Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.0% → 66.6%

Time: 37.7s

Alternatives: 8

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = -7.769687906319002e+303
  2. w = 6.195639414201533e-213
  3. h = 2.6110266054466837e-101
  4. D = -4.66505019648093e-309
  5. d = -2.657130228691967e+225
  6. M = 1.7156557152097113e-289

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.0% 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: 66.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.5%

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

      1. times-frac 76.1%

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

      2. fma-def 76.1%

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

      3. times-frac 76.3%

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

      4. difference-of-squares 76.3%

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

   3. Simplified 75.1%

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

      1. *-un-lft-identity 75.1%

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

      2. associate-*l* 75.1%

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

      3. div-inv 75.1%

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

      4. clear-num 75.1%

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

      5. associate-*r/ 76.3%

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

      6. *-commutative 76.3%

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

      7. times-frac 76.3%

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

      8. frac-times 77.3%

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

      9. pow2 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. *-lft-identity 77.3%

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

      2. *-commutative 77.3%

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

      3. associate-*l/ 77.2%

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

      4. associate-/l* 77.2%

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

      5. *-commutative 77.2%

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

   7. Simplified 77.2%

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

      1. fma-udef 77.2%

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

      2. *-commutative 77.2%

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

      3. associate-*r/ 77.2%

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

      4. associate-/l* 77.2%

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

   9. Applied egg-rr 77.2%

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

   10. Taylor expanded in c0 around inf 80.8%

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

       1. associate-/r* 80.8%

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

       2. unpow2 80.8%

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

       3. unpow2 80.8%

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

   12. Simplified 80.8%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in c0 around -inf 2.9%

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

      1. fma-def 2.9%

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

      2. associate-/l* 2.9%

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

      3. unpow2 2.9%

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

      4. unpow2 2.9%

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

      5. *-commutative 2.9%

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

      6. unpow2 2.9%

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

      7. associate-*r* 2.9%

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

   4. Simplified 36.4%

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

   5. Taylor expanded in c0 around 0 46.7%

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

      1. associate-/l* 46.8%

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

      2. *-commutative 46.8%

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

      3. unpow2 46.8%

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

      4. unpow2 46.8%

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

      5. *-commutative 46.8%

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

      6. unpow2 46.8%

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

   7. Simplified 46.8%

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

      1. *-un-lft-identity 46.8%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

      2. associate-/l* 53.1%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\frac{D}{\frac{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}{D}}}\right) \]

      3. associate-/l* 63.7%

         \[\leadsto 0.25 \cdot \left(1 \cdot \frac{D}{\frac{\color{blue}{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}}{D}}\right) \]

   9. Applied egg-rr 63.7%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D}{\frac{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}{D}}\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 63.7%

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

       2. associate-/r/ 63.7%

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

       3. unpow2 63.7%

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

       4. associate-/l* 66.9%

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

       5. unpow2 66.9%

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

   11. Simplified 66.9%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{D}{\frac{d}{\frac{h}{\frac{d}{M \cdot M}}}} \cdot D\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

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

Alternative 2: 43.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{D}\\ t_1 := 0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{\frac{h}{\frac{d}{M \cdot M}}}}\right)\\ t_2 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;h \leq -6.8 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -1.6 \cdot 10^{-224}:\\ \;\;\;\;t_0 \cdot \frac{\frac{c0}{w} \cdot \frac{\frac{c0}{h}}{w}}{D}\\ \mathbf{elif}\;h \leq 5.5 \cdot 10^{-200}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{d \cdot \frac{d}{t_2}}{D}}\\ \mathbf{elif}\;h \leq 7.6 \cdot 10^{-140}:\\ \;\;\;\;t_0 \cdot \frac{c0 \cdot c0}{D \cdot \left(w \cdot \left(w \cdot h\right)\right)}\\ \mathbf{elif}\;h \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;0.25 \cdot \left(t_2 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{elif}\;h \leq 420000000000 \lor \neg \left(h \leq 5 \cdot 10^{+88}\right):\\ \;\;\;\;d \cdot \left(\frac{d}{D} \cdot \left(\frac{c0}{h} \cdot \frac{\frac{c0}{w \cdot w}}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d D)))
        (t_1 (* 0.25 (* D (/ D (/ d (/ h (/ d (* M M))))))))
        (t_2 (* h (* M M))))
   (if (<= h -6.8e-19)
     t_1
     (if (<= h -1.6e-224)
       (* t_0 (/ (* (/ c0 w) (/ (/ c0 h) w)) D))
       (if (<= h 5.5e-200)
         (* 0.25 (/ D (/ (* d (/ d t_2)) D)))
         (if (<= h 7.6e-140)
           (* t_0 (/ (* c0 c0) (* D (* w (* w h)))))
           (if (<= h 4.5e-8)
             (* 0.25 (* t_2 (* (/ D d) (/ D d))))
             (if (or (<= h 420000000000.0) (not (<= h 5e+88)))
               (* d (* (/ d D) (* (/ c0 h) (/ (/ c0 (* w w)) D))))
               t_1))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / D);
	double t_1 = 0.25 * (D * (D / (d / (h / (d / (M * M))))));
	double t_2 = h * (M * M);
	double tmp;
	if (h <= -6.8e-19) {
		tmp = t_1;
	} else if (h <= -1.6e-224) {
		tmp = t_0 * (((c0 / w) * ((c0 / h) / w)) / D);
	} else if (h <= 5.5e-200) {
		tmp = 0.25 * (D / ((d * (d / t_2)) / D));
	} else if (h <= 7.6e-140) {
		tmp = t_0 * ((c0 * c0) / (D * (w * (w * h))));
	} else if (h <= 4.5e-8) {
		tmp = 0.25 * (t_2 * ((D / d) * (D / d)));
	} else if ((h <= 420000000000.0) || !(h <= 5e+88)) {
		tmp = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d_1 * (d_1 / d)
    t_1 = 0.25d0 * (d * (d / (d_1 / (h / (d_1 / (m * m))))))
    t_2 = h * (m * m)
    if (h <= (-6.8d-19)) then
        tmp = t_1
    else if (h <= (-1.6d-224)) then
        tmp = t_0 * (((c0 / w) * ((c0 / h) / w)) / d)
    else if (h <= 5.5d-200) then
        tmp = 0.25d0 * (d / ((d_1 * (d_1 / t_2)) / d))
    else if (h <= 7.6d-140) then
        tmp = t_0 * ((c0 * c0) / (d * (w * (w * h))))
    else if (h <= 4.5d-8) then
        tmp = 0.25d0 * (t_2 * ((d / d_1) * (d / d_1)))
    else if ((h <= 420000000000.0d0) .or. (.not. (h <= 5d+88))) then
        tmp = d_1 * ((d_1 / d) * ((c0 / h) * ((c0 / (w * w)) / d)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / D);
	double t_1 = 0.25 * (D * (D / (d / (h / (d / (M * M))))));
	double t_2 = h * (M * M);
	double tmp;
	if (h <= -6.8e-19) {
		tmp = t_1;
	} else if (h <= -1.6e-224) {
		tmp = t_0 * (((c0 / w) * ((c0 / h) / w)) / D);
	} else if (h <= 5.5e-200) {
		tmp = 0.25 * (D / ((d * (d / t_2)) / D));
	} else if (h <= 7.6e-140) {
		tmp = t_0 * ((c0 * c0) / (D * (w * (w * h))));
	} else if (h <= 4.5e-8) {
		tmp = 0.25 * (t_2 * ((D / d) * (D / d)));
	} else if ((h <= 420000000000.0) || !(h <= 5e+88)) {
		tmp = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = d * (d / D)
	t_1 = 0.25 * (D * (D / (d / (h / (d / (M * M))))))
	t_2 = h * (M * M)
	tmp = 0
	if h <= -6.8e-19:
		tmp = t_1
	elif h <= -1.6e-224:
		tmp = t_0 * (((c0 / w) * ((c0 / h) / w)) / D)
	elif h <= 5.5e-200:
		tmp = 0.25 * (D / ((d * (d / t_2)) / D))
	elif h <= 7.6e-140:
		tmp = t_0 * ((c0 * c0) / (D * (w * (w * h))))
	elif h <= 4.5e-8:
		tmp = 0.25 * (t_2 * ((D / d) * (D / d)))
	elif (h <= 420000000000.0) or not (h <= 5e+88):
		tmp = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)))
	else:
		tmp = t_1
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / D))
	t_1 = Float64(0.25 * Float64(D * Float64(D / Float64(d / Float64(h / Float64(d / Float64(M * M)))))))
	t_2 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (h <= -6.8e-19)
		tmp = t_1;
	elseif (h <= -1.6e-224)
		tmp = Float64(t_0 * Float64(Float64(Float64(c0 / w) * Float64(Float64(c0 / h) / w)) / D));
	elseif (h <= 5.5e-200)
		tmp = Float64(0.25 * Float64(D / Float64(Float64(d * Float64(d / t_2)) / D)));
	elseif (h <= 7.6e-140)
		tmp = Float64(t_0 * Float64(Float64(c0 * c0) / Float64(D * Float64(w * Float64(w * h)))));
	elseif (h <= 4.5e-8)
		tmp = Float64(0.25 * Float64(t_2 * Float64(Float64(D / d) * Float64(D / d))));
	elseif ((h <= 420000000000.0) || !(h <= 5e+88))
		tmp = Float64(d * Float64(Float64(d / D) * Float64(Float64(c0 / h) * Float64(Float64(c0 / Float64(w * w)) / D))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = d * (d / D);
	t_1 = 0.25 * (D * (D / (d / (h / (d / (M * M))))));
	t_2 = h * (M * M);
	tmp = 0.0;
	if (h <= -6.8e-19)
		tmp = t_1;
	elseif (h <= -1.6e-224)
		tmp = t_0 * (((c0 / w) * ((c0 / h) / w)) / D);
	elseif (h <= 5.5e-200)
		tmp = 0.25 * (D / ((d * (d / t_2)) / D));
	elseif (h <= 7.6e-140)
		tmp = t_0 * ((c0 * c0) / (D * (w * (w * h))));
	elseif (h <= 4.5e-8)
		tmp = 0.25 * (t_2 * ((D / d) * (D / d)));
	elseif ((h <= 420000000000.0) || ~((h <= 5e+88)))
		tmp = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(D * N[(D / N[(d / N[(h / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -6.8e-19], t$95$1, If[LessEqual[h, -1.6e-224], N[(t$95$0 * N[(N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 5.5e-200], N[(0.25 * N[(D / N[(N[(d * N[(d / t$95$2), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 7.6e-140], N[(t$95$0 * N[(N[(c0 * c0), $MachinePrecision] / N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 4.5e-8], N[(0.25 * N[(t$95$2 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[h, 420000000000.0], N[Not[LessEqual[h, 5e+88]], $MachinePrecision]], N[(d * N[(N[(d / D), $MachinePrecision] * N[(N[(c0 / h), $MachinePrecision] * N[(N[(c0 / N[(w * w), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if h < -6.8000000000000004e-19 or 4.2e11 < h < 4.99999999999999997e88

