Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.7% → 47.0%

Time: 19.6s

Alternatives: 12

Speedup: 156.0×

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: 24.7% 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: 47.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{\left(D \cdot M\_m\right) \cdot h}{d}\\ \mathbf{if}\;M\_m \leq 5.5 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{elif}\;M\_m \leq 7.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, -0.25 \cdot t\_0, \left(\frac{c0}{D} \cdot \frac{d}{w}\right) \cdot \frac{\frac{d}{w} \cdot c0}{D}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* (* D M_m) h) d)))
   (if (<= M_m 5.5e-121)
     0.0
     (if (<= M_m 7.5e+170)
       (/
        (fma t_0 (* -0.25 t_0) (* (* (/ c0 D) (/ d w)) (/ (* (/ d w) c0) D)))
        h)
       (/ (/ (* (* d c0) (* d c0)) (* (* (* D D) h) w)) w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = ((D * M_m) * h) / d;
	double tmp;
	if (M_m <= 5.5e-121) {
		tmp = 0.0;
	} else if (M_m <= 7.5e+170) {
		tmp = fma(t_0, (-0.25 * t_0), (((c0 / D) * (d / w)) * (((d / w) * c0) / D))) / h;
	} else {
		tmp = (((d * c0) * (d * c0)) / (((D * D) * h) * w)) / w;
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(Float64(D * M_m) * h) / d)
	tmp = 0.0
	if (M_m <= 5.5e-121)
		tmp = 0.0;
	elseif (M_m <= 7.5e+170)
		tmp = Float64(fma(t_0, Float64(-0.25 * t_0), Float64(Float64(Float64(c0 / D) * Float64(d / w)) * Float64(Float64(Float64(d / w) * c0) / D))) / h);
	else
		tmp = Float64(Float64(Float64(Float64(d * c0) * Float64(d * c0)) / Float64(Float64(Float64(D * D) * h) * w)) / w);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(N[(D * M$95$m), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[M$95$m, 5.5e-121], 0.0, If[LessEqual[M$95$m, 7.5e+170], N[(N[(t$95$0 * N[(-0.25 * t$95$0), $MachinePrecision] + N[(N[(N[(c0 / D), $MachinePrecision] * N[(d / w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(d / w), $MachinePrecision] * c0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 5.50000000000000031e-121

   1. Initial program 22.9%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 37.7

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

   5. Applied rewrites 37.7%

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

   if 5.50000000000000031e-121 < M < 7.5000000000000002e170

   1. Initial program 23.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {h}^{2}\right)}{{d}^{2}} + \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot {w}^{2}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {h}^{2}\right)}{{d}^{2}} + \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot {w}^{2}}}{h}} \]

   5. Applied rewrites 25.4%

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

   7. Applied rewrites 45.4%

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

   9. Applied rewrites 56.6%

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

   11. Applied rewrites 56.7%

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

   if 7.5000000000000002e170 < M

   1. Initial program 0.0%

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

   4. Applied rewrites 38.1%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 45.6

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

   7. Applied rewrites 45.6%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.5 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 7.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{d}, -0.25 \cdot \frac{\left(D \cdot M\right) \cdot h}{d}, \left(\frac{c0}{D} \cdot \frac{d}{w}\right) \cdot \frac{\frac{d}{w} \cdot c0}{D}\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.1% accurate, 0.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* (* d d) c0)) (t_1 (/ t_0 (* (* h w) (* D D)))))
   (if (<=
        (* (+ (sqrt (- (* t_1 t_1) (* M_m M_m))) t_1) (/ c0 (* w 2.0)))
        INFINITY)
     (* (/ t_0 (* (* (* h w) D) (* D w))) c0)
     0.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(h * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $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 76.1%

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 64.4

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

   7. Applied rewrites 64.4%

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

   9. Applied rewrites 73.1%

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

   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%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 38.2

