Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.4% → 53.7%

Time: 38.1s

Alternatives: 6

Speedup: 21.6×

• could not determine a ground truth

  1. c0 = -1.005529408269663e+273
  2. w = 1.7867969414658926e-244
  3. h = 3.900598749169759e-248
  4. D = 1.0887629019134947e-222
  5. d = 2.537547526380428e+295
  6. M = 3.0521563503797626e-165

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.4% 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: 53.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 70.6%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in c0 around inf 74.1%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Simplified 17.6%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. distribute-lft-in 0.1%

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

      2. mul-1-neg 0.1%

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

      3. distribute-rgt-neg-in 0.1%

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

      4. associate-/l* 0.1%

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

      5. mul-1-neg 0.1%

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

      6. associate-/l* 0.1%

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

      7. distribute-lft1-in 0.1%

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

      8. metadata-eval 0.1%

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

      9. mul0-lft 44.8%

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

      10. metadata-eval 44.8%

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

   7. Simplified 44.8%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

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

Alternative 2: 53.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \left(\left(c0 \cdot \left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)\right) \cdot \frac{1}{2 \cdot w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (* c0 (* (* c0 (* 2.0 (/ (pow (/ d D) 2.0) (* w h)))) (/ 1.0 (* 2.0 w))))
     (* c0 (/ 0.0 (* 2.0 w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * ((c0 * (2.0 * (pow((d / D), 2.0) / (w * h)))) * (1.0 / (2.0 * w)));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * ((c0 * (2.0 * (Math.pow((d / D), 2.0) / (w * h)))) * (1.0 / (2.0 * w)));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = c0 * ((c0 * (2.0 * (math.pow((d / D), 2.0) / (w * h)))) * (1.0 / (2.0 * w)))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(D * D) * Float64(w * h)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(Float64(c0 * Float64(2.0 * Float64((Float64(d / D) ^ 2.0) / Float64(w * h)))) * Float64(1.0 / Float64(2.0 * w))));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = c0 * ((c0 * (2.0 * (((d / D) ^ 2.0) / (w * h)))) * (1.0 / (2.0 * w)));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(N[(c0 * N[(2.0 * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 70.6%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in c0 around -inf 6.2%

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

      1. associate-*r/ 6.2%

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

      2. neg-mul-1 6.2%

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

      3. distribute-rgt-neg-in 6.2%

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

   7. Simplified 6.2%

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

      1. times-frac 4.5%

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

      2. *-commutative 4.5%

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

   9. Applied egg-rr 4.5%

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

       1. fma-undefine 4.5%

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

       2. associate-*r/ 4.7%

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

       3. pow2 4.7%

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

       4. associate-*r* 4.7%

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

       5. *-commutative 4.7%

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

       6. frac-times 6.1%

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

       7. add-sqr-sqrt 1.4%

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

       8. sqrt-unprod 68.8%

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

       9. sqr-neg 68.8%

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

       10. sqrt-unprod 70.4%

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

       11. add-sqr-sqrt 70.4%

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

       12. *-commutative 70.4%

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

       13. frac-times 68.8%

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

       14. pow2 68.8%

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

       15. pow2 68.8%

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

       16. frac-times 69.1%

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

       17. pow2 69.1%

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

   11. Applied egg-rr 69.1%

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

       1. +-commutative 69.1%

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

       2. associate-*l/ 68.1%

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

       3. associate-/l* 70.5%

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

       4. distribute-lft-out 70.5%

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

       5. *-commutative 70.5%

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

       6. associate-*l* 70.5%

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

       7. associate-/r* 71.4%

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

       8. unpow2 71.4%

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

       9. associate-*r/ 71.4%

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

       10. associate-*r* 72.6%

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

       11. *-commutative 72.6%

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

       12. associate-/r* 72.7%

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

       13. associate-*l/ 73.8%

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

       14. unpow2 73.8%

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

       15. *-commutative 73.8%

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

   13. Simplified 73.8%

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

       1. div-inv 73.8%

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

       2. count-2 73.8%

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

   15. Applied egg-rr 73.8%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Simplified 17.6%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. distribute-lft-in 0.1%

