePrecision], 1.65e+50], N[(N[(2.0 * N[(N[(d * N[(d * N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * M), $MachinePrecision], 5e+130], t$95$0, N[(N[(c0 * N[(N[(2.0 * N[(c0 / h), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M M) < 2.69999999999999981e-155 or 1.65e50 < (*.f64 M M) < 4.9999999999999996e130

   1. Initial program 29.5%

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

      1. associate-*l* 26.7%

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

      2. difference-of-squares 26.7%

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

      3. associate-*l* 28.2%

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

      4. associate-*l* 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in c0 around -inf 5.5%

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

      1. *-commutative 5.5%

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

      2. unpow2 5.5%

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

      3. distribute-rgt1-in 5.5%

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

      4. metadata-eval 5.5%

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

      5. mul0-lft 34.6%

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

   6. Simplified 34.6%

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

   7. Taylor expanded in c0 around 0 52.4%

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

   if 2.69999999999999981e-155 < (*.f64 M M) < 1.65e50

   1. Initial program 39.3%

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

      1. associate-*l* 39.3%

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

      2. difference-of-squares 39.3%

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

      3. associate-*l* 39.3%

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

      4. associate-*l* 41.7%

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

   3. Simplified 41.7%

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

      1. flip-+ 0.1%

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

   5. Applied egg-rr 0.2%

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

      1. unpow2 0.2%

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

      2. associate--r- 7.4%

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

      3. +-inverses 17.8%

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

      4. unpow2 17.8%

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

      5. associate-/l/ 18.0%

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

      6. associate-/l/ 20.4%

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

   7. Simplified 20.4%

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

   8. Taylor expanded in M around 0 44.5%

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

      1. associate-/r* 44.6%

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

      2. *-commutative 44.6%

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

      3. associate-*r/ 42.2%

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

      4. unpow2 42.2%

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

      5. unpow2 42.2%

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

      6. times-frac 45.0%

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

      7. unpow2 45.0%

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

      8. *-commutative 45.0%

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

      9. times-frac 47.2%

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

   10. Simplified 47.2%

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

       1. associate-*l/ 47.4%

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

       2. associate-*r* 47.4%

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

   12. Applied egg-rr 47.4%

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

   13. Taylor expanded in c0 around 0 41.9%

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

       1. unpow2 41.9%

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

       2. associate-*l* 49.6%

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

       3. unpow2 49.6%

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

       4. unpow2 49.6%

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

       5. associate-*r* 54.7%

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

   15. Simplified 54.7%

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

   if 4.9999999999999996e130 < (*.f64 M M)

   1. Initial program 14.7%

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

      1. associate-*l* 14.7%

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

      2. difference-of-squares 33.2%

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

      3. associate-*l* 33.2%

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

      4. associate-*l* 36.8%

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

   3. Simplified 36.8%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. unpow2 0.0%

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

      2. associate--r- 1.2%

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

      3. +-inverses 6.8%

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

      4. unpow2 6.8%

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

      5. associate-/l/ 6.8%

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

      6. associate-/l/ 6.8%

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

   7. Simplified 6.8%

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

   8. Taylor expanded in M around 0 38.0%

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

      1. associate-/r* 37.9%

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

      2. *-commutative 37.9%

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

      3. associate-*r/ 38.0%

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

      4. unpow2 38.0%

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

      5. unpow2 38.0%

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

      6. times-frac 42.9%

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

      7. unpow2 42.9%

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

      8. *-commutative 42.9%

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

      9. times-frac 44.2%

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

   10. Simplified 44.2%

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

       1. associate-*l/ 44.3%

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

       2. associate-*r* 44.3%

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

   12. Applied egg-rr 44.3%

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

       1. unpow2 44.2%

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

   14. Applied egg-rr 44.3%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 2.7 \cdot 10^{-155}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{elif}\;M \cdot M \leq 1.65 \cdot 10^{+50}:\\ \;\;\;\;\frac{2 \cdot \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}}{2 \cdot w}\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{+130}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(\left(2 \cdot \frac{c0}{h}\right) \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)}{2 \cdot w}\\ \end{array} \]