   1. Initial program 12.2%

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

   2. Taylor expanded in c0 around -inf 7.7%

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

      1. fma-def 7.7%

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

      2. associate-/l* 7.7%

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

      3. unpow2 7.7%

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

      4. unpow2 7.7%

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

      5. *-commutative 7.7%

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

      6. unpow2 7.7%

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

      7. associate-*r* 7.7%

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

   4. Simplified 33.7%

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

   5. Taylor expanded in c0 around 0 42.9%

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

      1. associate-/l* 42.8%

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

      2. *-commutative 42.8%

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

      3. unpow2 42.8%

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

      4. unpow2 42.8%

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

      5. *-commutative 42.8%

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

      6. unpow2 42.8%

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

   7. Simplified 42.8%

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

      1. *-un-lft-identity 42.8%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

      2. associate-/l* 46.1%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\frac{D}{\frac{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}{D}}}\right) \]

      3. associate-/l* 57.1%

         \[\leadsto 0.25 \cdot \left(1 \cdot \frac{D}{\frac{\color{blue}{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}}{D}}\right) \]

   9. Applied egg-rr 57.1%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D}{\frac{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}{D}}\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 57.1%

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

       2. associate-/r/ 57.1%

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

       3. unpow2 57.1%

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

       4. associate-/l* 63.0%

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

       5. unpow2 63.0%

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

   11. Simplified 63.0%

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

   if -6.8000000000000004e-19 < h < -1.5999999999999999e-224

   1. Initial program 34.9%

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

      1. times-frac 34.9%

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

      2. fma-def 34.9%

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

      3. times-frac 35.1%

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

      4. difference-of-squares 46.0%

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

   3. Simplified 44.6%

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

      1. *-un-lft-identity 44.6%

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

      2. associate-*l* 44.6%

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

      3. div-inv 44.6%

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

      4. clear-num 44.6%

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

      5. associate-*r/ 46.0%

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

      6. *-commutative 46.0%

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

      7. times-frac 46.1%

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

      8. frac-times 52.1%

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

      9. pow2 52.1%

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

   5. Applied egg-rr 52.1%

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

      1. *-lft-identity 52.1%

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

      2. *-commutative 52.1%

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

      3. associate-*l/ 52.1%

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

      4. associate-/l* 52.1%

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

      5. *-commutative 52.1%

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

   7. Simplified 52.1%

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

   8. Taylor expanded in c0 around inf 33.3%

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

      1. times-frac 34.8%

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

      2. unpow2 34.8%

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

      3. unpow2 34.8%

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

      4. unpow2 34.8%

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

      5. unpow2 34.8%

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

   10. Simplified 34.8%

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

       1. associate-*l/ 33.4%

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

       2. times-frac 38.2%

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

   12. Applied egg-rr 38.2%

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

       1. unpow2 38.2%

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

       2. times-frac 49.2%

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

       3. unpow2 49.2%

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

       4. associate-*r/ 50.8%

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

   14. Simplified 50.8%

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

       1. associate-*l/ 46.0%

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

   16. Applied egg-rr 46.0%

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

       1. times-frac 60.3%

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

   18. Simplified 60.3%

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

   if -1.5999999999999999e-224 < h < 5.4999999999999996e-200

   1. Initial program 15.5%

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

   2. Taylor expanded in c0 around -inf 1.9%

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

      1. fma-def 1.9%

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

      2. associate-/l* 1.9%

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

      3. unpow2 1.9%

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

      4. unpow2 1.9%

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

      5. *-commutative 1.9%

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

      6. unpow2 1.9%

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

      7. associate-*r* 1.9%

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

   4. Simplified 36.4%

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

   5. Taylor expanded in c0 around 0 43.5%

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

      1. associate-/l* 44.2%

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

      2. *-commutative 44.2%

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

      3. unpow2 44.2%

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

      4. unpow2 44.2%

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

      5. *-commutative 44.2%

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

      6. unpow2 44.2%

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

   7. Simplified 44.2%

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

      1. *-un-lft-identity 44.2%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

      2. associate-/l* 46.3%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\frac{D}{\frac{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}{D}}}\right) \]

      3. associate-/l* 52.4%

         \[\leadsto 0.25 \cdot \left(1 \cdot \frac{D}{\frac{\color{blue}{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}}{D}}\right) \]

   9. Applied egg-rr 52.4%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D}{\frac{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}{D}}\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 52.4%