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

   5. Applied rewrites 38.2%

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

4. Final simplification 47.9%

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

Alternative 3: 55.7% accurate, 0.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* (* d d) c0) (* (* h w) (* D D)))))
   (if (<=
        (* (+ (sqrt (- (* t_0 t_0) (* M_m M_m))) t_0) (/ c0 (* w 2.0)))
        INFINITY)
     (* (/ (* (* d c0) d) (* (* (* h w) D) (* D w))) c0)
     0.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(d * c0), $MachinePrecision] * d), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $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 76.1%

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 64.4

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

   7. Applied rewrites 64.4%

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

   9. Applied rewrites 64.4%

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

   11. Applied rewrites 73.1%

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

   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%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 38.2

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

   5. Applied rewrites 38.2%

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

4. Final simplification 47.9%

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

Alternative 4: 51.7% accurate, 0.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* (* d d) c0) (* (* h w) (* D D)))))
   (if (<=
        (* (+ (sqrt (- (* t_0 t_0) (* M_m M_m))) t_0) (/ c0 (* w 2.0)))
        INFINITY)
     (* (/ (* (* d c0) d) (* (* w w) (* (* D D) h))) c0)
     0.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(d * c0), $MachinePrecision] * d), $MachinePrecision] / N[(N[(w * w), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $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 76.1%

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 64.4

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

   7. Applied rewrites 64.4%

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

   9. Applied rewrites 64.4%

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

   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%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 38.2

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

   5. Applied rewrites 38.2%

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

4. Final simplification 45.5%

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

Alternative 5: 47.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{\left(D \cdot M\_m\right) \cdot h}{d}\\ t_1 := \frac{c0}{D} \cdot \frac{d}{w}\\ \mathbf{if}\;M\_m \leq 5.5 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{elif}\;M\_m \leq 1.05 \cdot 10^{+170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, -0.25 \cdot t\_0, t\_1 \cdot t\_1\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* (* D M_m) h) d)) (t_1 (* (/ c0 D) (/ d w))))
   (if (<= M_m 5.5e-121)
     0.0
     (if (<= M_m 1.05e+170)
       (/ (fma t_0 (* -0.25 t_0) (* t_1 t_1)) h)
       (/ (/ (* (* d c0) (* d c0)) (* (* (* D D) h) w)) w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = ((D * M_m) * h) / d;
	double t_1 = (c0 / D) * (d / w);
	double tmp;
	if (M_m <= 5.5e-121) {
		tmp = 0.0;
	} else if (M_m <= 1.05e+170) {
		tmp = fma(t_0, (-0.25 * t_0), (t_1 * t_1)) / h;
	} else {
		tmp = (((d * c0) * (d * c0)) / (((D * D) * h) * w)) / w;
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(Float64(D * M_m) * h) / d)
	t_1 = Float64(Float64(c0 / D) * Float64(d / w))
	tmp = 0.0
	if (M_m <= 5.5e-121)
		tmp = 0.0;
	elseif (M_m <= 1.05e+170)
		tmp = Float64(fma(t_0, Float64(-0.25 * t_0), Float64(t_1 * t_1)) / h);
	else
		tmp = Float64(Float64(Float64(Float64(d * c0) * Float64(d * c0)) / Float64(Float64(Float64(D * D) * h) * w)) / w);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(N[(D * M$95$m), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 / D), $MachinePrecision] * N[(d / w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 5.5e-121], 0.0, If[LessEqual[M$95$m, 1.05e+170], N[(N[(t$95$0 * N[(-0.25 * t$95$0), $MachinePrecision] + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 5.50000000000000031e-121

   1. Initial program 22.9%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 37.7

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

   5. Applied rewrites 37.7%

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

   if 5.50000000000000031e-121 < M < 1.04999999999999999e170

   1. Initial program 23.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {h}^{2}\right)}{{d}^{2}} + \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot {w}^{2}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {h}^{2}\right)}{{d}^{2}} + \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot {w}^{2}}}{h}} \]