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

      2. mul-1-neg 0.1%

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

      3. distribute-rgt-neg-in 0.1%

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

      4. associate-/l* 0.1%

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

      5. mul-1-neg 0.1%

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

      6. associate-/l* 0.1%

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

      7. distribute-lft1-in 0.1%

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

      8. metadata-eval 0.1%

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

      9. mul0-lft 44.8%

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

      10. metadata-eval 44.8%

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

   7. Simplified 44.8%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \left(\left(c0 \cdot \left(2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\right)\right) \cdot \frac{1}{2 \cdot w}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-109} \lor \neg \left(M \leq 7 \cdot 10^{-32}\right) \land M \leq 64000000:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(\frac{c0}{h} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w}\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= M 1.7e-109) (and (not (<= M 7e-32)) (<= M 64000000.0)))
   (* c0 (/ 0.0 (* 2.0 w)))
   (* c0 (/ (* 2.0 (* (/ c0 h) (/ (pow (/ d D) 2.0) w))) (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M <= 1.7e-109) || (!(M <= 7e-32) && (M <= 64000000.0))) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * ((2.0 * ((c0 / h) * (pow((d / D), 2.0) / w))) / (2.0 * w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= 1.7d-109) .or. (.not. (m <= 7d-32)) .and. (m <= 64000000.0d0)) then
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    else
        tmp = c0 * ((2.0d0 * ((c0 / h) * (((d_1 / d) ** 2.0d0) / w))) / (2.0d0 * w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M <= 1.7e-109) || (!(M <= 7e-32) && (M <= 64000000.0))) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * ((2.0 * ((c0 / h) * (Math.pow((d / D), 2.0) / w))) / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M <= 1.7e-109) or (not (M <= 7e-32) and (M <= 64000000.0)):
		tmp = c0 * (0.0 / (2.0 * w))
	else:
		tmp = c0 * ((2.0 * ((c0 / h) * (math.pow((d / D), 2.0) / w))) / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((M <= 1.7e-109) || (!(M <= 7e-32) && (M <= 64000000.0)))
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(c0 / h) * Float64((Float64(d / D) ^ 2.0) / w))) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M <= 1.7e-109) || (~((M <= 7e-32)) && (M <= 64000000.0)))
		tmp = c0 * (0.0 / (2.0 * w));
	else
		tmp = c0 * ((2.0 * ((c0 / h) * (((d / D) ^ 2.0) / w))) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[M, 1.7e-109], And[N[Not[LessEqual[M, 7e-32]], $MachinePrecision], LessEqual[M, 64000000.0]]], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(2.0 * N[(N[(c0 / h), $MachinePrecision] * N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.70000000000000006e-109 or 6.9999999999999997e-32 < M < 6.4e7