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

       2. associate-/r/ 52.4%

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

   11. Simplified 52.4%

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

   if 5.4999999999999996e-200 < h < 7.59999999999999997e-140

   1. Initial program 38.6%

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

      1. times-frac 38.6%

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

      2. fma-def 38.6%

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

      3. times-frac 38.6%

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

      4. difference-of-squares 54.0%

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

   3. Simplified 54.0%

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

      1. *-un-lft-identity 54.0%

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

      2. associate-*l* 54.0%

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

      3. div-inv 54.0%

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

      4. clear-num 54.0%

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

      5. associate-*r/ 54.0%

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

      6. *-commutative 54.0%

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

      7. times-frac 61.7%

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

      8. frac-times 77.0%

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

      9. pow2 77.0%

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

   5. Applied egg-rr 77.0%

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

      1. *-lft-identity 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-*l/ 77.0%

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

      4. associate-/l* 77.0%

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

      5. *-commutative 77.0%

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

   7. Simplified 77.0%

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

   8. Taylor expanded in c0 around inf 61.7%

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

      1. times-frac 61.7%

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

      2. unpow2 61.7%

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

      3. unpow2 61.7%

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

      4. unpow2 61.7%

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

      5. unpow2 61.7%

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

   10. Simplified 61.7%

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

       1. associate-*l/ 61.7%

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

       2. times-frac 61.7%

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

   12. Applied egg-rr 61.7%

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

       1. unpow2 61.7%

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

       2. times-frac 61.7%

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

       3. unpow2 61.7%

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

       4. associate-*r/ 77.0%

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

   14. Simplified 77.0%

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

   15. Taylor expanded in c0 around 0 77.0%

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

       1. unpow2 77.0%

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

       2. unpow2 77.0%

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

       3. associate-*l* 84.7%

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

   17. Simplified 84.7%

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

   if 7.59999999999999997e-140 < h < 4.49999999999999993e-8

   1. Initial program 19.8%

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

   2. Taylor expanded in c0 around -inf 11.2%

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

      1. fma-def 11.2%

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

      2. associate-/l* 7.6%

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

      3. unpow2 7.6%

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

      4. unpow2 7.6%

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

      5. *-commutative 7.6%

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

      6. unpow2 7.6%

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

      7. associate-*r* 7.6%

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

   4. Simplified 31.5%

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

   5. Taylor expanded in c0 around 0 51.0%

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

      1. associate-/l* 50.9%

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

      2. *-commutative 50.9%

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

      3. unpow2 50.9%

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

      4. unpow2 50.9%

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

      5. *-commutative 50.9%

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

      6. unpow2 50.9%

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

   7. Simplified 50.9%

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

      1. associate-/r/ 50.8%

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

      2. times-frac 58.4%

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

   9. Applied egg-rr 58.4%

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

   if 4.49999999999999993e-8 < h < 4.2e11 or 4.99999999999999997e88 < h

   1. Initial program 50.5%

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

      1. times-frac 50.4%

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

      2. fma-def 50.4%

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

      3. times-frac 50.4%

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

      4. difference-of-squares 59.2%

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

   3. Simplified 59.2%

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

      1. *-un-lft-identity 59.2%

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

      2. associate-*l* 59.2%

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

      3. div-inv 59.2%

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

      4. clear-num 59.2%

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

      5. associate-*r/ 59.2%

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

      6. *-commutative 59.2%

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

      7. times-frac 59.2%

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

      8. frac-times 68.4%

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

      9. pow2 68.4%

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

   5. Applied egg-rr 68.4%

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

      1. *-lft-identity 68.4%

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

      2. *-commutative 68.4%

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

      3. associate-*l/ 68.3%

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

      4. associate-/l* 68.3%

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

      5. *-commutative 68.3%

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

   7. Simplified 68.3%

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

   8. Taylor expanded in c0 around inf 59.0%

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

      1. times-frac 65.1%

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

      2. unpow2 65.1%

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

      3. unpow2 65.1%

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

      4. unpow2 65.1%

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

      5. unpow2 65.1%

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

   10. Simplified 65.1%

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

       1. associate-*l/ 62.3%

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

       2. times-frac 62.3%

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

   12. Applied egg-rr 62.3%

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

       1. unpow2 62.3%

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

       2. times-frac 68.1%

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

       3. unpow2 68.1%

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

       4. associate-*r/ 71.1%

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

   14. Simplified 71.1%

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

       1. pow1 71.1%

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

       2. associate-*l* 74.3%

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

       3. associate-/l* 71.4%

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

   16. Applied egg-rr 71.4%

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

       1. unpow1 71.4%

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

       2. associate-/r/ 74.3%

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

   18. Simplified 74.3%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6.8 \cdot 10^{-19}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{\frac{h}{\frac{d}{M \cdot M}}}}\right)\\ \mathbf{elif}\;h \leq -1.6 \cdot 10^{-224}:\\ \;\;\;\;\left(d \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w} \cdot \frac{\frac{c0}{h}}{w}}{D}\\ \mathbf{elif}\;h \leq 5.5 \cdot 10^{-200}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{d \cdot \frac{d}{h \cdot \left(M \cdot M\right)}}{D}}\\ \mathbf{elif}\;h \leq 7.6 \cdot 10^{-140}:\\ \;\;\;\;\left(d \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{D \cdot \left(w \cdot \left(w \cdot h\right)\right)}\\ \mathbf{elif}\;h \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{elif}\;h \leq 420000000000 \lor \neg \left(h \leq 5 \cdot 10^{+88}\right):\\ \;\;\;\;d \cdot \left(\frac{d}{D} \cdot \left(\frac{c0}{h} \cdot \frac{\frac{c0}{w \cdot w}}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{\frac{h}{\frac{d}{M \cdot M}}}}\right)\\ \end{array} \]