   5. Applied rewrites 25.4%

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

   7. Applied rewrites 45.4%

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

   9. Applied rewrites 56.6%

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

   if 1.04999999999999999e170 < M

   1. Initial program 0.0%

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

   4. Applied rewrites 38.1%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 45.6

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

   7. Applied rewrites 45.6%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.5 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 1.05 \cdot 10^{+170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{d}, -0.25 \cdot \frac{\left(D \cdot M\right) \cdot h}{d}, \left(\frac{c0}{D} \cdot \frac{d}{w}\right) \cdot \left(\frac{c0}{D} \cdot \frac{d}{w}\right)\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{\left(D \cdot M\_m\right) \cdot h}{d}\\ \mathbf{if}\;M\_m \leq 6 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{elif}\;M\_m \leq 7.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, -0.25 \cdot t\_0, \left(\frac{c0}{D \cdot w} \cdot d\right) \cdot \left(\frac{c0}{D} \cdot \frac{d}{w}\right)\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* (* D M_m) h) d)))
   (if (<= M_m 6e-121)
     0.0
     (if (<= M_m 7.5e+170)
       (/
        (fma t_0 (* -0.25 t_0) (* (* (/ c0 (* D w)) d) (* (/ c0 D) (/ d w))))
        h)
       (/ (/ (* (* d c0) (* d c0)) (* (* (* D D) h) w)) w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = ((D * M_m) * h) / d;
	double tmp;
	if (M_m <= 6e-121) {
		tmp = 0.0;
	} else if (M_m <= 7.5e+170) {
		tmp = fma(t_0, (-0.25 * t_0), (((c0 / (D * w)) * d) * ((c0 / D) * (d / w)))) / h;
	} else {
		tmp = (((d * c0) * (d * c0)) / (((D * D) * h) * w)) / w;
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(Float64(D * M_m) * h) / d)
	tmp = 0.0
	if (M_m <= 6e-121)
		tmp = 0.0;
	elseif (M_m <= 7.5e+170)
		tmp = Float64(fma(t_0, Float64(-0.25 * t_0), Float64(Float64(Float64(c0 / Float64(D * w)) * d) * Float64(Float64(c0 / D) * Float64(d / w)))) / h);
	else
		tmp = Float64(Float64(Float64(Float64(d * c0) * Float64(d * c0)) / Float64(Float64(Float64(D * D) * h) * w)) / w);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(N[(D * M$95$m), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[M$95$m, 6e-121], 0.0, If[LessEqual[M$95$m, 7.5e+170], N[(N[(t$95$0 * N[(-0.25 * t$95$0), $MachinePrecision] + N[(N[(N[(c0 / N[(D * w), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * N[(N[(c0 / D), $MachinePrecision] * N[(d / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 5.9999999999999999e-121

   1. Initial program 22.9%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 37.7

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

   5. Applied rewrites 37.7%

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

   if 5.9999999999999999e-121 < M < 7.5000000000000002e170

   1. Initial program 23.4%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {h}^{2}\right)}{{d}^{2}} + \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot {w}^{2}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {h}^{2}\right)}{{d}^{2}} + \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot {w}^{2}}}{h}} \]