   1. Initial program 22.4%

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

   3. Simplified 31.2%

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

   5. Taylor expanded in c0 around -inf 3.2%

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

      1. distribute-lft-in 2.1%

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

      2. mul-1-neg 2.1%

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

      3. distribute-rgt-neg-in 2.1%

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

      4. associate-/l* 2.2%

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

      5. mul-1-neg 2.2%

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

      6. associate-/l* 3.2%

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

      7. distribute-lft1-in 3.2%

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

      8. metadata-eval 3.2%

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

      9. mul0-lft 39.8%

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

      10. metadata-eval 39.8%

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

   7. Simplified 39.8%

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

   if 1.70000000000000006e-109 < M < 6.9999999999999997e-32 or 6.4e7 < M

   1. Initial program 21.3%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in c0 around -inf 0.5%

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

      1. associate-*r/ 0.5%

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

      2. neg-mul-1 0.5%

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

      3. distribute-rgt-neg-in 0.5%

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

   7. Simplified 0.5%

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

      1. times-frac 0.0%

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

      2. *-commutative 0.0%

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

   9. Applied egg-rr 0.0%

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

       1. fma-undefine 0.0%

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

       2. associate-*r/ 0.0%

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

       3. pow2 0.0%

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

       4. associate-*r* 0.0%

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

       5. *-commutative 0.0%

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

       6. frac-times 0.3%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 35.8%

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

       9. sqr-neg 35.8%

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

       10. sqrt-unprod 40.1%

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

       11. add-sqr-sqrt 40.1%

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

       12. *-commutative 40.1%

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

       13. frac-times 41.5%

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

       14. pow2 41.5%

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

       15. pow2 41.5%

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

       16. frac-times 42.3%

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

       17. pow2 42.3%

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

   11. Applied egg-rr 42.3%

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

       1. +-commutative 42.3%

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

       2. associate-*l/ 40.9%

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

       3. associate-/l* 42.3%

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

       4. distribute-lft-out 42.3%

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

       5. *-commutative 42.3%

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

       6. associate-*l* 42.2%

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

       7. associate-/r* 47.5%

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

       8. unpow2 47.5%

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

       9. associate-*r/ 50.2%

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

       10. associate-*r* 50.2%

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

       11. *-commutative 50.2%

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

       12. associate-/r* 50.2%

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

       13. associate-*l/ 50.2%

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

       14. unpow2 50.2%

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

       15. *-commutative 50.2%

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

   13. Simplified 50.2%

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

       1. distribute-lft-in 50.2%

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

   15. Applied egg-rr 50.2%

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

       1. count-2 50.2%

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

       2. associate-*r/ 50.2%

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

       3. times-frac 55.7%

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

   17. Simplified 55.7%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-109} \lor \neg \left(M \leq 7 \cdot 10^{-32}\right) \land M \leq 64000000:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(\frac{c0}{h} \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w}\right)}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 38.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}\\ t_1 := c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{if}\;M \leq 1.45 \cdot 10^{-109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{-33}:\\ \;\;\;\;c0 \cdot \left(c0 \cdot \frac{t\_0}{w}\right)\\ \mathbf{elif}\;M \leq 550:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\frac{c0}{2} \cdot \frac{2 \cdot t\_0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (pow (/ d D) 2.0) (* w h))) (t_1 (* c0 (/ 0.0 (* 2.0 w)))))
   (if (<= M 1.45e-109)
     t_1
     (if (<= M 2.2e-33)
       (* c0 (* c0 (/ t_0 w)))
       (if (<= M 550.0) t_1 (* c0 (* (/ c0 2.0) (/ (* 2.0 t_0) w))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((d / D), 2.0) / (w * h);
	double t_1 = c0 * (0.0 / (2.0 * w));
	double tmp;
	if (M <= 1.45e-109) {
		tmp = t_1;
	} else if (M <= 2.2e-33) {
		tmp = c0 * (c0 * (t_0 / w));
	} else if (M <= 550.0) {
		tmp = t_1;
	} else {
		tmp = c0 * ((c0 / 2.0) * ((2.0 * t_0) / w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((d_1 / d) ** 2.0d0) / (w * h)
    t_1 = c0 * (0.0d0 / (2.0d0 * w))
    if (m <= 1.45d-109) then
        tmp = t_1
    else if (m <= 2.2d-33) then
        tmp = c0 * (c0 * (t_0 / w))
    else if (m <= 550.0d0) then
        tmp = t_1
    else
        tmp = c0 * ((c0 / 2.0d0) * ((2.0d0 * t_0) / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow((d / D), 2.0) / (w * h);
	double t_1 = c0 * (0.0 / (2.0 * w));
	double tmp;
	if (M <= 1.45e-109) {
		tmp = t_1;
	} else if (M <= 2.2e-33) {
		tmp = c0 * (c0 * (t_0 / w));
	} else if (M <= 550.0) {
		tmp = t_1;
	} else {
		tmp = c0 * ((c0 / 2.0) * ((2.0 * t_0) / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((d / D), 2.0) / (w * h)
	t_1 = c0 * (0.0 / (2.0 * w))
	tmp = 0
	if M <= 1.45e-109:
		tmp = t_1
	elif M <= 2.2e-33:
		tmp = c0 * (c0 * (t_0 / w))
	elif M <= 550.0:
		tmp = t_1
	else:
		tmp = c0 * ((c0 / 2.0) * ((2.0 * t_0) / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64((Float64(d / D) ^ 2.0) / Float64(w * h))
	t_1 = Float64(c0 * Float64(0.0 / Float64(2.0 * w)))
	tmp = 0.0
	if (M <= 1.45e-109)
		tmp = t_1;
	elseif (M <= 2.2e-33)
		tmp = Float64(c0 * Float64(c0 * Float64(t_0 / w)));
	elseif (M <= 550.0)
		tmp = t_1;
	else
		tmp = Float64(c0 * Float64(Float64(c0 / 2.0) * Float64(Float64(2.0 * t_0) / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d / D) ^ 2.0) / (w * h);
	t_1 = c0 * (0.0 / (2.0 * w));
	tmp = 0.0;
	if (M <= 1.45e-109)
		tmp = t_1;
	elseif (M <= 2.2e-33)
		tmp = c0 * (c0 * (t_0 / w));
	elseif (M <= 550.0)
		tmp = t_1;
	else
		tmp = c0 * ((c0 / 2.0) * ((2.0 * t_0) / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 1.45e-109], t$95$1, If[LessEqual[M, 2.2e-33], N[(c0 * N[(c0 * N[(t$95$0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 550.0], t$95$1, N[(c0 * N[(N[(c0 / 2.0), $MachinePrecision] * N[(N[(2.0 * t$95$0), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.45e-109 or 2.20000000000000005e-33 < M < 550