Alternative 3: 43.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot \frac{d}{D}\\ t_1 := 0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{\frac{h}{\frac{d}{M \cdot M}}}}\right)\\ t_2 := h \cdot \left(M \cdot M\right)\\ \mathbf{if}\;h \leq -1.8 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;h \leq -7 \cdot 10^{-223}:\\ \;\;\;\;t_0 \cdot \frac{\frac{c0}{w} \cdot \frac{\frac{c0}{h}}{w}}{D}\\ \mathbf{elif}\;h \leq 9 \cdot 10^{-207}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{d \cdot \frac{d}{t_2}}{D}}\\ \mathbf{elif}\;h \leq 8.4 \cdot 10^{-140}:\\ \;\;\;\;t_0 \cdot \frac{c0 \cdot c0}{D \cdot \left(w \cdot \left(w \cdot h\right)\right)}\\ \mathbf{elif}\;h \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;0.25 \cdot \left(t_2 \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{elif}\;h \leq 450000000000:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;h \leq 5.2 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\frac{d}{D} \cdot \left(\frac{c0}{h} \cdot \frac{\frac{c0}{w \cdot w}}{D}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* d (/ d D)))
        (t_1 (* 0.25 (* D (/ D (/ d (/ h (/ d (* M M))))))))
        (t_2 (* h (* M M))))
   (if (<= h -1.8e-20)
     t_1
     (if (<= h -7e-223)
       (* t_0 (/ (* (/ c0 w) (/ (/ c0 h) w)) D))
       (if (<= h 9e-207)
         (* 0.25 (/ D (/ (* d (/ d t_2)) D)))
         (if (<= h 8.4e-140)
           (* t_0 (/ (* c0 c0) (* D (* w (* w h)))))
           (if (<= h 1.25e-5)
             (* 0.25 (* t_2 (* (/ D d) (/ D d))))
             (if (<= h 450000000000.0)
               (* (* (/ d D) (/ d D)) (/ (* c0 c0) (* h (* w w))))
               (if (<= h 5.2e+88)
                 t_1
                 (* d (* (/ d D) (* (/ c0 h) (/ (/ c0 (* w w)) D)))))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / D);
	double t_1 = 0.25 * (D * (D / (d / (h / (d / (M * M))))));
	double t_2 = h * (M * M);
	double tmp;
	if (h <= -1.8e-20) {
		tmp = t_1;
	} else if (h <= -7e-223) {
		tmp = t_0 * (((c0 / w) * ((c0 / h) / w)) / D);
	} else if (h <= 9e-207) {
		tmp = 0.25 * (D / ((d * (d / t_2)) / D));
	} else if (h <= 8.4e-140) {
		tmp = t_0 * ((c0 * c0) / (D * (w * (w * h))));
	} else if (h <= 1.25e-5) {
		tmp = 0.25 * (t_2 * ((D / d) * (D / d)));
	} else if (h <= 450000000000.0) {
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	} else if (h <= 5.2e+88) {
		tmp = t_1;
	} else {
		tmp = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d_1 * (d_1 / d)
    t_1 = 0.25d0 * (d * (d / (d_1 / (h / (d_1 / (m * m))))))
    t_2 = h * (m * m)
    if (h <= (-1.8d-20)) then
        tmp = t_1
    else if (h <= (-7d-223)) then
        tmp = t_0 * (((c0 / w) * ((c0 / h) / w)) / d)
    else if (h <= 9d-207) then
        tmp = 0.25d0 * (d / ((d_1 * (d_1 / t_2)) / d))
    else if (h <= 8.4d-140) then
        tmp = t_0 * ((c0 * c0) / (d * (w * (w * h))))
    else if (h <= 1.25d-5) then
        tmp = 0.25d0 * (t_2 * ((d / d_1) * (d / d_1)))
    else if (h <= 450000000000.0d0) then
        tmp = ((d_1 / d) * (d_1 / d)) * ((c0 * c0) / (h * (w * w)))
    else if (h <= 5.2d+88) then
        tmp = t_1
    else
        tmp = d_1 * ((d_1 / d) * ((c0 / h) * ((c0 / (w * w)) / d)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d * (d / D);
	double t_1 = 0.25 * (D * (D / (d / (h / (d / (M * M))))));
	double t_2 = h * (M * M);
	double tmp;
	if (h <= -1.8e-20) {
		tmp = t_1;
	} else if (h <= -7e-223) {
		tmp = t_0 * (((c0 / w) * ((c0 / h) / w)) / D);
	} else if (h <= 9e-207) {
		tmp = 0.25 * (D / ((d * (d / t_2)) / D));
	} else if (h <= 8.4e-140) {
		tmp = t_0 * ((c0 * c0) / (D * (w * (w * h))));
	} else if (h <= 1.25e-5) {
		tmp = 0.25 * (t_2 * ((D / d) * (D / d)));
	} else if (h <= 450000000000.0) {
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	} else if (h <= 5.2e+88) {
		tmp = t_1;
	} else {
		tmp = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = d * (d / D)
	t_1 = 0.25 * (D * (D / (d / (h / (d / (M * M))))))
	t_2 = h * (M * M)
	tmp = 0
	if h <= -1.8e-20:
		tmp = t_1
	elif h <= -7e-223:
		tmp = t_0 * (((c0 / w) * ((c0 / h) / w)) / D)
	elif h <= 9e-207:
		tmp = 0.25 * (D / ((d * (d / t_2)) / D))
	elif h <= 8.4e-140:
		tmp = t_0 * ((c0 * c0) / (D * (w * (w * h))))
	elif h <= 1.25e-5:
		tmp = 0.25 * (t_2 * ((D / d) * (D / d)))
	elif h <= 450000000000.0:
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)))
	elif h <= 5.2e+88:
		tmp = t_1
	else:
		tmp = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * Float64(d / D))
	t_1 = Float64(0.25 * Float64(D * Float64(D / Float64(d / Float64(h / Float64(d / Float64(M * M)))))))
	t_2 = Float64(h * Float64(M * M))
	tmp = 0.0
	if (h <= -1.8e-20)
		tmp = t_1;
	elseif (h <= -7e-223)
		tmp = Float64(t_0 * Float64(Float64(Float64(c0 / w) * Float64(Float64(c0 / h) / w)) / D));
	elseif (h <= 9e-207)
		tmp = Float64(0.25 * Float64(D / Float64(Float64(d * Float64(d / t_2)) / D)));
	elseif (h <= 8.4e-140)
		tmp = Float64(t_0 * Float64(Float64(c0 * c0) / Float64(D * Float64(w * Float64(w * h)))));
	elseif (h <= 1.25e-5)
		tmp = Float64(0.25 * Float64(t_2 * Float64(Float64(D / d) * Float64(D / d))));
	elseif (h <= 450000000000.0)
		tmp = Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(Float64(c0 * c0) / Float64(h * Float64(w * w))));
	elseif (h <= 5.2e+88)
		tmp = t_1;
	else
		tmp = Float64(d * Float64(Float64(d / D) * Float64(Float64(c0 / h) * Float64(Float64(c0 / Float64(w * w)) / D))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = d * (d / D);
	t_1 = 0.25 * (D * (D / (d / (h / (d / (M * M))))));
	t_2 = h * (M * M);
	tmp = 0.0;
	if (h <= -1.8e-20)
		tmp = t_1;
	elseif (h <= -7e-223)
		tmp = t_0 * (((c0 / w) * ((c0 / h) / w)) / D);
	elseif (h <= 9e-207)
		tmp = 0.25 * (D / ((d * (d / t_2)) / D));
	elseif (h <= 8.4e-140)
		tmp = t_0 * ((c0 * c0) / (D * (w * (w * h))));
	elseif (h <= 1.25e-5)
		tmp = 0.25 * (t_2 * ((D / d) * (D / d)));
	elseif (h <= 450000000000.0)
		tmp = ((d / D) * (d / D)) * ((c0 * c0) / (h * (w * w)));
	elseif (h <= 5.2e+88)
		tmp = t_1;
	else
		tmp = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(D * N[(D / N[(d / N[(h / N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.8e-20], t$95$1, If[LessEqual[h, -7e-223], N[(t$95$0 * N[(N[(N[(c0 / w), $MachinePrecision] * N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 9e-207], N[(0.25 * N[(D / N[(N[(d * N[(d / t$95$2), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 8.4e-140], N[(t$95$0 * N[(N[(c0 * c0), $MachinePrecision] / N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.25e-5], N[(0.25 * N[(t$95$2 * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 450000000000.0], N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * c0), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 5.2e+88], t$95$1, N[(d * N[(N[(d / D), $MachinePrecision] * N[(N[(c0 / h), $MachinePrecision] * N[(N[(c0 / N[(w * w), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if h < -1.79999999999999987e-20 or 4.5e11 < h < 5.2000000000000001e88

   1. Initial program 12.2%

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

   2. Taylor expanded in c0 around -inf 7.7%

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

      1. fma-def 7.7%

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

      2. associate-/l* 7.7%

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

      3. unpow2 7.7%

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

      4. unpow2 7.7%

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

      5. *-commutative 7.7%

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

      6. unpow2 7.7%

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

      7. associate-*r* 7.7%

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

   4. Simplified 33.7%

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

   5. Taylor expanded in c0 around 0 42.9%

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

      1. associate-/l* 42.8%

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

      2. *-commutative 42.8%

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

      3. unpow2 42.8%

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

      4. unpow2 42.8%

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

      5. *-commutative 42.8%

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

      6. unpow2 42.8%

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

   7. Simplified 42.8%

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

      1. *-un-lft-identity 42.8%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

      2. associate-/l* 46.1%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\frac{D}{\frac{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}{D}}}\right) \]

      3. associate-/l* 57.1%

         \[\leadsto 0.25 \cdot \left(1 \cdot \frac{D}{\frac{\color{blue}{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}}{D}}\right) \]

   9. Applied egg-rr 57.1%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D}{\frac{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}{D}}\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 57.1%