   5. Applied rewrites 25.4%

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

   7. Applied rewrites 45.4%

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

   9. Applied rewrites 56.6%

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

   11. Applied rewrites 54.7%

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

   if 7.5000000000000002e170 < M

   1. Initial program 0.0%

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

   4. Applied rewrites 38.1%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 45.6

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

   7. Applied rewrites 45.6%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 7.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{d}, -0.25 \cdot \frac{\left(D \cdot M\right) \cdot h}{d}, \left(\frac{c0}{D \cdot w} \cdot d\right) \cdot \left(\frac{c0}{D} \cdot \frac{d}{w}\right)\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.1 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{elif}\;M\_m \leq 2 \cdot 10^{+164}:\\ \;\;\;\;\left(\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{c0}{D}\right) \cdot \frac{d}{w}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 2.1e-123)
   0.0
   (if (<= M_m 2e+164)
     (* (* (* (/ d (* (* h w) D)) (/ c0 D)) (/ d w)) c0)
     (/ (/ (* (* d c0) (* d c0)) (* (* (* D D) h) w)) w))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 2.1e-123) {
		tmp = 0.0;
	} else if (M_m <= 2e+164) {
		tmp = (((d / ((h * w) * D)) * (c0 / D)) * (d / w)) * c0;
	} else {
		tmp = (((d * c0) * (d * c0)) / (((D * D) * h) * w)) / w;
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 2.1d-123) then
        tmp = 0.0d0
    else if (m_m <= 2d+164) then
        tmp = (((d_1 / ((h * w) * d)) * (c0 / d)) * (d_1 / w)) * c0
    else
        tmp = (((d_1 * c0) * (d_1 * c0)) / (((d * d) * h) * w)) / w
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 2.1e-123) {
		tmp = 0.0;
	} else if (M_m <= 2e+164) {
		tmp = (((d / ((h * w) * D)) * (c0 / D)) * (d / w)) * c0;
	} else {
		tmp = (((d * c0) * (d * c0)) / (((D * D) * h) * w)) / w;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 2.1e-123:
		tmp = 0.0
	elif M_m <= 2e+164:
		tmp = (((d / ((h * w) * D)) * (c0 / D)) * (d / w)) * c0
	else:
		tmp = (((d * c0) * (d * c0)) / (((D * D) * h) * w)) / w
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 2.1e-123)
		tmp = 0.0;
	elseif (M_m <= 2e+164)
		tmp = Float64(Float64(Float64(Float64(d / Float64(Float64(h * w) * D)) * Float64(c0 / D)) * Float64(d / w)) * c0);
	else
		tmp = Float64(Float64(Float64(Float64(d * c0) * Float64(d * c0)) / Float64(Float64(Float64(D * D) * h) * w)) / w);
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 2.1e-123)
		tmp = 0.0;
	elseif (M_m <= 2e+164)
		tmp = (((d / ((h * w) * D)) * (c0 / D)) * (d / w)) * c0;
	else
		tmp = (((d * c0) * (d * c0)) / (((D * D) * h) * w)) / w;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 2.1e-123], 0.0, If[LessEqual[M$95$m, 2e+164], N[(N[(N[(N[(d / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(c0 / D), $MachinePrecision]), $MachinePrecision] * N[(d / w), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < 2.0999999999999999e-123