   1. Initial program 22.1%

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

   3. Simplified 30.9%

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

   5. Taylor expanded in c0 around -inf 3.2%

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

      1. distribute-lft-in 2.1%

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

      2. mul-1-neg 2.1%

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

      3. distribute-rgt-neg-in 2.1%

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

      4. associate-/l* 2.2%

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

      5. mul-1-neg 2.2%

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

      6. associate-/l* 3.2%

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

      7. distribute-lft1-in 3.2%

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

      8. metadata-eval 3.2%

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

      9. mul0-lft 40.3%

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

      10. metadata-eval 40.3%

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

   7. Simplified 40.3%

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

   if 1.45e-109 < M < 2.20000000000000005e-33

   1. Initial program 37.0%

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

   3. Simplified 42.4%

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

   5. Taylor expanded in c0 around -inf 1.3%

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

      1. associate-*r/ 1.3%

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

      2. neg-mul-1 1.3%

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

      3. distribute-rgt-neg-in 1.3%

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

   7. Simplified 1.3%

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

      1. times-frac 0.1%

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

      2. *-commutative 0.1%

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

   9. Applied egg-rr 0.1%

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

       1. fma-undefine 0.0%

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

       2. associate-*r/ 0.0%

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

       3. pow2 0.0%

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

       4. associate-*r* 0.0%

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

       5. *-commutative 0.0%

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

       6. frac-times 0.6%

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

       7. add-sqr-sqrt 0.1%

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

       8. sqrt-unprod 37.8%

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

       9. sqr-neg 37.8%

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

       10. sqrt-unprod 37.8%

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

       11. add-sqr-sqrt 37.8%

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

       12. *-commutative 37.8%

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

       13. frac-times 37.9%

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

       14. pow2 37.9%

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

       15. pow2 37.9%

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

       16. frac-times 38.8%

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

       17. pow2 38.8%

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

   11. Applied egg-rr 38.8%

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

       1. +-commutative 38.8%

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

       2. associate-*l/ 38.4%

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

       3. associate-/l* 38.8%

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

       4. distribute-lft-out 38.8%

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

       5. *-commutative 38.8%

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

       6. associate-*l* 38.6%

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

       7. associate-/r* 49.4%

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

       8. unpow2 49.4%

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

       9. associate-*r/ 53.8%

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

       10. associate-*r* 53.8%

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

       11. *-commutative 53.8%

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

       12. associate-/r* 53.8%

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

       13. associate-*l/ 54.1%

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

       14. unpow2 54.1%

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

       15. *-commutative 54.1%

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

   13. Simplified 54.1%

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

       1. associate-/l* 54.1%

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

       2. count-2 54.1%

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

   15. Applied egg-rr 54.1%

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

       1. times-frac 54.1%

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

       2. metadata-eval 54.1%

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

   17. Simplified 54.1%

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

   if 550 < M

   1. Initial program 16.9%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in c0 around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