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

       2. associate-/r/ 57.1%

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

       3. unpow2 57.1%

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

       4. associate-/l* 63.0%

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

       5. unpow2 63.0%

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

   11. Simplified 63.0%

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

   if -1.79999999999999987e-20 < h < -7.00000000000000018e-223

   1. Initial program 34.9%

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

      1. times-frac 34.9%

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

      2. fma-def 34.9%

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

      3. times-frac 35.1%

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

      4. difference-of-squares 46.0%

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

   3. Simplified 44.6%

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

      1. *-un-lft-identity 44.6%

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

      2. associate-*l* 44.6%

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

      3. div-inv 44.6%

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

      4. clear-num 44.6%

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

      5. associate-*r/ 46.0%

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

      6. *-commutative 46.0%

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

      7. times-frac 46.1%

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

      8. frac-times 52.1%

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

      9. pow2 52.1%

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

   5. Applied egg-rr 52.1%

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

      1. *-lft-identity 52.1%

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

      2. *-commutative 52.1%

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

      3. associate-*l/ 52.1%

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

      4. associate-/l* 52.1%

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

      5. *-commutative 52.1%

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

   7. Simplified 52.1%

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

   8. Taylor expanded in c0 around inf 33.3%

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

      1. times-frac 34.8%

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

      2. unpow2 34.8%

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

      3. unpow2 34.8%

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

      4. unpow2 34.8%

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

      5. unpow2 34.8%

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

   10. Simplified 34.8%

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

       1. associate-*l/ 33.4%

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

       2. times-frac 38.2%

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

   12. Applied egg-rr 38.2%

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

       1. unpow2 38.2%

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

       2. times-frac 49.2%

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

       3. unpow2 49.2%

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

       4. associate-*r/ 50.8%

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

   14. Simplified 50.8%

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

       1. associate-*l/ 46.0%

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

   16. Applied egg-rr 46.0%

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

       1. times-frac 60.3%

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

   18. Simplified 60.3%

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

   if -7.00000000000000018e-223 < h < 8.99999999999999984e-207

   1. Initial program 15.5%

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

   2. Taylor expanded in c0 around -inf 1.9%

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

      1. fma-def 1.9%

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

      2. associate-/l* 1.9%

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

      3. unpow2 1.9%

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

      4. unpow2 1.9%

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

      5. *-commutative 1.9%

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

      6. unpow2 1.9%

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

      7. associate-*r* 1.9%

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

   4. Simplified 36.4%

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

   5. Taylor expanded in c0 around 0 43.5%

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

      1. associate-/l* 44.2%

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

      2. *-commutative 44.2%

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

      3. unpow2 44.2%

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

      4. unpow2 44.2%

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

      5. *-commutative 44.2%

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

      6. unpow2 44.2%

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

   7. Simplified 44.2%

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

      1. *-un-lft-identity 44.2%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

      2. associate-/l* 46.3%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\frac{D}{\frac{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}{D}}}\right) \]

      3. associate-/l* 52.4%

         \[\leadsto 0.25 \cdot \left(1 \cdot \frac{D}{\frac{\color{blue}{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}}{D}}\right) \]

   9. Applied egg-rr 52.4%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D}{\frac{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}{D}}\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 52.4%

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

       2. associate-/r/ 52.4%

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

   11. Simplified 52.4%

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

   if 8.99999999999999984e-207 < h < 8.4000000000000007e-140

   1. Initial program 38.6%

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

      1. times-frac 38.6%

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

      2. fma-def 38.6%

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

      3. times-frac 38.6%

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

      4. difference-of-squares 54.0%

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

   3. Simplified 54.0%

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

      1. *-un-lft-identity 54.0%

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

      2. associate-*l* 54.0%

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

      3. div-inv 54.0%

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

      4. clear-num 54.0%

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

      5. associate-*r/ 54.0%

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

      6. *-commutative 54.0%

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

      7. times-frac 61.7%

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

      8. frac-times 77.0%

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

      9. pow2 77.0%

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

   5. Applied egg-rr 77.0%

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

      1. *-lft-identity 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-*l/ 77.0%

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

      4. associate-/l* 77.0%

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

      5. *-commutative 77.0%

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

   7. Simplified 77.0%

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

   8. Taylor expanded in c0 around inf 61.7%

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

      1. times-frac 61.7%

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

      2. unpow2 61.7%

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

      3. unpow2 61.7%

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

      4. unpow2 61.7%

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

      5. unpow2 61.7%

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

   10. Simplified 61.7%

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

       1. associate-*l/ 61.7%

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

       2. times-frac 61.7%

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

   12. Applied egg-rr 61.7%

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

       1. unpow2 61.7%

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

       2. times-frac 61.7%

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

       3. unpow2 61.7%

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

       4. associate-*r/ 77.0%

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

   14. Simplified 77.0%

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

   15. Taylor expanded in c0 around 0 77.0%

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

       1. unpow2 77.0%

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

       2. unpow2 77.0%

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

       3. associate-*l* 84.7%

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

   17. Simplified 84.7%

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

   if 8.4000000000000007e-140 < h < 1.25000000000000006e-5

   1. Initial program 19.8%

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

   2. Taylor expanded in c0 around -inf 11.2%

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

      1. fma-def 11.2%

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

      2. associate-/l* 7.6%

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

      3. unpow2 7.6%

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

      4. unpow2 7.6%

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

      5. *-commutative 7.6%

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

      6. unpow2 7.6%

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

      7. associate-*r* 7.6%

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

   4. Simplified 31.5%

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

   5. Taylor expanded in c0 around 0 51.0%

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

      1. associate-/l* 50.9%

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

      2. *-commutative 50.9%

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

      3. unpow2 50.9%

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

      4. unpow2 50.9%

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

      5. *-commutative 50.9%

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

      6. unpow2 50.9%

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

   7. Simplified 50.9%

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

      1. associate-/r/ 50.8%

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

      2. times-frac 58.4%

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

   9. Applied egg-rr 58.4%

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

   if 1.25000000000000006e-5 < h < 4.5e11

   1. Initial program 71.4%

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

      1. times-frac 71.4%

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

      2. fma-def 71.4%

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

      3. times-frac 71.4%

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

      4. difference-of-squares 71.4%

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

   3. Simplified 71.4%

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

      1. *-un-lft-identity 71.4%

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

      2. associate-*l* 71.4%

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

      3. div-inv 71.4%

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

      4. clear-num 71.4%

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

      5. associate-*r/ 71.4%

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

      6. *-commutative 71.4%

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

      7. times-frac 71.4%

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

      8. frac-times 87.4%

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

      9. pow2 87.4%

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

   5. Applied egg-rr 87.4%

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

      1. *-lft-identity 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-*l/ 87.4%

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

      4. associate-/l* 87.4%

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

      5. *-commutative 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in c0 around inf 42.9%

