Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.1% → 54.2%

Time: 43.6s

Alternatives: 6

Speedup: 151.0×

• could not determine a ground truth

  1. c0 = -5.8638369941451835e+280
  2. w = -9.566886910937678e-297
  3. h = -2.37000286026661e-10
  4. D = -3.3768420508776844e-201
  5. d = -7.193917391482734e+300
  6. M = -4.0749014428286635e+149

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 54.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{t\_0 + \sqrt{{t\_0}^{2} - {M}^{2}}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 78.4%

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

   3. Simplified 73.6%

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

      1. fma-undefine 78.3%

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

      2. associate-*r/ 78.3%

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

      3. *-commutative 78.3%

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

      4. associate-*r* 79.5%

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

      5. associate-*r* 76.1%

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

      6. associate-/l* 72.6%

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

      7. frac-times 71.2%

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

      8. times-frac 73.4%

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

      9. pow2 73.4%

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

   6. Applied egg-rr 79.4%

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

   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 2.3%

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

      1. distribute-lft-in 2.3%

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

      2. *-commutative 2.3%

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

      3. times-frac 2.3%

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

      4. pow2 2.3%

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

      5. *-commutative 2.3%

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

   6. Applied egg-rr 13.3%

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

   7. Taylor expanded in c0 around -inf 0.7%

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

      1. mul-1-neg 0.7%

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

   9. Simplified 0.7%

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

       1. fma-define 0.8%

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

       2. distribute-rgt-neg-out 0.8%

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

       3. pow2 0.8%

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

       4. *-commutative 0.8%

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

       5. *-commutative 0.8%

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

       6. frac-times 1.4%

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

       7. pow2 1.4%

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

       8. frac-times 2.2%

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

       9. pow2 2.2%

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

   11. Applied egg-rr 3.7%

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

       1. +-inverses 43.4%

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

   13. Simplified 43.4%

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

4. Final simplification 54.8%

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

Alternative 2: 54.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.0;
	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[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, 0.0]]]

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 78.4%

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

   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 2.3%

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

      1. distribute-lft-in 2.3%

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

      2. *-commutative 2.3%

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

      3. times-frac 2.3%

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

      4. pow2 2.3%

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

      5. *-commutative 2.3%

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

   6. Applied egg-rr 13.3%

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

   7. Taylor expanded in c0 around -inf 0.7%

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

      1. mul-1-neg 0.7%

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

   9. Simplified 0.7%

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

       1. fma-define 0.8%

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

       2. distribute-rgt-neg-out 0.8%

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

       3. pow2 0.8%

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

       4. *-commutative 0.8%

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

       5. *-commutative 0.8%

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

       6. frac-times 1.4%

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

       7. pow2 1.4%

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

       8. frac-times 2.2%

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

       9. pow2 2.2%

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

   11. Applied egg-rr 3.7%

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

       1. +-inverses 43.4%

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

   13. Simplified 43.4%

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

4. Final simplification 54.5%

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

Alternative 3: 43.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 6 \cdot 10^{-192}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 2 \cdot 10^{-123} \lor \neg \left(M \cdot M \leq 2 \cdot 10^{-57}\right):\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 6e-192)
   0.0
   (if (or (<= (* M M) 2e-123) (not (<= (* M M) 2e-57)))
     (* c0 (/ (* 2.0 (* (/ c0 (* w h)) (pow (/ d D) 2.0))) (* 2.0 w)))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 6e-192) {
		tmp = 0.0;
	} else if (((M * M) <= 2e-123) || !((M * M) <= 2e-57)) {
		tmp = c0 * ((2.0 * ((c0 / (w * h)) * pow((d / D), 2.0))) / (2.0 * w));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m * m) <= 6d-192) then
        tmp = 0.0d0
    else if (((m * m) <= 2d-123) .or. (.not. ((m * m) <= 2d-57))) then
        tmp = c0 * ((2.0d0 * ((c0 / (w * h)) * ((d_1 / d) ** 2.0d0))) / (2.0d0 * w))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 6e-192) {
		tmp = 0.0;
	} else if (((M * M) <= 2e-123) || !((M * M) <= 2e-57)) {
		tmp = c0 * ((2.0 * ((c0 / (w * h)) * Math.pow((d / D), 2.0))) / (2.0 * w));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 6e-192:
		tmp = 0.0
	elif ((M * M) <= 2e-123) or not ((M * M) <= 2e-57):
		tmp = c0 * ((2.0 * ((c0 / (w * h)) * math.pow((d / D), 2.0))) / (2.0 * w))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 6e-192)
		tmp = 0.0;
	elseif ((Float64(M * M) <= 2e-123) || !(Float64(M * M) <= 2e-57))
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * (Float64(d / D) ^ 2.0))) / Float64(2.0 * w)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 6e-192)
		tmp = 0.0;
	elseif (((M * M) <= 2e-123) || ~(((M * M) <= 2e-57)))
		tmp = c0 * ((2.0 * ((c0 / (w * h)) * ((d / D) ^ 2.0))) / (2.0 * w));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 6e-192], 0.0, If[Or[LessEqual[N[(M * M), $MachinePrecision], 2e-123], N[Not[LessEqual[N[(M * M), $MachinePrecision], 2e-57]], $MachinePrecision]], N[(c0 * N[(N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 5.9999999999999998e-192 or 2.0000000000000001e-123 < (*.f64 M M) < 1.99999999999999991e-57

