Henrywood and Agarwal, Equation (13)

Percentage Accurate: 23.7% → 53.2%

Time: 17.5s

Alternatives: 8

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = -5.865894831559731e+285
  2. w = -5.543050975883308e-96
  3. h = -3.2886318487620183e-286
  4. D = 1.315350307595602e-283
  5. d = -5.901685168173225e+301
  6. M = -7.077086883950598e-252

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: 23.7% 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: 53.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ \begin{array}{l} t_0 := \frac{c0}{h \cdot w}\\ t_1 := {\left(\frac{d\_m}{D\_m}\right)}^{2}\\ t_2 := \frac{c0}{w \cdot 2}\\ t_3 := \frac{\left(d\_m \cdot d\_m\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D\_m \cdot D\_m\right)}\\ \mathbf{if}\;\left(\sqrt{t\_3 \cdot t\_3 - M \cdot M} + t\_3\right) \cdot t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{t\_0} \cdot \frac{d\_m}{D\_m}, \sqrt{\mathsf{fma}\left(t\_1, t\_0, M\right)}, \frac{t\_1}{h \cdot w} \cdot c0\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ c0 (* h w)))
        (t_1 (pow (/ d_m D_m) 2.0))
        (t_2 (/ c0 (* w 2.0)))
        (t_3 (/ (* (* d_m d_m) c0) (* (* h w) (* D_m D_m)))))
   (if (<= (* (+ (sqrt (- (* t_3 t_3) (* M M))) t_3) t_2) INFINITY)
     (*
      (fma
       (* (sqrt t_0) (/ d_m D_m))
       (sqrt (fma t_1 t_0 M))
       (* (/ t_1 (* h w)) c0))
      t_2)
     0.0)))

C

d_m = fabs(d);
D_m = fabs(D);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = c0 / (h * w);
	double t_1 = pow((d_m / D_m), 2.0);
	double t_2 = c0 / (w * 2.0);
	double t_3 = ((d_m * d_m) * c0) / ((h * w) * (D_m * D_m));
	double tmp;
	if (((sqrt(((t_3 * t_3) - (M * M))) + t_3) * t_2) <= ((double) INFINITY)) {
		tmp = fma((sqrt(t_0) * (d_m / D_m)), sqrt(fma(t_1, t_0, M)), ((t_1 / (h * w)) * c0)) * t_2;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

d_m = abs(d)
D_m = abs(D)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(c0 / Float64(h * w))
	t_1 = Float64(d_m / D_m) ^ 2.0
	t_2 = Float64(c0 / Float64(w * 2.0))
	t_3 = Float64(Float64(Float64(d_m * d_m) * c0) / Float64(Float64(h * w) * Float64(D_m * D_m)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))) + t_3) * t_2) <= Inf)
		tmp = Float64(fma(Float64(sqrt(t_0) * Float64(d_m / D_m)), sqrt(fma(t_1, t_0, M)), Float64(Float64(t_1 / Float64(h * w)) * c0)) * t_2);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d$95$m / D$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(d$95$m * d$95$m), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision], Infinity], N[(N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$1 * t$95$0 + M), $MachinePrecision]], $MachinePrecision] + N[(N[(t$95$1 / N[(h * w), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.3%

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in w around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   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 34.8%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 49.5%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

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

Alternative 2: 54.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ \begin{array}{l} t_0 := \frac{\left(d\_m \cdot d\_m\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D\_m \cdot D\_m\right)}\\ t_1 := \sqrt{t\_0 \cdot t\_0 - M \cdot M} + t\_0\\ \mathbf{if}\;t\_1 \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\left(\frac{0.5}{w} \cdot c0\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ (* (* d_m d_m) c0) (* (* h w) (* D_m D_m))))
        (t_1 (+ (sqrt (- (* t_0 t_0) (* M M))) t_0)))
   (if (<= (* t_1 (/ c0 (* w 2.0))) INFINITY) (* (* (/ 0.5 w) c0) t_1) 0.0)))

C

d_m = fabs(d);
D_m = fabs(D);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = ((d_m * d_m) * c0) / ((h * w) * (D_m * D_m));
	double t_1 = sqrt(((t_0 * t_0) - (M * M))) + t_0;
	double tmp;
	if ((t_1 * (c0 / (w * 2.0))) <= ((double) INFINITY)) {
		tmp = ((0.5 / w) * c0) * t_1;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
public static double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = ((d_m * d_m) * c0) / ((h * w) * (D_m * D_m));
	double t_1 = Math.sqrt(((t_0 * t_0) - (M * M))) + t_0;
	double tmp;
	if ((t_1 * (c0 / (w * 2.0))) <= Double.POSITIVE_INFINITY) {
		tmp = ((0.5 / w) * c0) * t_1;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
D_m = math.fabs(D)
def code(c0, w, h, D_m, d_m, M):
	t_0 = ((d_m * d_m) * c0) / ((h * w) * (D_m * D_m))
	t_1 = math.sqrt(((t_0 * t_0) - (M * M))) + t_0
	tmp = 0
	if (t_1 * (c0 / (w * 2.0))) <= math.inf:
		tmp = ((0.5 / w) * c0) * t_1
	else:
		tmp = 0.0
	return tmp