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

      1. times-frac 71.4%

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

      2. unpow2 71.4%

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

      3. unpow2 71.4%

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

      4. unpow2 71.4%

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

      5. unpow2 71.4%

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

   10. Simplified 71.4%

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

       1. frac-times 86.2%

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

   12. Applied egg-rr 86.2%

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

   if 5.2000000000000001e88 < h

   1. Initial program 45.0%

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

      1. times-frac 44.9%

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

      2. fma-def 44.9%

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

      3. times-frac 44.9%

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

      4. difference-of-squares 56.0%

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

   3. Simplified 56.0%

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

      1. *-un-lft-identity 56.0%

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

      2. associate-*l* 56.0%

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

      3. div-inv 56.0%

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

      4. clear-num 56.0%

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

      5. associate-*r/ 56.0%

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

      6. *-commutative 56.0%

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

      7. times-frac 56.0%

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

      8. frac-times 63.4%

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

      9. pow2 63.4%

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

   5. Applied egg-rr 63.4%

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

      1. *-lft-identity 63.4%

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

      2. *-commutative 63.4%

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

      3. associate-*l/ 63.4%

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

      4. associate-/l* 63.4%

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

      5. *-commutative 63.4%

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

   7. Simplified 63.4%

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

   8. Taylor expanded in c0 around inf 63.2%

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

      1. times-frac 63.5%

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

      2. unpow2 63.5%

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

      3. unpow2 63.5%

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

      4. unpow2 63.5%

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

      5. unpow2 63.5%

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

   10. Simplified 63.5%

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

       1. associate-*l/ 60.0%

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

       2. times-frac 60.0%

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

   12. Applied egg-rr 60.0%

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

       1. unpow2 60.0%

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

       2. times-frac 67.3%

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

       3. unpow2 67.3%

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

       4. associate-*r/ 67.3%

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

   14. Simplified 67.3%

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

       1. pow1 67.3%

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

       2. associate-*l* 71.3%

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

       3. associate-/l* 67.5%

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

   16. Applied egg-rr 67.5%

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

       1. unpow1 67.5%

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

       2. associate-/r/ 71.2%

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

   18. Simplified 71.2%

       \[\leadsto \color{blue}{d \cdot \left(\frac{d}{D} \cdot \left(\frac{\frac{c0}{w \cdot w}}{D} \cdot \frac{c0}{h}\right)\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.8 \cdot 10^{-20}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{\frac{h}{\frac{d}{M \cdot M}}}}\right)\\ \mathbf{elif}\;h \leq -7 \cdot 10^{-223}:\\ \;\;\;\;\left(d \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w} \cdot \frac{\frac{c0}{h}}{w}}{D}\\ \mathbf{elif}\;h \leq 9 \cdot 10^{-207}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{d \cdot \frac{d}{h \cdot \left(M \cdot M\right)}}{D}}\\ \mathbf{elif}\;h \leq 8.4 \cdot 10^{-140}:\\ \;\;\;\;\left(d \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{D \cdot \left(w \cdot \left(w \cdot h\right)\right)}\\ \mathbf{elif}\;h \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{elif}\;h \leq 450000000000:\\ \;\;\;\;\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0 \cdot c0}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;h \leq 5.2 \cdot 10^{+88}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{\frac{h}{\frac{d}{M \cdot M}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left(\frac{d}{D} \cdot \left(\frac{c0}{h} \cdot \frac{\frac{c0}{w \cdot w}}{D}\right)\right)\\ \end{array} \]

Alternative 4: 46.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{M \cdot M}\\ t_1 := d \cdot \left(\frac{d}{D} \cdot \left(\frac{c0}{h} \cdot \frac{\frac{c0}{w \cdot w}}{D}\right)\right)\\ t_2 := 0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{\frac{h}{t_0}}}\right)\\ \mathbf{if}\;w \leq -2.2 \cdot 10^{-130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;w \leq -1.26 \cdot 10^{-222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;w \leq 1.25 \cdot 10^{-275}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{t_0 \cdot \frac{d}{h}}\\ \mathbf{elif}\;w \leq 2.55 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ d (* M M)))
        (t_1 (* d (* (/ d D) (* (/ c0 h) (/ (/ c0 (* w w)) D)))))
        (t_2 (* 0.25 (* D (/ D (/ d (/ h t_0)))))))
   (if (<= w -2.2e-130)
     t_2
     (if (<= w -1.26e-222)
       t_1
       (if (<= w 1.25e-275)
         (/ (* (* D D) 0.25) (* t_0 (/ d h)))
         (if (<= w 2.55e+16) t_1 t_2))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d / (M * M);
	double t_1 = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)));
	double t_2 = 0.25 * (D * (D / (d / (h / t_0))));
	double tmp;
	if (w <= -2.2e-130) {
		tmp = t_2;
	} else if (w <= -1.26e-222) {
		tmp = t_1;
	} else if (w <= 1.25e-275) {
		tmp = ((D * D) * 0.25) / (t_0 * (d / h));
	} else if (w <= 2.55e+16) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = d_1 / (m * m)
    t_1 = d_1 * ((d_1 / d) * ((c0 / h) * ((c0 / (w * w)) / d)))
    t_2 = 0.25d0 * (d * (d / (d_1 / (h / t_0))))
    if (w <= (-2.2d-130)) then
        tmp = t_2
    else if (w <= (-1.26d-222)) then
        tmp = t_1
    else if (w <= 1.25d-275) then
        tmp = ((d * d) * 0.25d0) / (t_0 * (d_1 / h))
    else if (w <= 2.55d+16) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = d / (M * M);
	double t_1 = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)));
	double t_2 = 0.25 * (D * (D / (d / (h / t_0))));
	double tmp;
	if (w <= -2.2e-130) {
		tmp = t_2;
	} else if (w <= -1.26e-222) {
		tmp = t_1;
	} else if (w <= 1.25e-275) {
		tmp = ((D * D) * 0.25) / (t_0 * (d / h));
	} else if (w <= 2.55e+16) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = d / (M * M)
	t_1 = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)))
	t_2 = 0.25 * (D * (D / (d / (h / t_0))))
	tmp = 0
	if w <= -2.2e-130:
		tmp = t_2
	elif w <= -1.26e-222:
		tmp = t_1
	elif w <= 1.25e-275:
		tmp = ((D * D) * 0.25) / (t_0 * (d / h))
	elif w <= 2.55e+16:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d / Float64(M * M))
	t_1 = Float64(d * Float64(Float64(d / D) * Float64(Float64(c0 / h) * Float64(Float64(c0 / Float64(w * w)) / D))))
	t_2 = Float64(0.25 * Float64(D * Float64(D / Float64(d / Float64(h / t_0)))))
	tmp = 0.0
	if (w <= -2.2e-130)
		tmp = t_2;
	elseif (w <= -1.26e-222)
		tmp = t_1;
	elseif (w <= 1.25e-275)
		tmp = Float64(Float64(Float64(D * D) * 0.25) / Float64(t_0 * Float64(d / h)));
	elseif (w <= 2.55e+16)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = d / (M * M);
	t_1 = d * ((d / D) * ((c0 / h) * ((c0 / (w * w)) / D)));
	t_2 = 0.25 * (D * (D / (d / (h / t_0))));
	tmp = 0.0;
	if (w <= -2.2e-130)
		tmp = t_2;
	elseif (w <= -1.26e-222)
		tmp = t_1;
	elseif (w <= 1.25e-275)
		tmp = ((D * D) * 0.25) / (t_0 * (d / h));
	elseif (w <= 2.55e+16)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(d / N[(M * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d * N[(N[(d / D), $MachinePrecision] * N[(N[(c0 / h), $MachinePrecision] * N[(N[(c0 / N[(w * w), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.25 * N[(D * N[(D / N[(d / N[(h / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -2.2e-130], t$95$2, If[LessEqual[w, -1.26e-222], t$95$1, If[LessEqual[w, 1.25e-275], N[(N[(N[(D * D), $MachinePrecision] * 0.25), $MachinePrecision] / N[(t$95$0 * N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 2.55e+16], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -2.1999999999999999e-130 or 2.55e16 < w

   1. Initial program 13.6%

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

   2. Taylor expanded in c0 around -inf 6.1%

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

      1. fma-def 6.1%

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

      2. associate-/l* 5.3%

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

      3. unpow2 5.3%

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

      4. unpow2 5.3%

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

      5. *-commutative 5.3%

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

      6. unpow2 5.3%

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

      7. associate-*r* 5.3%

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

   4. Simplified 31.8%

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

   5. Taylor expanded in c0 around 0 39.3%

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

      1. associate-/l* 39.6%

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

      2. *-commutative 39.6%

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

      3. unpow2 39.6%

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

      4. unpow2 39.6%

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

      5. *-commutative 39.6%

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

      6. unpow2 39.6%

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

   7. Simplified 39.6%

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

      1. *-un-lft-identity 39.6%

         \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

      2. associate-/l* 46.0%

         \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\frac{D}{\frac{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}{D}}}\right) \]

      3. associate-/l* 53.7%

         \[\leadsto 0.25 \cdot \left(1 \cdot \frac{D}{\frac{\color{blue}{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}}{D}}\right) \]

   9. Applied egg-rr 53.7%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D}{\frac{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}{D}}\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 53.7%

