Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 93.0%

Time: 15.5s

Alternatives: 3

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{M\_m \cdot D\_m}{2 \cdot d\_m}\\ t_1 := {t\_0}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+297}:\\ \;\;\;\;\sqrt{\frac{h \cdot -0.25}{\ell}} \cdot \left(D\_m \cdot \frac{M\_m \cdot w0}{d\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{\left(M\_m \cdot D\_m\right) \cdot 0.5}{d\_m} \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (* 2.0 d_m))) (t_1 (* (pow t_0 2.0) (/ h l))))
   (if (<= t_1 -4e+297)
     (* (sqrt (/ (* h -0.25) l)) (* D_m (/ (* M_m w0) d_m)))
     (if (<= t_1 1.0)
       (* w0 (sqrt (- 1.0 (* (/ h l) (* (/ (* (* M_m D_m) 0.5) d_m) t_0)))))
       w0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double t_1 = pow(t_0, 2.0) * (h / l);
	double tmp;
	if (t_1 <= -4e+297) {
		tmp = sqrt(((h * -0.25) / l)) * (D_m * ((M_m * w0) / d_m));
	} else if (t_1 <= 1.0) {
		tmp = w0 * sqrt((1.0 - ((h / l) * ((((M_m * D_m) * 0.5) / d_m) * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m_m * d_m) / (2.0d0 * d_m_1)
    t_1 = (t_0 ** 2.0d0) * (h / l)
    if (t_1 <= (-4d+297)) then
        tmp = sqrt(((h * (-0.25d0)) / l)) * (d_m * ((m_m * w0) / d_m_1))
    else if (t_1 <= 1.0d0) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((((m_m * d_m) * 0.5d0) / d_m_1) * t_0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double t_1 = Math.pow(t_0, 2.0) * (h / l);
	double tmp;
	if (t_1 <= -4e+297) {
		tmp = Math.sqrt(((h * -0.25) / l)) * (D_m * ((M_m * w0) / d_m));
	} else if (t_1 <= 1.0) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * ((((M_m * D_m) * 0.5) / d_m) * t_0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = (M_m * D_m) / (2.0 * d_m)
	t_1 = math.pow(t_0, 2.0) * (h / l)
	tmp = 0
	if t_1 <= -4e+297:
		tmp = math.sqrt(((h * -0.25) / l)) * (D_m * ((M_m * w0) / d_m))
	elif t_1 <= 1.0:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * ((((M_m * D_m) * 0.5) / d_m) * t_0))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(2.0 * d_m))
	t_1 = Float64((t_0 ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_1 <= -4e+297)
		tmp = Float64(sqrt(Float64(Float64(h * -0.25) / l)) * Float64(D_m * Float64(Float64(M_m * w0) / d_m)));
	elseif (t_1 <= 1.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(M_m * D_m) * 0.5) / d_m) * t_0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (M_m * D_m) / (2.0 * d_m);
	t_1 = (t_0 ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_1 <= -4e+297)
		tmp = sqrt(((h * -0.25) / l)) * (D_m * ((M_m * w0) / d_m));
	elseif (t_1 <= 1.0)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((((M_m * D_m) * 0.5) / d_m) * t_0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+297], N[(N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(D$95$m * N[(N[(M$95$m * w0), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4.0000000000000001e297

   1. Initial program 57.6%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 57.6%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 57.6%

         \[\leadsto w0 \cdot \sqrt{1 - {\left(M \cdot \color{blue}{\frac{1}{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot \frac{h}{\ell}} \]

      2. un-div-inv 57.6%

         \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]

      3. associate-/l* 57.6%

         \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M}{\color{blue}{2 \cdot \frac{d}{D}}}\right)}^{2} \cdot \frac{h}{\ell}} \]

   6. Applied egg-rr 57.6%

      \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]
   7. Step-by-step derivation

      1. associate-*r/ 59.1%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2} \cdot h}{\ell}}} \]

      2. associate-*r/ 59.1%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{\color{blue}{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot h}{\ell}} \]

      3. *-commutative 59.1%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{\frac{\color{blue}{d \cdot 2}}{D}}\right)}^{2} \cdot h}{\ell}} \]

   8. Applied egg-rr 59.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot h}{\ell}}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 59.1%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot h} \cdot \sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot h}\right) \cdot \sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot h}}}{\ell}} \]

      2. pow3 59.1%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot h}\right)}^{3}}}{\ell}} \]

      3. associate-/r/ 59.1%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\sqrt[3]{{\color{blue}{\left(\frac{M}{d \cdot 2} \cdot D\right)}}^{2} \cdot h}\right)}^{3}}{\ell}} \]

   10. Applied egg-rr 59.1%

       \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\sqrt[3]{{\left(\frac{M}{d \cdot 2} \cdot D\right)}^{2} \cdot h}\right)}^{3}}}{\ell}} \]