   1. Initial program 22.9%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 37.7

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

   5. Applied rewrites 37.7%

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

   if 2.0999999999999999e-123 < M < 2e164

   1. Initial program 24.8%

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

   4. Applied rewrites 37.4%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 27.2

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

   7. Applied rewrites 27.2%

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

   9. Applied rewrites 44.3%

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

   11. Applied rewrites 50.5%

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

   if 2e164 < M

   1. Initial program 0.0%

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

   4. Applied rewrites 33.8%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 44.7

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

   7. Applied rewrites 44.7%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{-123}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \leq 2 \cdot 10^{+164}:\\ \;\;\;\;\left(\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{c0}{D}\right) \cdot \frac{d}{w}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4.5 \cdot 10^{-195}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{d}{D}}{w} \cdot \frac{\frac{d \cdot c0}{h}}{D \cdot w}\right) \cdot c0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 4.5e-195)
   0.0
   (* (* (/ (/ d D) w) (/ (/ (* d c0) h) (* D w))) c0)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 4.5e-195) {
		tmp = 0.0;
	} else {
		tmp = (((d / D) / w) * (((d * c0) / h) / (D * w))) * c0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 4.5d-195) then
        tmp = 0.0d0
    else
        tmp = (((d_1 / d) / w) * (((d_1 * c0) / h) / (d * w))) * c0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 4.5e-195) {
		tmp = 0.0;
	} else {
		tmp = (((d / D) / w) * (((d * c0) / h) / (D * w))) * c0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 4.5e-195:
		tmp = 0.0
	else:
		tmp = (((d / D) / w) * (((d * c0) / h) / (D * w))) * c0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 4.5e-195)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(d / D) / w) * Float64(Float64(Float64(d * c0) / h) / Float64(D * w))) * c0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 4.5e-195)
		tmp = 0.0;
	else
		tmp = (((d / D) / w) * (((d * c0) / h) / (D * w))) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 4.5e-195], 0.0, N[(N[(N[(N[(d / D), $MachinePrecision] / w), $MachinePrecision] * N[(N[(N[(d * c0), $MachinePrecision] / h), $MachinePrecision] / N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.5e-195

   1. Initial program 23.1%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 36.2

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

   5. Applied rewrites 36.2%

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

   if 4.5e-195 < M

   1. Initial program 17.4%

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

   4. Applied rewrites 37.1%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 30.5

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

   7. Applied rewrites 30.5%

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

   9. Applied rewrites 46.1%

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

   11. Applied rewrites 47.8%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.5 \cdot 10^{-195}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{d}{D}}{w} \cdot \frac{\frac{d \cdot c0}{h}}{D \cdot w}\right) \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3.2 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{d \cdot c0}{h} \cdot d}{D \cdot w}}{D \cdot w} \cdot c0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 3.2e-184)
   0.0
   (* (/ (/ (* (/ (* d c0) h) d) (* D w)) (* D w)) c0)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 3.2e-184) {
		tmp = 0.0;
	} else {
		tmp = (((((d * c0) / h) * d) / (D * w)) / (D * w)) * c0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 3.2d-184) then
        tmp = 0.0d0
    else
        tmp = (((((d_1 * c0) / h) * d_1) / (d * w)) / (d * w)) * c0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 3.2e-184) {
		tmp = 0.0;
	} else {
		tmp = (((((d * c0) / h) * d) / (D * w)) / (D * w)) * c0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 3.2e-184:
		tmp = 0.0
	else:
		tmp = (((((d * c0) / h) * d) / (D * w)) / (D * w)) * c0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 3.2e-184)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(d * c0) / h) * d) / Float64(D * w)) / Float64(D * w)) * c0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 3.2e-184)
		tmp = 0.0;
	else
		tmp = (((((d * c0) / h) * d) / (D * w)) / (D * w)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 3.2e-184], 0.0, N[(N[(N[(N[(N[(N[(d * c0), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision] / N[(D * w), $MachinePrecision]), $MachinePrecision] / N[(D * w), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.2e-184