Alternative 5: 38.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{-78} \lor \neg \left(M \leq 4.7 \cdot 10^{+38}\right) \land M \leq 2.7 \cdot 10^{+65}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= M 2.6e-78) (and (not (<= M 4.7e+38)) (<= M 2.7e+65)))
   (* -0.5 (/ 0.0 w))
   (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ c0 h) (/ (* (/ d D) (/ d D)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M <= 2.6e-78) || (!(M <= 4.7e+38) && (M <= 2.7e+65))) {
		tmp = -0.5 * (0.0 / w);
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / h) * (((d / D) * (d / D)) / w)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= 2.6d-78) .or. (.not. (m <= 4.7d+38)) .and. (m <= 2.7d+65)) then
        tmp = (-0.5d0) * (0.0d0 / w)
    else
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * ((c0 / h) * (((d_1 / d) * (d_1 / d)) / w)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M <= 2.6e-78) || (!(M <= 4.7e+38) && (M <= 2.7e+65))) {
		tmp = -0.5 * (0.0 / w);
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / h) * (((d / D) * (d / D)) / w)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M <= 2.6e-78) or (not (M <= 4.7e+38) and (M <= 2.7e+65)):
		tmp = -0.5 * (0.0 / w)
	else:
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / h) * (((d / D) * (d / D)) / w)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((M <= 2.6e-78) || (!(M <= 4.7e+38) && (M <= 2.7e+65)))
		tmp = Float64(-0.5 * Float64(0.0 / w));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(c0 / h) * Float64(Float64(Float64(d / D) * Float64(d / D)) / w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M <= 2.6e-78) || (~((M <= 4.7e+38)) && (M <= 2.7e+65)))
		tmp = -0.5 * (0.0 / w);
	else
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / h) * (((d / D) * (d / D)) / w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[M, 2.6e-78], And[N[Not[LessEqual[M, 4.7e+38]], $MachinePrecision], LessEqual[M, 2.7e+65]]], N[(-0.5 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(c0 / h), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.6000000000000001e-78 or 4.6999999999999999e38 < M < 2.70000000000000019e65

   1. Initial program 26.9%

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

      1. associate-*l* 24.8%

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

      2. difference-of-squares 28.1%

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

      3. associate-*l* 29.2%

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

      4. associate-*l* 33.4%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in c0 around -inf 4.1%

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

      1. *-commutative 4.1%

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

      2. unpow2 4.1%

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

      3. distribute-rgt1-in 4.1%

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

      4. metadata-eval 4.1%

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

      5. mul0-lft 27.9%

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

   6. Simplified 27.9%

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

   7. Taylor expanded in c0 around 0 41.8%

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

   if 2.6000000000000001e-78 < M < 4.6999999999999999e38 or 2.70000000000000019e65 < M

   1. Initial program 25.1%

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

      1. associate-*l* 25.1%

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

      2. difference-of-squares 37.1%

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

      3. associate-*l* 37.1%

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

      4. associate-*l* 39.7%

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

   3. Simplified 39.7%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. unpow2 0.0%

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

      2. associate--r- 1.4%

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

      3. +-inverses 9.9%

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

      4. unpow2 9.9%

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

      5. associate-/l/ 10.0%

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

      6. associate-/l/ 10.0%

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

   7. Simplified 10.0%

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

   8. Taylor expanded in M around 0 40.6%

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

      1. associate-/r* 40.6%

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

      2. *-commutative 40.6%

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

      3. associate-*r/ 39.3%

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

      4. unpow2 39.3%

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

      5. unpow2 39.3%

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

      6. times-frac 42.2%

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

      7. unpow2 42.2%

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

      8. *-commutative 42.2%

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

      9. times-frac 43.4%

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

   10. Simplified 43.4%

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

       1. unpow2 43.4%

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

   12. Applied egg-rr 43.4%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{-78} \lor \neg \left(M \leq 4.7 \cdot 10^{+38}\right) \land M \leq 2.7 \cdot 10^{+65}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)\right)\\ \end{array} \]