   11. Taylor expanded in h around -inf 0.0%

       \[\leadsto \color{blue}{-1 \cdot \left(\frac{D \cdot \left(M \cdot \left(w0 \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}{d} \cdot \sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 0.0%

          \[\leadsto \color{blue}{-\frac{D \cdot \left(M \cdot \left(w0 \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}{d} \cdot \sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}} \]

       2. distribute-rgt-neg-in 0.0%

          \[\leadsto \color{blue}{\frac{D \cdot \left(M \cdot \left(w0 \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}{d} \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right)} \]

       3. associate-/l* 0.0%

          \[\leadsto \color{blue}{\left(D \cdot \frac{M \cdot \left(w0 \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right) \]

       4. *-commutative 0.0%

          \[\leadsto \left(D \cdot \frac{M \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot w0\right)}}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right) \]

       5. unpow2 0.0%

          \[\leadsto \left(D \cdot \frac{M \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot w0\right)}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right) \]

       6. rem-square-sqrt 51.1%

          \[\leadsto \left(D \cdot \frac{M \cdot \left(\color{blue}{-1} \cdot w0\right)}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right) \]

       7. mul-1-neg 51.1%

          \[\leadsto \left(D \cdot \frac{M \cdot \color{blue}{\left(-w0\right)}}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right) \]

       8. rem-cube-cbrt 51.1%

          \[\leadsto \left(D \cdot \frac{M \cdot \left(-w0\right)}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot \color{blue}{-0.25}}{\ell}}\right) \]

   13. Simplified 51.1%

       \[\leadsto \color{blue}{\left(D \cdot \frac{M \cdot \left(-w0\right)}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot -0.25}{\ell}}\right)} \]

   if -4.0000000000000001e297 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 1

   1. Initial program 99.9%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

      2. unpow2 99.9%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}} \]

      3. associate-/r* 99.9%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \]

      4. frac-times 79.9%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2} \cdot \left(M \cdot D\right)}{d \cdot \left(2 \cdot d\right)}} \cdot \frac{h}{\ell}} \]

      5. associate-/l* 79.9%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(M \cdot \frac{D}{2}\right)} \cdot \left(M \cdot D\right)}{d \cdot \left(2 \cdot d\right)} \cdot \frac{h}{\ell}} \]

      6. div-inv 79.9%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(M \cdot \color{blue}{\left(D \cdot \frac{1}{2}\right)}\right) \cdot \left(M \cdot D\right)}{d \cdot \left(2 \cdot d\right)} \cdot \frac{h}{\ell}} \]

      7. metadata-eval 79.9%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\left(M \cdot \left(D \cdot \color{blue}{0.5}\right)\right) \cdot \left(M \cdot D\right)}{d \cdot \left(2 \cdot d\right)} \cdot \frac{h}{\ell}} \]

   6. Applied egg-rr 79.9%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(M \cdot \left(D \cdot 0.5\right)\right) \cdot \left(M \cdot D\right)}{d \cdot \left(2 \cdot d\right)}} \cdot \frac{h}{\ell}} \]
   7. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot \left(D \cdot 0.5\right)}{d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}} \]

      2. associate-*r* 99.9%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\color{blue}{\left(M \cdot D\right) \cdot 0.5}}{d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \]

      3. *-commutative 99.9%

         \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{\left(M \cdot D\right) \cdot 0.5}{d} \cdot \frac{M \cdot D}{\color{blue}{d \cdot 2}}\right) \cdot \frac{h}{\ell}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{\left(M \cdot D\right) \cdot 0.5}{d} \cdot \frac{M \cdot D}{d \cdot 2}\right)} \cdot \frac{h}{\ell}} \]

   if 1 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 0.0%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 71.1%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+297}:\\ \;\;\;\;\sqrt{\frac{h \cdot -0.25}{\ell}} \cdot \left(D \cdot \frac{M \cdot w0}{d}\right)\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 1:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{\left(M \cdot D\right) \cdot 0.5}{d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell} \leq -4000:\\ \;\;\;\;w0 \cdot \left(\sqrt{\frac{h \cdot -0.25}{\ell}} \cdot \left(D\_m \cdot \frac{M\_m}{d\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0) (/ h l)) -4000.0)
   (* w0 (* (sqrt (/ (* h -0.25) l)) (* D_m (/ M_m d_m))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -4000.0) {
		tmp = w0 * (sqrt(((h * -0.25) / l)) * (D_m * (M_m / d_m)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0) * (h / l)) <= (-4000.0d0)) then
        tmp = w0 * (sqrt(((h * (-0.25d0)) / l)) * (d_m * (m_m / d_m_1)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -4000.0) {
		tmp = w0 * (Math.sqrt(((h * -0.25) / l)) * (D_m * (M_m / d_m)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0) * (h / l)) <= -4000.0:
		tmp = w0 * (math.sqrt(((h * -0.25) / l)) * (D_m * (M_m / d_m)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)) <= -4000.0)
		tmp = Float64(w0 * Float64(sqrt(Float64(Float64(h * -0.25) / l)) * Float64(D_m * Float64(M_m / d_m))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d_m)) ^ 2.0) * (h / l)) <= -4000.0)
		tmp = w0 * (sqrt(((h * -0.25) / l)) * (D_m * (M_m / d_m)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4000.0], N[(w0 * N[(N[Sqrt[N[(N[(h * -0.25), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < -4e3