      2. neg-mul-1 0.2%

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

      3. distribute-rgt-neg-in 0.2%

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

   7. Simplified 0.2%

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

      1. times-frac 0.0%

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

      2. *-commutative 0.0%

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

   9. Applied egg-rr 0.0%

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

       1. fma-undefine 0.0%

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

       2. associate-*r/ 0.0%

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

       3. pow2 0.0%

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

       4. associate-*r* 0.0%

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

       5. *-commutative 0.0%

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

       6. frac-times 0.2%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 35.6%

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

       9. sqr-neg 35.6%

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

       10. sqrt-unprod 41.2%

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

       11. add-sqr-sqrt 41.2%

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

       12. *-commutative 41.2%

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

       13. frac-times 43.0%

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

       14. pow2 43.0%

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

       15. pow2 43.0%

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

       16. frac-times 43.7%

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

       17. pow2 43.7%

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

   11. Applied egg-rr 43.7%

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

       1. +-commutative 43.7%

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

       2. associate-*l/ 42.1%

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

       3. associate-/l* 43.8%

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

       4. distribute-lft-out 43.8%

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

       5. *-commutative 43.8%

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

       6. associate-*l* 43.8%

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

       7. associate-/r* 47.0%

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

       8. unpow2 47.0%

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

       9. associate-*r/ 48.9%

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

       10. associate-*r* 49.2%

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

       11. *-commutative 49.2%

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

       12. associate-/r* 49.2%

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

       13. associate-*l/ 50.7%

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

       14. unpow2 50.7%

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

       15. *-commutative 50.7%

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

   13. Simplified 50.7%

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

       1. times-frac 49.0%

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

       2. count-2 49.0%

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

   15. Applied egg-rr 49.0%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.45 \cdot 10^{-109}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{elif}\;M \leq 2.2 \cdot 10^{-33}:\\ \;\;\;\;c0 \cdot \left(c0 \cdot \frac{\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}}{w}\right)\\ \mathbf{elif}\;M \leq 550:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\frac{c0}{2} \cdot \frac{2 \cdot \frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}}{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{-108} \lor \neg \left(M \leq 4.1 \cdot 10^{-33}\right) \land M \leq 34000:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(c0 \cdot \frac{\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= M 1.05e-108) (and (not (<= M 4.1e-33)) (<= M 34000.0)))
   (* c0 (/ 0.0 (* 2.0 w)))
   (* c0 (* c0 (/ (/ (pow (/ d D) 2.0) (* w h)) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M <= 1.05e-108) || (!(M <= 4.1e-33) && (M <= 34000.0))) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * (c0 * ((pow((d / D), 2.0) / (w * h)) / w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= 1.05d-108) .or. (.not. (m <= 4.1d-33)) .and. (m <= 34000.0d0)) then
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    else
        tmp = c0 * (c0 * ((((d_1 / d) ** 2.0d0) / (w * h)) / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M <= 1.05e-108) || (!(M <= 4.1e-33) && (M <= 34000.0))) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * (c0 * ((Math.pow((d / D), 2.0) / (w * h)) / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M <= 1.05e-108) or (not (M <= 4.1e-33) and (M <= 34000.0)):
		tmp = c0 * (0.0 / (2.0 * w))
	else:
		tmp = c0 * (c0 * ((math.pow((d / D), 2.0) / (w * h)) / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((M <= 1.05e-108) || (!(M <= 4.1e-33) && (M <= 34000.0)))
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(c0 * Float64(Float64((Float64(d / D) ^ 2.0) / Float64(w * h)) / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M <= 1.05e-108) || (~((M <= 4.1e-33)) && (M <= 34000.0)))
		tmp = c0 * (0.0 / (2.0 * w));
	else
		tmp = c0 * (c0 * ((((d / D) ^ 2.0) / (w * h)) / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[M, 1.05e-108], And[N[Not[LessEqual[M, 4.1e-33]], $MachinePrecision], LessEqual[M, 34000.0]]], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(c0 * N[(N[(N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.05e-108 or 4.1e-33 < M < 34000