Alternative 6: 38.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{0}{w}\\ \mathbf{if}\;M \leq 1.8 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq 1.3 \cdot 10^{+25}:\\ \;\;\;\;\frac{2 \cdot \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}}{2 \cdot w}\\ \mathbf{elif}\;M \leq 3.8 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ 0.0 w))))
   (if (<= M 1.8e-80)
     t_0
     (if (<= M 1.3e+25)
       (/ (* 2.0 (/ (* d (* d (* c0 c0))) (* D (* (* w h) D)))) (* 2.0 w))
       (if (<= M 3.8e+64)
         t_0
         (*
          (/ c0 (* 2.0 w))
          (* 2.0 (* (/ c0 h) (/ (* (/ d D) (/ d D)) w)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -0.5 * (0.0 / w);
	double tmp;
	if (M <= 1.8e-80) {
		tmp = t_0;
	} else if (M <= 1.3e+25) {
		tmp = (2.0 * ((d * (d * (c0 * c0))) / (D * ((w * h) * D)))) / (2.0 * w);
	} else if (M <= 3.8e+64) {
		tmp = t_0;
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / h) * (((d / D) * (d / D)) / w)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (0.0d0 / w)
    if (m <= 1.8d-80) then
        tmp = t_0
    else if (m <= 1.3d+25) then
        tmp = (2.0d0 * ((d_1 * (d_1 * (c0 * c0))) / (d * ((w * h) * d)))) / (2.0d0 * w)
    else if (m <= 3.8d+64) then
        tmp = t_0
    else
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * ((c0 / h) * (((d_1 / d) * (d_1 / d)) / w)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = -0.5 * (0.0 / w);
	double tmp;
	if (M <= 1.8e-80) {
		tmp = t_0;
	} else if (M <= 1.3e+25) {
		tmp = (2.0 * ((d * (d * (c0 * c0))) / (D * ((w * h) * D)))) / (2.0 * w);
	} else if (M <= 3.8e+64) {
		tmp = t_0;
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / h) * (((d / D) * (d / D)) / w)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = -0.5 * (0.0 / w)
	tmp = 0
	if M <= 1.8e-80:
		tmp = t_0
	elif M <= 1.3e+25:
		tmp = (2.0 * ((d * (d * (c0 * c0))) / (D * ((w * h) * D)))) / (2.0 * w)
	elif M <= 3.8e+64:
		tmp = t_0
	else:
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / h) * (((d / D) * (d / D)) / w)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(-0.5 * Float64(0.0 / w))
	tmp = 0.0
	if (M <= 1.8e-80)
		tmp = t_0;
	elseif (M <= 1.3e+25)
		tmp = Float64(Float64(2.0 * Float64(Float64(d * Float64(d * Float64(c0 * c0))) / Float64(D * Float64(Float64(w * h) * D)))) / Float64(2.0 * w));
	elseif (M <= 3.8e+64)
		tmp = t_0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(c0 / h) * Float64(Float64(Float64(d / D) * Float64(d / D)) / w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = -0.5 * (0.0 / w);
	tmp = 0.0;
	if (M <= 1.8e-80)
		tmp = t_0;
	elseif (M <= 1.3e+25)
		tmp = (2.0 * ((d * (d * (c0 * c0))) / (D * ((w * h) * D)))) / (2.0 * w);
	elseif (M <= 3.8e+64)
		tmp = t_0;
	else
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / h) * (((d / D) * (d / D)) / w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(-0.5 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 1.8e-80], t$95$0, If[LessEqual[M, 1.3e+25], N[(N[(2.0 * N[(N[(d * N[(d * N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 3.8e+64], t$95$0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(c0 / h), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.8e-80 or 1.2999999999999999e25 < M < 3.8000000000000001e64