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

       2. associate-/r/ 53.7%

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

       3. unpow2 53.7%

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

       4. associate-/l* 56.5%

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

       5. unpow2 56.5%

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

   11. Simplified 56.5%

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

   if -2.1999999999999999e-130 < w < -1.25999999999999997e-222 or 1.24999999999999996e-275 < w < 2.55e16

   1. Initial program 42.1%

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

      1. times-frac 40.2%

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

      2. fma-def 40.2%

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

      3. times-frac 40.2%

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

      4. difference-of-squares 46.2%

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

   3. Simplified 46.2%

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

      1. *-un-lft-identity 46.2%

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

      2. associate-*l* 46.2%

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

      3. div-inv 46.2%

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

      4. clear-num 46.2%

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

      5. associate-*r/ 46.2%

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

      6. *-commutative 46.2%

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

      7. times-frac 46.2%

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

      8. frac-times 52.3%

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

      9. pow2 52.3%

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

   5. Applied egg-rr 52.3%

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

      1. *-lft-identity 52.3%

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

      2. *-commutative 52.3%

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

      3. associate-*l/ 52.2%

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

      4. associate-/l* 52.3%

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

      5. *-commutative 52.3%

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

   7. Simplified 52.3%

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

   8. Taylor expanded in c0 around inf 41.6%

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

      1. times-frac 41.6%

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

      2. unpow2 41.6%

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

      3. unpow2 41.6%

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

      4. unpow2 41.6%

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

      5. unpow2 41.6%

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

   10. Simplified 41.6%

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

       1. associate-*l/ 41.7%

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

       2. times-frac 48.7%

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

   12. Applied egg-rr 48.7%

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

       1. unpow2 48.7%

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

       2. times-frac 53.8%

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

       3. unpow2 53.8%

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

       4. associate-*r/ 55.8%

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

   14. Simplified 55.8%

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

       1. pow1 55.8%

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

       2. associate-*l* 57.9%

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

       3. associate-/l* 57.8%

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

   16. Applied egg-rr 57.8%

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

       1. unpow1 57.8%

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

       2. associate-/r/ 57.9%

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

   18. Simplified 57.9%

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

   if -1.25999999999999997e-222 < w < 1.24999999999999996e-275

   1. Initial program 20.2%

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

   2. Taylor expanded in c0 around -inf 2.5%

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

      1. fma-def 2.5%

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

      2. associate-/l* 2.5%

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

      3. unpow2 2.5%

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

      4. unpow2 2.5%

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

      5. *-commutative 2.5%

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

      6. unpow2 2.5%

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

      7. associate-*r* 2.5%

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

   4. Simplified 30.4%

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

      1. add-cbrt-cube 30.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \color{blue}{\sqrt[3]{\left(\frac{D \cdot D}{\frac{\left(d \cdot d\right) \cdot c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}} \cdot \frac{D \cdot D}{\frac{\left(d \cdot d\right) \cdot c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\right) \cdot \frac{D \cdot D}{\frac{\left(d \cdot d\right) \cdot c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}}}}, c0 \cdot 0\right) \]

      2. associate-/r/ 22.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \sqrt[3]{\left(\color{blue}{\left(\frac{D \cdot D}{\left(d \cdot d\right) \cdot c0} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)} \cdot \frac{D \cdot D}{\frac{\left(d \cdot d\right) \cdot c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\right) \cdot \frac{D \cdot D}{\frac{\left(d \cdot d\right) \cdot c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}}}, c0 \cdot 0\right) \]

      3. *-commutative 22.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \sqrt[3]{\left(\left(\frac{D \cdot D}{\color{blue}{c0 \cdot \left(d \cdot d\right)}} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \frac{D \cdot D}{\frac{\left(d \cdot d\right) \cdot c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}}\right) \cdot \frac{D \cdot D}{\frac{\left(d \cdot d\right) \cdot c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}}}, c0 \cdot 0\right) \]

      4. associate-/r/ 22.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \sqrt[3]{\left(\left(\frac{D \cdot D}{c0 \cdot \left(d \cdot d\right)} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{D \cdot D}{\left(d \cdot d\right) \cdot c0} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}\right) \cdot \frac{D \cdot D}{\frac{\left(d \cdot d\right) \cdot c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}}}, c0 \cdot 0\right) \]

      5. *-commutative 22.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \sqrt[3]{\left(\left(\frac{D \cdot D}{c0 \cdot \left(d \cdot d\right)} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \left(\frac{D \cdot D}{\color{blue}{c0 \cdot \left(d \cdot d\right)}} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\right) \cdot \frac{D \cdot D}{\frac{\left(d \cdot d\right) \cdot c0}{w \cdot \left(h \cdot \left(M \cdot M\right)\right)}}}, c0 \cdot 0\right) \]

      6. associate-/r/ 22.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \sqrt[3]{\left(\left(\frac{D \cdot D}{c0 \cdot \left(d \cdot d\right)} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \left(\frac{D \cdot D}{c0 \cdot \left(d \cdot d\right)} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{D \cdot D}{\left(d \cdot d\right) \cdot c0} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}}, c0 \cdot 0\right) \]

   6. Applied egg-rr 22.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \color{blue}{\sqrt[3]{\left(\left(\frac{D \cdot D}{c0 \cdot \left(d \cdot d\right)} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \left(\frac{D \cdot D}{c0 \cdot \left(d \cdot d\right)} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\right) \cdot \left(\frac{D \cdot D}{c0 \cdot \left(d \cdot d\right)} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}}, c0 \cdot 0\right) \]
   7. Step-by-step derivation

      1. associate-*l* 22.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(0.5, \sqrt[3]{\color{blue}{\left(\frac{D \cdot D}{c0 \cdot \left(d \cdot d\right)} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \left(\left(\frac{D \cdot D}{c0 \cdot \left(d \cdot d\right)} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right) \cdot \left(\frac{D \cdot D}{c0 \cdot \left(d \cdot d\right)} \cdot \left(w \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)\right)}}, c0 \cdot 0\right) \]

      2. cube-unmult 22.9%

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

      3. *-commutative 22.9%

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

      4. unpow2 22.9%

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

      5. times-frac 20.6%

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

      6. unpow2 20.6%

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

   8. Simplified 20.6%

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

   9. Taylor expanded in c0 around 0 43.4%

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

       1. associate-/l* 43.3%

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

       2. associate-*r/ 43.3%

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

       3. unpow2 43.3%

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

       4. *-commutative 43.3%

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

       5. associate-/r* 45.8%

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

       6. unpow2 45.8%

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

       7. associate-*r/ 55.9%

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

       8. unpow2 55.9%

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

       9. associate-*l/ 58.2%

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

   11. Simplified 58.2%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.2 \cdot 10^{-130}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{\frac{h}{\frac{d}{M \cdot M}}}}\right)\\ \mathbf{elif}\;w \leq -1.26 \cdot 10^{-222}:\\ \;\;\;\;d \cdot \left(\frac{d}{D} \cdot \left(\frac{c0}{h} \cdot \frac{\frac{c0}{w \cdot w}}{D}\right)\right)\\ \mathbf{elif}\;w \leq 1.25 \cdot 10^{-275}:\\ \;\;\;\;\frac{\left(D \cdot D\right) \cdot 0.25}{\frac{d}{M \cdot M} \cdot \frac{d}{h}}\\ \mathbf{elif}\;w \leq 2.55 \cdot 10^{+16}:\\ \;\;\;\;d \cdot \left(\frac{d}{D} \cdot \left(\frac{c0}{h} \cdot \frac{\frac{c0}{w \cdot w}}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{D}{\frac{d}{\frac{h}{\frac{d}{M \cdot M}}}}\right)\\ \end{array} \]