Julia

d_m = abs(d)
D_m = abs(D)
function code(c0, w, h, D_m, d_m, M)
	t_0 = Float64(Float64(Float64(d_m * d_m) * c0) / Float64(Float64(h * w) * Float64(D_m * D_m)))
	t_1 = Float64(sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))) + t_0)
	tmp = 0.0
	if (Float64(t_1 * Float64(c0 / Float64(w * 2.0))) <= Inf)
		tmp = Float64(Float64(Float64(0.5 / w) * c0) * t_1);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

d_m = abs(d);
D_m = abs(D);
function tmp_2 = code(c0, w, h, D_m, d_m, M)
	t_0 = ((d_m * d_m) * c0) / ((h * w) * (D_m * D_m));
	t_1 = sqrt(((t_0 * t_0) - (M * M))) + t_0;
	tmp = 0.0;
	if ((t_1 * (c0 / (w * 2.0))) <= Inf)
		tmp = ((0.5 / w) * c0) * t_1;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(N[(N[(d$95$m * d$95$m), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(0.5 / w), $MachinePrecision] * c0), $MachinePrecision] * t$95$1), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 78.3

         \[\leadsto \left(\color{blue}{\frac{0.5}{w}} \cdot c0\right) \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   4. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\left(\frac{0.5}{w} \cdot c0\right)} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   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 34.8%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 49.5%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\left(\frac{0.5}{w} \cdot c0\right) \cdot \left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(N[(N[(d$95$m * d$95$m), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(c0 / D$95$m), $MachinePrecision] * d$95$m), $MachinePrecision] * N[(c0 / D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] / d$95$m), $MachinePrecision] * 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 78.3%

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

   3. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 75.1%

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

   9. Applied rewrites 76.5%

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

   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 34.8%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 49.5%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

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

Alternative 4: 53.1% accurate, 0.7× speedup

Math

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

FPCore

d_m = (fabs.f64 d)
D_m = (fabs.f64 D)
(FPCore (c0 w h D_m d_m M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w 2.0)))
        (t_1 (* (* d_m d_m) c0))
        (t_2 (/ t_1 (* (* h w) (* D_m D_m)))))
   (if (<= (* (+ (sqrt (- (* t_2 t_2) (* M M))) t_2) t_0) INFINITY)
     (* (/ (* t_1 2.0) (* (* (* D_m D_m) h) w)) t_0)
     0.0)))

C

d_m = fabs(d);
D_m = fabs(D);
double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = c0 / (w * 2.0);
	double t_1 = (d_m * d_m) * c0;
	double t_2 = t_1 / ((h * w) * (D_m * D_m));
	double tmp;
	if (((sqrt(((t_2 * t_2) - (M * M))) + t_2) * t_0) <= ((double) INFINITY)) {
		tmp = ((t_1 * 2.0) / (((D_m * D_m) * h) * w)) * t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

d_m = Math.abs(d);
D_m = Math.abs(D);
public static double code(double c0, double w, double h, double D_m, double d_m, double M) {
	double t_0 = c0 / (w * 2.0);
	double t_1 = (d_m * d_m) * c0;
	double t_2 = t_1 / ((h * w) * (D_m * D_m));
	double tmp;
	if (((Math.sqrt(((t_2 * t_2) - (M * M))) + t_2) * t_0) <= Double.POSITIVE_INFINITY) {
		tmp = ((t_1 * 2.0) / (((D_m * D_m) * h) * w)) * t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := Block[{t$95$0 = N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d$95$m * d$95$m), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(h * w), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision], Infinity], N[(N[(N[(t$95$1 * 2.0), $MachinePrecision] / N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.3%

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

   3. Taylor expanded in w around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

   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 34.8%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 49.5%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

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

Alternative 5: 53.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.3%

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

   3. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 75.1%

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

   9. Applied rewrites 74.0%

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

   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 34.8%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 49.5%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

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

Alternative 6: 52.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.3%

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

   3. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 70.2%

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

   9. Applied rewrites 72.7%

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

   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. *-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 34.8%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 49.5%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

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

Alternative 7: 50.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.3%

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

   3. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 70.2%

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

   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 34.8%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 49.5%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.9%

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

Alternative 8: 33.8% accurate, 156.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ D_m = \left|D\right| \\ 0 \end{array} \]

FPCore

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

C

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

Fortran

d_m = abs(d)
D_m = abs(d)
real(8) function code(c0, w, h, d_m, d_m_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

d_m = N[Abs[d], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M_] := 0.0

Derivation

1. Initial program 24.2%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. 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 26.4%

   \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

6. Taylor expanded in w around 0

   \[\leadsto 0 \]
7. Step-by-step derivation

8. Applied rewrites 36.9%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(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))))))