   1. Initial program 27.9%

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

      1. associate-*l* 25.9%

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

      2. difference-of-squares 29.1%

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

      3. associate-*l* 30.2%

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

      4. associate-*l* 34.3%

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

   3. Simplified 34.3%

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

   4. Taylor expanded in c0 around -inf 4.0%

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

      1. *-commutative 4.0%

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

      2. unpow2 4.0%

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

      3. distribute-rgt1-in 4.0%

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

      4. metadata-eval 4.0%

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

      5. mul0-lft 27.9%

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

   6. Simplified 27.9%

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

   7. Taylor expanded in c0 around 0 41.4%

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

   if 1.8e-80 < M < 1.2999999999999999e25

   1. Initial program 37.6%

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

      1. associate-*l* 37.6%

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

      2. difference-of-squares 37.6%

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

      3. associate-*l* 37.6%

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

      4. associate-*l* 41.8%

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

   3. Simplified 41.8%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.1%

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

      1. unpow2 0.1%

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

      2. associate--r- 4.3%

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

      3. +-inverses 17.3%

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

      4. unpow2 17.3%

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

      5. associate-/l/ 17.5%

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

      6. associate-/l/ 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in M around 0 42.2%

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

      1. associate-/r* 42.3%

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

      2. *-commutative 42.3%

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

      3. associate-*r/ 38.1%

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

      4. unpow2 38.1%

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

      5. unpow2 38.1%

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

      6. times-frac 42.7%

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

      7. unpow2 42.7%

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

      8. *-commutative 42.7%

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

      9. times-frac 46.6%

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

   10. Simplified 46.6%

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

       1. associate-*l/ 46.9%

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

       2. associate-*r* 46.9%

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

   12. Applied egg-rr 46.9%

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

   13. Taylor expanded in c0 around 0 38.1%

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

       1. unpow2 38.1%

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

       2. associate-*l* 46.6%

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

       3. unpow2 46.6%

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

       4. unpow2 46.6%

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

       5. associate-*r* 50.9%

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

   15. Simplified 50.9%

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

   if 3.8000000000000001e64 < M

   1. Initial program 14.7%

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

      1. associate-*l* 14.7%

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

      2. difference-of-squares 33.7%

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

      3. associate-*l* 33.7%

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

      4. associate-*l* 35.7%

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

   3. Simplified 35.7%

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

      1. flip-+ 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. unpow2 0.0%

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

      2. associate--r- 0.0%

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

      3. +-inverses 4.9%

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

      4. unpow2 4.9%

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

      5. associate-/l/ 4.9%

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

      6. associate-/l/ 4.9%

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

   7. Simplified 4.9%

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

   8. Taylor expanded in M around 0 37.0%

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

      1. associate-/r* 36.9%

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

      2. *-commutative 36.9%

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

      3. associate-*r/ 36.9%

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

      4. unpow2 36.9%

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

      5. unpow2 36.9%

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

      6. times-frac 39.2%

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

      7. unpow2 39.2%

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

      8. *-commutative 39.2%

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

      9. times-frac 39.1%

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

   10. Simplified 39.1%

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

       1. unpow2 39.1%

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

   12. Applied egg-rr 39.1%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-80}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{elif}\;M \leq 1.3 \cdot 10^{+25}:\\ \;\;\;\;\frac{2 \cdot \frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}}{2 \cdot w}\\ \mathbf{elif}\;M \leq 3.8 \cdot 10^{+64}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)\right)\\ \end{array} \]