Alternative 5: 38.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.45 \cdot 10^{+248}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{\frac{D \cdot D}{d}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 1.45e+248) (* 0.25 (* (* h (* M M)) (/ (/ (* D D) d) d))) 0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.45e+248) {
		tmp = 0.25 * ((h * (M * M)) * (((D * D) / d) / d));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 1.45d+248) then
        tmp = 0.25d0 * ((h * (m * m)) * (((d * d) / d_1) / d_1))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.45e+248) {
		tmp = 0.25 * ((h * (M * M)) * (((D * D) / d) / d));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 1.45e+248:
		tmp = 0.25 * ((h * (M * M)) * (((D * D) / d) / d))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 1.45e+248)
		tmp = Float64(0.25 * Float64(Float64(h * Float64(M * M)) * Float64(Float64(Float64(D * D) / d) / d)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 1.45e+248)
		tmp = 0.25 * ((h * (M * M)) * (((D * D) / d) / d));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 1.45e+248], N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if d < 1.45e248

   1. Initial program 24.9%

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

   2. Taylor expanded in c0 around -inf 4.8%

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

      1. fma-def 4.8%

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

      2. associate-/l* 4.4%

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

      3. unpow2 4.4%

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

      4. unpow2 4.4%

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

      5. *-commutative 4.4%

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

      6. unpow2 4.4%

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

      7. associate-*r* 4.4%

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

   4. Simplified 28.2%

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

   5. Taylor expanded in c0 around 0 36.3%

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

      1. associate-/l* 36.5%

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

      2. *-commutative 36.5%

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

      3. unpow2 36.5%

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

      4. unpow2 36.5%

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

      5. *-commutative 36.5%

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

      6. unpow2 36.5%

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

   7. Simplified 36.5%

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

      1. associate-/r/ 35.2%

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

      2. times-frac 42.0%

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

   9. Applied egg-rr 42.0%

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

       1. *-commutative 42.0%

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

       2. associate-*l/ 42.0%

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

       3. associate-*r/ 41.5%

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

   11. Simplified 41.5%

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

   if 1.45e248 < d

   1. Initial program 34.8%

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

      1. times-frac 34.8%

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

      2. fma-def 34.8%

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

      3. associate-/r* 34.8%

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

      4. difference-of-squares 43.5%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt1-in 0.0%

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

      3. metadata-eval 0.0%

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

      4. mul0-lft 39.9%

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

      5. metadata-eval 39.9%

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

      6. mul0-lft 0.0%

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

      7. metadata-eval 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. *-commutative 0.0%

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

      10. distribute-lft1-in 0.0%

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

      11. metadata-eval 0.0%

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

      12. mul0-lft 39.9%

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

   6. Simplified 39.9%

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

   7. Taylor expanded in c0 around 0 44.5%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.45 \cdot 10^{+248}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \frac{\frac{D \cdot D}{d}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 42.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{+212}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 9.5e+212) (* 0.25 (* (* h (* M M)) (* (/ D d) (/ D d)))) 0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 9.5e+212) {
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 9.5d+212) then
        tmp = 0.25d0 * ((h * (m * m)) * ((d / d_1) * (d / d_1)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 9.5e+212) {
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 9.5e+212:
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 9.5e+212)
		tmp = Float64(0.25 * Float64(Float64(h * Float64(M * M)) * Float64(Float64(D / d) * Float64(D / d))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 9.5e+212)
		tmp = 0.25 * ((h * (M * M)) * ((D / d) * (D / d)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 9.5e+212], N[(0.25 * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if d < 9.4999999999999993e212

   1. Initial program 23.8%

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

   2. Taylor expanded in c0 around -inf 5.2%

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

      1. fma-def 5.2%

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

      2. associate-/l* 4.7%

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

      3. unpow2 4.7%

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

      4. unpow2 4.7%

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

      5. *-commutative 4.7%

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

      6. unpow2 4.7%

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

      7. associate-*r* 4.7%

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

   4. Simplified 28.7%

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

   5. Taylor expanded in c0 around 0 36.5%

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

      1. associate-/l* 36.6%

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

      2. *-commutative 36.6%

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

      3. unpow2 36.6%

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

      4. unpow2 36.6%

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

      5. *-commutative 36.6%

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

      6. unpow2 36.6%

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

   7. Simplified 36.6%

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

      1. associate-/r/ 35.2%

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

      2. times-frac 42.1%

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

   9. Applied egg-rr 42.1%

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

   if 9.4999999999999993e212 < d

   1. Initial program 36.8%

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

      1. times-frac 36.8%

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

      2. fma-def 36.8%

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

      3. associate-/r* 36.8%

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

      4. difference-of-squares 42.1%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in c0 around -inf 0.0%

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

      1. associate-*r* 0.0%

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

      2. distribute-rgt1-in 0.0%

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

      3. metadata-eval 0.0%

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

      4. mul0-lft 35.0%

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

      5. metadata-eval 35.0%

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

      6. mul0-lft 0.0%

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

      7. metadata-eval 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. *-commutative 0.0%

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

      10. distribute-lft1-in 0.0%

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

      11. metadata-eval 0.0%

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

      12. mul0-lft 35.0%

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

   6. Simplified 35.0%

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

   7. Taylor expanded in c0 around 0 43.1%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{+212}:\\ \;\;\;\;0.25 \cdot \left(\left(h \cdot \left(M \cdot M\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 47.8% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.25d0 * (d * (d / (d_1 / (h / (d_1 / (m * m))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.8%

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

2. Taylor expanded in c0 around -inf 4.4%

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

   1. fma-def 4.4%

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

   2. associate-/l* 4.0%

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

   3. unpow2 4.0%

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

   4. unpow2 4.0%

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

   5. *-commutative 4.0%

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

   6. unpow2 4.0%

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

   7. associate-*r* 4.0%

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

4. Simplified 26.8%

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

5. Taylor expanded in c0 around 0 35.1%

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

   1. associate-/l* 35.2%

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

   2. *-commutative 35.2%

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

   3. unpow2 35.2%

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

   4. unpow2 35.2%

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

   5. *-commutative 35.2%

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

   6. unpow2 35.2%

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

7. Simplified 35.2%

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

   1. *-un-lft-identity 35.2%

      \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D \cdot D}{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}\right)} \]

   2. associate-/l* 39.8%

      \[\leadsto 0.25 \cdot \left(1 \cdot \color{blue}{\frac{D}{\frac{\frac{d \cdot d}{h \cdot \left(M \cdot M\right)}}{D}}}\right) \]

   3. associate-/l* 47.4%

      \[\leadsto 0.25 \cdot \left(1 \cdot \frac{D}{\frac{\color{blue}{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}}{D}}\right) \]

9. Applied egg-rr 47.4%

   \[\leadsto 0.25 \cdot \color{blue}{\left(1 \cdot \frac{D}{\frac{\frac{d}{\frac{h \cdot \left(M \cdot M\right)}{d}}}{D}}\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 47.4%

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

    2. associate-/r/ 47.4%

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

    3. unpow2 47.4%

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

    4. associate-/l* 48.7%

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

    5. unpow2 48.7%

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

11. Simplified 48.7%

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

12. Final simplification 48.7%

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

Alternative 8: 33.6% accurate, 151.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(c0, w, h, D, d, M):
	return 0.0

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 25.8%

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

   1. times-frac 25.0%

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

   2. fma-def 25.0%

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

   3. associate-/r* 25.0%

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

   4. difference-of-squares 31.3%

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

3. Simplified 37.2%

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

4. Taylor expanded in c0 around -inf 3.0%

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

   1. associate-*r* 3.0%

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

   2. distribute-rgt1-in 3.0%

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

   3. metadata-eval 3.0%

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

   4. mul0-lft 31.3%

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

   5. metadata-eval 31.3%

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

   6. mul0-lft 2.9%

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

   7. metadata-eval 2.9%

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

   8. distribute-lft1-in 2.9%

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

   9. *-commutative 2.9%

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

   10. distribute-lft1-in 2.9%

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

   11. metadata-eval 2.9%

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

   12. mul0-lft 31.3%

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

6. Simplified 31.3%

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

7. Taylor expanded in c0 around 0 36.3%

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

8. Final simplification 36.3%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023242 
(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))))))