   1. Initial program 26.6%

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

   3. Simplified 25.9%

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

      1. distribute-lft-in 25.1%

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

      2. *-commutative 25.1%

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

      3. times-frac 25.2%

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

      4. pow2 25.2%

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

      5. *-commutative 25.2%

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

   6. Applied egg-rr 37.1%

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

   7. Taylor expanded in c0 around -inf 5.6%

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

      1. mul-1-neg 5.6%

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

   9. Simplified 5.6%

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

       1. fma-define 4.3%

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

       2. distribute-rgt-neg-out 4.3%

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

       3. pow2 4.3%

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

       4. *-commutative 4.3%

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

       5. *-commutative 4.3%

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

       6. frac-times 4.4%

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

       7. pow2 4.4%

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

       8. frac-times 6.9%

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

       9. pow2 6.9%

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

   11. Applied egg-rr 11.7%

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

       1. +-inverses 54.2%

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

   13. Simplified 54.2%

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

   if 5.9999999999999998e-192 < (*.f64 M M) < 2.0000000000000001e-123 or 1.99999999999999991e-57 < (*.f64 M M)

   1. Initial program 22.9%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in c0 around inf 39.4%

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

      1. associate-/l* 40.3%

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

   7. Simplified 40.3%

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

   8. Taylor expanded in c0 around 0 39.4%

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

      1. *-commutative 39.4%

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

      2. *-commutative 39.4%

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

      3. times-frac 41.8%

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

      4. unpow2 41.8%

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

      5. unpow2 41.8%

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

      6. times-frac 48.5%

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

      7. unpow2 48.5%

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

   10. Simplified 48.5%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 6 \cdot 10^{-192}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 2 \cdot 10^{-123} \lor \neg \left(M \cdot M \leq 2 \cdot 10^{-57}\right):\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 34.8% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.2 \cdot 10^{+47}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(M \cdot \frac{c0}{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 2.2e+47) 0.0 (* 0.5 (* M (/ c0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2.2e+47) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * (M * (c0 / w));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.2d+47) then
        tmp = 0.0d0
    else
        tmp = 0.5d0 * (m * (c0 / w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 2.2e+47) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * (M * (c0 / w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 2.2e+47:
		tmp = 0.0
	else:
		tmp = 0.5 * (M * (c0 / w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 2.2e+47)
		tmp = 0.0;
	else
		tmp = Float64(0.5 * Float64(M * Float64(c0 / w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 2.2e+47)
		tmp = 0.0;
	else
		tmp = 0.5 * (M * (c0 / w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 2.2e+47], 0.0, N[(0.5 * N[(M * N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.1999999999999999e47