   1. Initial program 23.7%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 36.6

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

   5. Applied rewrites 36.6%

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

   if 3.2e-184 < M

   1. Initial program 15.8%

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

   4. Applied rewrites 36.4%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 29.5

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

   7. Applied rewrites 29.5%

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

   9. Applied rewrites 44.8%

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

   11. Applied rewrites 46.7%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.2 \cdot 10^{-184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{d \cdot c0}{h} \cdot d}{D \cdot w}}{D \cdot w} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \cdot M\_m \leq 6 \cdot 10^{-242}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{d}{\left(D \cdot h\right) \cdot \left(D \cdot w\right)} \cdot \left(\frac{d}{w} \cdot c0\right)\right) \cdot c0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= (* M_m M_m) 6e-242)
   0.0
   (* (* (/ d (* (* D h) (* D w))) (* (/ d w) c0)) c0)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if ((M_m * M_m) <= 6e-242) {
		tmp = 0.0;
	} else {
		tmp = ((d / ((D * h) * (D * w))) * ((d / w) * c0)) * c0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if ((m_m * m_m) <= 6d-242) then
        tmp = 0.0d0
    else
        tmp = ((d_1 / ((d * h) * (d * w))) * ((d_1 / w) * c0)) * c0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if ((M_m * M_m) <= 6e-242) {
		tmp = 0.0;
	} else {
		tmp = ((d / ((D * h) * (D * w))) * ((d / w) * c0)) * c0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if (M_m * M_m) <= 6e-242:
		tmp = 0.0
	else:
		tmp = ((d / ((D * h) * (D * w))) * ((d / w) * c0)) * c0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (Float64(M_m * M_m) <= 6e-242)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(d / Float64(Float64(D * h) * Float64(D * w))) * Float64(Float64(d / w) * c0)) * c0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if ((M_m * M_m) <= 6e-242)
		tmp = 0.0;
	else
		tmp = ((d / ((D * h) * (D * w))) * ((d / w) * c0)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[N[(M$95$m * M$95$m), $MachinePrecision], 6e-242], 0.0, N[(N[(N[(d / N[(N[(D * h), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(d / w), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 6e-242

   1. Initial program 26.5%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 54.5

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

   5. Applied rewrites 54.5%

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

   if 6e-242 < (*.f64 M M)

   1. Initial program 18.1%

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

   4. Applied rewrites 34.8%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 32.3

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

   7. Applied rewrites 32.3%

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

   9. Applied rewrites 45.3%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 6 \cdot 10^{-242}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{d}{\left(D \cdot h\right) \cdot \left(D \cdot w\right)} \cdot \left(\frac{d}{w} \cdot c0\right)\right) \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 6 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{d \cdot c0}{\left(D \cdot h\right) \cdot \left(D \cdot w\right)} \cdot \frac{d}{w}\right) \cdot c0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 6e-121) 0.0 (* (* (/ (* d c0) (* (* D h) (* D w))) (/ d w)) c0)))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 6e-121) {
		tmp = 0.0;
	} else {
		tmp = (((d * c0) / ((D * h) * (D * w))) * (d / w)) * c0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 6d-121) then
        tmp = 0.0d0
    else
        tmp = (((d_1 * c0) / ((d * h) * (d * w))) * (d_1 / w)) * c0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 6e-121) {
		tmp = 0.0;
	} else {
		tmp = (((d * c0) / ((D * h) * (D * w))) * (d / w)) * c0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 6e-121:
		tmp = 0.0
	else:
		tmp = (((d * c0) / ((D * h) * (D * w))) * (d / w)) * c0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 6e-121)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(d * c0) / Float64(Float64(D * h) * Float64(D * w))) * Float64(d / w)) * c0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 6e-121)
		tmp = 0.0;
	else
		tmp = (((d * c0) / ((D * h) * (D * w))) * (d / w)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 6e-121], 0.0, N[(N[(N[(N[(d * c0), $MachinePrecision] / N[(N[(D * h), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / w), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.9999999999999999e-121

   1. Initial program 22.9%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 37.7

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

   5. Applied rewrites 37.7%

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

   if 5.9999999999999999e-121 < M

   1. Initial program 16.7%

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

   4. Applied rewrites 36.3%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 32.5

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

   7. Applied rewrites 32.5%

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

   9. Applied rewrites 45.7%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6 \cdot 10^{-121}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{d \cdot c0}{\left(D \cdot h\right) \cdot \left(D \cdot w\right)} \cdot \frac{d}{w}\right) \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.4% accurate, 156.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ 0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m) :precision binary64 0.0)

C

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

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    code = 0.0d0
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.0;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return 0.0

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := 0.0

Derivation

1. Initial program 21.1%

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

3. Taylor expanded in c0 around -inf

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

   1. associate-/l* N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

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

   5. div0 N/A

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

   6. mul0-rgt N/A

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

   7. metadata-eval 31.3

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

5. Applied rewrites 31.3%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024241 
(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))))))