   1. Initial program 22.1%

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

   3. Simplified 30.9%

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

   5. Taylor expanded in c0 around -inf 3.2%

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

      1. distribute-lft-in 2.1%

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

      2. mul-1-neg 2.1%

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

      3. distribute-rgt-neg-in 2.1%

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

      4. associate-/l* 2.2%

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

      5. mul-1-neg 2.2%

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

      6. associate-/l* 3.2%

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

      7. distribute-lft1-in 3.2%

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

      8. metadata-eval 3.2%

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

      9. mul0-lft 40.3%

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

      10. metadata-eval 40.3%

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

   7. Simplified 40.3%

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

   if 1.05e-108 < M < 4.1e-33 or 34000 < M

   1. Initial program 22.1%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in c0 around -inf 0.5%

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

      1. associate-*r/ 0.5%

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

      2. neg-mul-1 0.5%

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

      3. distribute-rgt-neg-in 0.5%

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

   7. Simplified 0.5%

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

      1. times-frac 0.0%

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

      2. *-commutative 0.0%

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

   9. Applied egg-rr 0.0%

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

       1. fma-undefine 0.0%

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

       2. associate-*r/ 0.0%

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

       3. pow2 0.0%

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

       4. associate-*r* 0.0%

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

       5. *-commutative 0.0%

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

       6. frac-times 0.3%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 36.2%

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

       9. sqr-neg 36.2%

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

       10. sqrt-unprod 40.4%

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

       11. add-sqr-sqrt 40.4%

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

       12. *-commutative 40.4%

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

       13. frac-times 41.7%

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

       14. pow2 41.7%

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

       15. pow2 41.7%

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

       16. frac-times 42.5%

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

       17. pow2 42.5%

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

   11. Applied egg-rr 42.5%

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

       1. +-commutative 42.5%

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

       2. associate-*l/ 41.2%

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

       3. associate-/l* 42.5%

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

       4. distribute-lft-out 42.5%

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

       5. *-commutative 42.5%

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

       6. associate-*l* 42.4%

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

       7. associate-/r* 47.6%

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

       8. unpow2 47.6%

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

       9. associate-*r/ 50.2%

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

       10. associate-*r* 50.4%

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

       11. *-commutative 50.4%

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

       12. associate-/r* 50.4%

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

       13. associate-*l/ 51.6%

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

       14. unpow2 51.6%

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

       15. *-commutative 51.6%

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

   13. Simplified 51.6%

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

       1. associate-/l* 50.3%

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

       2. count-2 50.3%

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

   15. Applied egg-rr 50.3%

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

       1. times-frac 50.3%

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

       2. metadata-eval 50.3%

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

   17. Simplified 50.3%

       \[\leadsto c0 \cdot \color{blue}{\left(c0 \cdot \left(1 \cdot \frac{\frac{{\left(\frac{d}{D}\right)}^{2}}{h \cdot w}}{w}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{-108} \lor \neg \left(M \leq 4.1 \cdot 10^{-33}\right) \land M \leq 34000:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(c0 \cdot \frac{\frac{{\left(\frac{d}{D}\right)}^{2}}{w \cdot h}}{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \frac{0}{2 \cdot w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* c0 (/ 0.0 (* 2.0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = c0 * (0.0d0 / (2.0d0 * w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

Python

def code(c0, w, h, D, d, M):
	return c0 * (0.0 / (2.0 * w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(c0 * Float64(0.0 / Float64(2.0 * w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = c0 * (0.0 / (2.0 * w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.1%

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

3. Simplified 33.9%

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

5. Taylor expanded in c0 around -inf 2.3%

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

   1. distribute-lft-in 1.5%

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

   2. mul-1-neg 1.5%

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

   3. distribute-rgt-neg-in 1.5%

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

   4. associate-/l* 1.6%

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

   5. mul-1-neg 1.6%

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

   6. associate-/l* 2.3%

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

   7. distribute-lft1-in 2.3%

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

   8. metadata-eval 2.3%

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

   9. mul0-lft 33.5%

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

   10. metadata-eval 33.5%

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

7. Simplified 33.5%

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

8. Final simplification 33.5%

   \[\leadsto c0 \cdot \frac{0}{2 \cdot w} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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))))))