   1. Initial program 65.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 63.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 63.5%

         \[\leadsto w0 \cdot \sqrt{1 - {\left(M \cdot \color{blue}{\frac{1}{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot \frac{h}{\ell}} \]

      2. un-div-inv 63.5%

         \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]

      3. associate-/l* 63.5%

         \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M}{\color{blue}{2 \cdot \frac{d}{D}}}\right)}^{2} \cdot \frac{h}{\ell}} \]

   6. Applied egg-rr 63.5%

      \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]
   7. Step-by-step derivation

      1. associate-*r/ 64.7%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2} \cdot h}{\ell}}} \]

      2. associate-*r/ 64.7%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{\color{blue}{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot h}{\ell}} \]

      3. *-commutative 64.7%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{\frac{\color{blue}{d \cdot 2}}{D}}\right)}^{2} \cdot h}{\ell}} \]

   8. Applied egg-rr 64.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot h}{\ell}}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 64.6%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot h} \cdot \sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot h}\right) \cdot \sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot h}}}{\ell}} \]

      2. pow3 64.6%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2} \cdot h}\right)}^{3}}}{\ell}} \]

      3. associate-/r/ 64.6%

         \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\sqrt[3]{{\color{blue}{\left(\frac{M}{d \cdot 2} \cdot D\right)}}^{2} \cdot h}\right)}^{3}}{\ell}} \]

   10. Applied egg-rr 64.6%

       \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\sqrt[3]{{\left(\frac{M}{d \cdot 2} \cdot D\right)}^{2} \cdot h}\right)}^{3}}}{\ell}} \]

   11. Taylor expanded in h around -inf 0.0%

       \[\leadsto w0 \cdot \color{blue}{\left(-1 \cdot \left(\frac{D \cdot \left(M \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right)\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 0.0%

          \[\leadsto w0 \cdot \color{blue}{\left(-\frac{D \cdot \left(M \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right)} \]

       2. distribute-rgt-neg-in 0.0%

          \[\leadsto w0 \cdot \color{blue}{\left(\frac{D \cdot \left(M \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right)\right)} \]

       3. associate-/l* 0.0%

          \[\leadsto w0 \cdot \left(\color{blue}{\left(D \cdot \frac{M \cdot {\left(\sqrt{-1}\right)}^{2}}{d}\right)} \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right)\right) \]

       4. *-commutative 0.0%

          \[\leadsto w0 \cdot \left(\left(D \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot M}}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right)\right) \]

       5. unpow2 0.0%

          \[\leadsto w0 \cdot \left(\left(D \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot M}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right)\right) \]

       6. rem-square-sqrt 43.0%

          \[\leadsto w0 \cdot \left(\left(D \cdot \frac{\color{blue}{-1} \cdot M}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right)\right) \]

       7. mul-1-neg 43.0%

          \[\leadsto w0 \cdot \left(\left(D \cdot \frac{\color{blue}{-M}}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}}{\ell}}\right)\right) \]

       8. rem-cube-cbrt 43.0%

          \[\leadsto w0 \cdot \left(\left(D \cdot \frac{-M}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot \color{blue}{-0.25}}{\ell}}\right)\right) \]

   13. Simplified 43.0%

       \[\leadsto w0 \cdot \color{blue}{\left(\left(D \cdot \frac{-M}{d}\right) \cdot \left(-\sqrt{\frac{h \cdot -0.25}{\ell}}\right)\right)} \]

   if -4e3 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in M around 0 94.8%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4000:\\ \;\;\;\;w0 \cdot \left(\sqrt{\frac{h \cdot -0.25}{\ell}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
(FPCore (w0 M_m D_m h l d_m) :precision binary64 w0)

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
def code(w0, M_m, D_m, h, l, d_m):
	return w0

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
function code(w0, M_m, D_m, h, l, d_m)
	return w0
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
function tmp = code(w0, M_m, D_m, h, l, d_m)
	tmp = w0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 79.0%

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

3. Simplified 77.9%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
4. Add Preprocessing

5. Taylor expanded in M around 0 66.5%

   \[\leadsto \color{blue}{w0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))