Alternative 7: 36.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-78} \lor \neg \left(M \leq 1.9 \cdot 10^{+38}\right) \land M \leq 6.2 \cdot 10^{+91}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot d}{D \cdot D} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{c0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= M 4e-78) (and (not (<= M 1.9e+38)) (<= M 6.2e+91)))
   (* -0.5 (/ 0.0 w))
   (* (/ (* d d) (* D D)) (* (/ c0 (* w h)) (/ c0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M <= 4e-78) || (!(M <= 1.9e+38) && (M <= 6.2e+91))) {
		tmp = -0.5 * (0.0 / w);
	} else {
		tmp = ((d * d) / (D * D)) * ((c0 / (w * h)) * (c0 / w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= 4d-78) .or. (.not. (m <= 1.9d+38)) .and. (m <= 6.2d+91)) then
        tmp = (-0.5d0) * (0.0d0 / w)
    else
        tmp = ((d_1 * d_1) / (d * d)) * ((c0 / (w * h)) * (c0 / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M <= 4e-78) || (!(M <= 1.9e+38) && (M <= 6.2e+91))) {
		tmp = -0.5 * (0.0 / w);
	} else {
		tmp = ((d * d) / (D * D)) * ((c0 / (w * h)) * (c0 / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M <= 4e-78) or (not (M <= 1.9e+38) and (M <= 6.2e+91)):
		tmp = -0.5 * (0.0 / w)
	else:
		tmp = ((d * d) / (D * D)) * ((c0 / (w * h)) * (c0 / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((M <= 4e-78) || (!(M <= 1.9e+38) && (M <= 6.2e+91)))
		tmp = Float64(-0.5 * Float64(0.0 / w));
	else
		tmp = Float64(Float64(Float64(d * d) / Float64(D * D)) * Float64(Float64(c0 / Float64(w * h)) * Float64(c0 / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M <= 4e-78) || (~((M <= 1.9e+38)) && (M <= 6.2e+91)))
		tmp = -0.5 * (0.0 / w);
	else
		tmp = ((d * d) / (D * D)) * ((c0 / (w * h)) * (c0 / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[M, 4e-78], And[N[Not[LessEqual[M, 1.9e+38]], $MachinePrecision], LessEqual[M, 6.2e+91]]], N[(-0.5 * N[(0.0 / w), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4e-78 or 1.8999999999999999e38 < M < 6.19999999999999995e91

   1. Initial program 26.2%

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

      1. associate-*l* 24.1%

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

      2. difference-of-squares 27.4%

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

      3. associate-*l* 28.4%

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

      4. associate-*l* 32.5%

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

   3. Simplified 32.5%

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

   4. Taylor expanded in c0 around -inf 4.0%

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

      1. *-commutative 4.0%

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

      2. unpow2 4.0%

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

      3. distribute-rgt1-in 4.0%

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

      4. metadata-eval 4.0%

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

      5. mul0-lft 27.9%

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

   6. Simplified 27.9%

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

   7. Taylor expanded in c0 around 0 41.3%

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

   if 4e-78 < M < 1.8999999999999999e38 or 6.19999999999999995e91 < M

   1. Initial program 26.8%

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

      1. associate-*l* 26.8%

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

      2. difference-of-squares 39.7%

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

      3. associate-*l* 39.7%

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

      4. associate-*l* 42.5%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in c0 around inf 31.8%

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

      1. times-frac 31.8%

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

      2. unpow2 31.8%

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

      3. unpow2 31.8%

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

      4. unpow2 31.8%

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

      5. *-commutative 31.8%

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

      6. unpow2 31.8%

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

   6. Simplified 31.8%

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

   7. Taylor expanded in c0 around 0 31.8%

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

      1. unpow2 31.8%

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

      2. unpow2 31.8%

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

      3. associate-*l* 31.9%

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

      4. *-commutative 31.9%

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

      5. times-frac 41.7%

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

   9. Simplified 41.7%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-78} \lor \neg \left(M \leq 1.9 \cdot 10^{+38}\right) \land M \leq 6.2 \cdot 10^{+91}:\\ \;\;\;\;-0.5 \cdot \frac{0}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot d}{D \cdot D} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{c0}{w}\right)\\ \end{array} \]

Alternative 8: 33.5% accurate, 30.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 26.3%

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

   1. associate-*l* 24.9%

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

   2. difference-of-squares 30.8%

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

   3. associate-*l* 31.5%

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

   4. associate-*l* 35.3%

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

3. Simplified 35.3%

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

4. Taylor expanded in c0 around -inf 2.9%

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

   1. *-commutative 2.9%

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

   2. unpow2 2.9%

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

   3. distribute-rgt1-in 2.9%

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

   4. metadata-eval 2.9%

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

   5. mul0-lft 22.9%

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

6. Simplified 22.9%

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

7. Taylor expanded in c0 around 0 34.0%

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

8. Final simplification 34.0%

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

Reproduce

herbie shell --seed 2023228 
(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))))))