Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.7% → 46.5%

Time: 15.2s

Alternatives: 7

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: 46.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4.4 \cdot 10^{-261}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{{\left(\frac{c0}{D} \cdot d\right)}^{2}}{h}}{w}}{w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 4.4e-261) 0.0 (/ (/ (/ (pow (* (/ c0 D) d) 2.0) 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 <= 4.4e-261) {
		tmp = 0.0;
	} else {
		tmp = ((pow(((c0 / D) * d), 2.0) / 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 <= 4.4d-261) then
        tmp = 0.0d0
    else
        tmp = (((((c0 / d) * d_1) ** 2.0d0) / 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 <= 4.4e-261) {
		tmp = 0.0;
	} else {
		tmp = ((Math.pow(((c0 / D) * d), 2.0) / 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 <= 4.4e-261:
		tmp = 0.0
	else:
		tmp = ((math.pow(((c0 / D) * d), 2.0) / 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 <= 4.4e-261)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64((Float64(Float64(c0 / D) * d) ^ 2.0) / 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 <= 4.4e-261)
		tmp = 0.0;
	else
		tmp = (((((c0 / D) * d) ^ 2.0) / 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, 4.4e-261], 0.0, N[(N[(N[(N[Power[N[(N[(c0 / D), $MachinePrecision] * d), $MachinePrecision], 2.0], $MachinePrecision] / h), $MachinePrecision] / w), $MachinePrecision] / w), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.4000000000000003e-261

   1. Initial program 25.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. 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. *-commutative N/A

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

   5. Applied rewrites 24.9%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.8%

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

   if 4.4000000000000003e-261 < M

   1. Initial program 28.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 55.1%

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

Alternative 2: 53.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \frac{2 \cdot t\_0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* c0 (* d d)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ t_0 (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M_m M_m))))) INFINITY)
     (* t_1 (/ (* 2.0 t_0) (* (* (* D D) h) w)))
     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 = c0 * (d * d);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_1 * ((2.0 * t_0) / (((D * D) * h) * w));
	} 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 = c0 * (d * d);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * ((2.0 * t_0) / (((D * D) * h) * w));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 * (d * d)
	t_1 = c0 / (2.0 * w)
	t_2 = t_0 / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M_m * M_m))))) <= math.inf:
		tmp = t_1 * ((2.0 * t_0) / (((D * D) * h) * w))
	else:
		tmp = 0.0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_0 / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(2.0 * t_0) / Float64(Float64(Float64(D * D) * h) * w)));
	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 = c0 * (d * d);
	t_1 = c0 / (2.0 * w);
	t_2 = t_0 / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))))) <= Inf)
		tmp = t_1 * ((2.0 * t_0) / (((D * D) * h) * w));
	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[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(2.0 * t$95$0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 71.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   4. Applied rewrites 71.1%

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

   5. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 70.5

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

   7. Applied rewrites 70.5%

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

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

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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. *-commutative N/A

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

   5. Applied rewrites 30.5%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.4%

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

Alternative 3: 53.0% accurate, 0.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M_m M_m)))))
        INFINITY)
     (/ (/ (/ (* (* c0 d) (* c0 d)) (* (* D D) h)) w) w)
     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 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = ((((c0 * d) * (c0 * d)) / ((D * D) * h)) / w) / w;
	} 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 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = ((((c0 * d) * (c0 * d)) / ((D * D) * h)) / w) / w;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(Float64(Float64(Float64(Float64(c0 * d) * Float64(c0 * d)) / Float64(Float64(D * D) * h)) / w) / w);
	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 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Inf)
		tmp = ((((c0 * d) * (c0 * d)) / ((D * D) * h)) / w) / w;
	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[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(c0 * d), $MachinePrecision] * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / w), $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 71.7%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 64.4%

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

   9. Applied rewrites 67.5%

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

   10. Taylor expanded in c0 around inf

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

   12. Applied rewrites 67.3%

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

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

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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. *-commutative N/A

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

   5. Applied rewrites 30.5%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.4%

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

4. Final simplification 50.5%

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

Alternative 4: 50.4% accurate, 0.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M_m M_m)))))
        INFINITY)
     (/ (/ (* (* c0 d) (* c0 d)) (* (* D D) h)) (* w w))
     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 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = (((c0 * d) * (c0 * d)) / ((D * D) * h)) / (w * w);
	} 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 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = (((c0 * d) * (c0 * d)) / ((D * D) * h)) / (w * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(Float64(Float64(Float64(c0 * d) * Float64(c0 * d)) / Float64(Float64(D * D) * h)) / Float64(w * w));
	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 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Inf)
		tmp = (((c0 * d) * (c0 * d)) / ((D * D) * h)) / (w * w);
	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[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(c0 * d), $MachinePrecision] * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 71.7%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in c0 around inf

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

   10. Applied rewrites 63.5%

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

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

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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. *-commutative N/A

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

   5. Applied rewrites 30.5%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.4%

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

Alternative 5: 50.8% accurate, 0.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M_m M_m)))))
        INFINITY)
     (* (* c0 c0) (* (/ d (* (* w (* D D)) h)) (/ d w)))
     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 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = (c0 * c0) * ((d / ((w * (D * D)) * h)) * (d / w));
	} 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 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * c0) * ((d / ((w * (D * D)) * h)) * (d / w));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(Float64(c0 * c0) * Float64(Float64(d / Float64(Float64(w * Float64(D * D)) * h)) * Float64(d / w)));
	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 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Inf)
		tmp = (c0 * c0) * ((d / ((w * (D * D)) * h)) * (d / w));
	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[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * c0), $MachinePrecision] * N[(N[(d / N[(N[(w * N[(D * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(d / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 71.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 61.5%

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

   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. *-commutative N/A

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

   5. Applied rewrites 30.5%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.4%

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

Alternative 6: 49.4% accurate, 0.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M_m M_m)))))
        INFINITY)
     (* (* c0 c0) (/ (* d d) (* D (* D (* (* h w) w)))))
     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 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = (c0 * c0) * ((d * d) / (D * (D * ((h * w) * w))));
	} 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 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * c0) * ((d * d) / (D * (D * ((h * w) * w))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(Float64(c0 * c0) * Float64(Float64(d * d) / Float64(D * Float64(D * Float64(Float64(h * w) * w)))));
	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 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))))) <= Inf)
		tmp = (c0 * c0) * ((d * d) / (D * (D * ((h * w) * w))));
	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[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * N[(D * N[(N[(h * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 71.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 60.5%

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

   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. *-commutative N/A

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

   5. Applied rewrites 30.5%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.4%

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

Alternative 7: 33.6% 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 26.9%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. 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. *-commutative N/A

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

5. Applied rewrites 23.5%

   \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 29.9%

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

Reproduce

herbie shell --seed 2024338 
(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))))))