   1. Initial program 25.7%

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

   3. Simplified 26.6%

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

      1. distribute-lft-in 26.1%

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

      2. *-commutative 26.1%

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

      3. times-frac 26.2%

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

      4. pow2 26.2%

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

      5. *-commutative 26.2%

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

   6. Applied egg-rr 35.9%

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

   7. Taylor expanded in c0 around -inf 4.1%

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

      1. mul-1-neg 4.1%

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

   9. Simplified 4.1%

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

       1. fma-define 3.3%

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

       2. distribute-rgt-neg-out 3.3%

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

       3. pow2 3.3%

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

       4. *-commutative 3.3%

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

       5. *-commutative 3.3%

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

       6. frac-times 3.8%

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

       7. pow2 3.8%

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

       8. frac-times 4.5%

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

       9. pow2 4.5%

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

   11. Applied egg-rr 7.8%

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

       1. +-inverses 42.0%

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

   13. Simplified 42.0%

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

   if 2.1999999999999999e47 < M

   1. Initial program 20.2%

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

   3. Simplified 40.5%

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

      1. *-un-lft-identity 40.5%

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

      2. fma-undefine 40.5%

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

      3. associate-*r/ 40.5%

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

      4. *-commutative 40.5%

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

      5. associate-*r* 40.5%

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

      6. associate-*r* 35.5%

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

      7. associate-/l* 35.5%

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

      8. frac-times 35.3%

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

      9. fma-define 35.3%

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

   6. Applied egg-rr 42.9%

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

      1. fma-undefine 42.8%

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

      2. sqrt-prod 42.8%

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

      3. add-log-exp 42.8%

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

      4. fma-undefine 42.8%

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

      5. add-log-exp 42.8%

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

   8. Applied egg-rr 51.4%

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

   9. Taylor expanded in c0 around 0 37.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{M \cdot c0}{w}} \]
   10. Step-by-step derivation

       1. associate-/l* 31.8%

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

   11. Simplified 31.8%

       \[\leadsto \color{blue}{0.5 \cdot \left(M \cdot \frac{c0}{w}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.2 \cdot 10^{+47}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(M \cdot \frac{c0}{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{+47}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot M}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 1.8e+47) 0.0 (* 0.5 (/ (* c0 M) w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.8e+47) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * ((c0 * M) / 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 <= 1.8d+47) then
        tmp = 0.0d0
    else
        tmp = 0.5d0 * ((c0 * m) / 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 <= 1.8e+47) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * ((c0 * M) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 1.8e+47:
		tmp = 0.0
	else:
		tmp = 0.5 * ((c0 * M) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 1.8e+47)
		tmp = 0.0;
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * M) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 1.8e+47)
		tmp = 0.0;
	else
		tmp = 0.5 * ((c0 * M) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 1.8e+47], 0.0, N[(0.5 * N[(N[(c0 * M), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.80000000000000004e47

   1. Initial program 25.7%

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

   3. Simplified 26.6%

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

      1. distribute-lft-in 26.1%

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

      2. *-commutative 26.1%

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

      3. times-frac 26.2%

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

      4. pow2 26.2%

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

      5. *-commutative 26.2%

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

   6. Applied egg-rr 35.9%

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

   7. Taylor expanded in c0 around -inf 4.1%

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

      1. mul-1-neg 4.1%

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

   9. Simplified 4.1%

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

       1. fma-define 3.3%

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

       2. distribute-rgt-neg-out 3.3%

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

       3. pow2 3.3%

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

       4. *-commutative 3.3%

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

       5. *-commutative 3.3%

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

       6. frac-times 3.8%

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

       7. pow2 3.8%

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

       8. frac-times 4.5%

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

       9. pow2 4.5%

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

   11. Applied egg-rr 7.8%

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

       1. +-inverses 42.0%

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

   13. Simplified 42.0%

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

   if 1.80000000000000004e47 < M

   1. Initial program 20.2%

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

   3. Simplified 40.5%

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

      1. *-un-lft-identity 40.5%

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

      2. fma-undefine 40.5%

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

      3. associate-*r/ 40.5%

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

      4. *-commutative 40.5%

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

      5. associate-*r* 40.5%

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

      6. associate-*r* 35.5%

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

      7. associate-/l* 35.5%

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

      8. frac-times 35.3%

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

      9. fma-define 35.3%

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

   6. Applied egg-rr 42.9%

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

      1. fma-undefine 42.8%

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

      2. sqrt-prod 42.8%

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

      3. add-log-exp 42.8%

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

      4. fma-undefine 42.8%

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

      5. add-log-exp 42.8%

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

   8. Applied egg-rr 51.4%

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

   9. Taylor expanded in c0 around 0 37.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{M \cdot c0}{w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{+47}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot M}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.5% accurate, 151.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 24.8%

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

3. Simplified 25.6%

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

   1. distribute-lft-in 25.2%

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

   2. *-commutative 25.2%

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

   3. times-frac 25.3%

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

   4. pow2 25.3%

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

   5. *-commutative 25.3%

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

6. Applied egg-rr 33.8%

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

7. Taylor expanded in c0 around -inf 3.5%

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

   1. mul-1-neg 3.5%

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

9. Simplified 3.5%

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

    1. fma-define 2.8%

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

    2. distribute-rgt-neg-out 2.8%

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

    3. pow2 2.8%

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

    4. *-commutative 2.8%

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

    5. *-commutative 2.8%

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

    6. frac-times 3.2%

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

    7. pow2 3.2%

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

    8. frac-times 3.8%

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

    9. pow2 3.8%

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

11. Applied egg-rr 6.6%

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

    1. +-inverses 36.3%

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

13. Simplified 36.3%

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

14. Final simplification 36.3%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(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))))))