Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 87.8%

Time: 14.6s

Alternatives: 8

Speedup: 1.0×

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: 81.3% 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: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} d = |d|\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+296}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(\frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell} \cdot -0.25\right)\right)\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 2e+296)
   (* w0 (sqrt (+ 1.0 (/ -1.0 (/ l (* (pow (* 0.5 (/ (* M D) d)) 2.0) h))))))
   (*
    w0
    (pow
     (pow
      (exp 0.25)
      (fma -2.0 (log d) (log (* (/ (* h (pow (* M D) 2.0)) l) -0.25))))
     2.0))))

C

d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 2e+296) {
		tmp = w0 * sqrt((1.0 + (-1.0 / (l / (pow((0.5 * ((M * D) / d)), 2.0) * h)))));
	} else {
		tmp = w0 * pow(pow(exp(0.25), fma(-2.0, log(d), log((((h * pow((M * D), 2.0)) / l) * -0.25)))), 2.0);
	}
	return tmp;
}

Julia

d = abs(d)
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 2e+296)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64((Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0) * h))))));
	else
		tmp = Float64(w0 * ((exp(0.25) ^ fma(-2.0, log(d), log(Float64(Float64(Float64(h * (Float64(M * D) ^ 2.0)) / l) * -0.25)))) ^ 2.0));
	end
	return tmp
end

Wolfram

NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 2e+296], N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(-2.0 * N[Log[d], $MachinePrecision] + N[Log[N[(N[(N[(h * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 1.99999999999999996e296

   1. Initial program 90.1%

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

   3. Simplified 89.6%

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

      1. associate-*r/ 96.1%

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

      2. frac-times 96.6%

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

      3. *-commutative 96.6%

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

      4. clear-num 96.6%

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

      5. unpow2 96.6%

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

      6. unpow2 96.6%

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

      7. *-un-lft-identity 96.6%

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

      8. times-frac 96.6%

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

      9. metadata-eval 96.6%

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

      10. *-commutative 96.6%

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

   5. Applied egg-rr 96.6%

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

   if 1.99999999999999996e296 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 49.2%

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

   3. Simplified 54.2%

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

      1. associate-*r/ 57.9%

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

      2. frac-times 52.9%

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

      3. *-commutative 52.9%

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

      4. clear-num 52.9%

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

      5. unpow2 52.9%

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

      6. unpow2 52.9%

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

      7. *-un-lft-identity 52.9%

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

      8. times-frac 52.9%

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

      9. metadata-eval 52.9%

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

      10. *-commutative 52.9%

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

   5. Applied egg-rr 52.9%

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

      1. clear-num 52.9%

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

      2. add-sqr-sqrt 52.9%

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

      3. pow2 52.9%

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

   7. Applied egg-rr 52.4%

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

   8. Taylor expanded in d around 0 30.3%

      \[\leadsto w0 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}}^{2} \]
   9. Step-by-step derivation

      1. exp-prod 30.3%

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

      2. +-commutative 30.3%

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

      3. fma-def 30.3%

         \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\color{blue}{\left(\mathsf{fma}\left(-2, \log d, \log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)\right)\right)}}\right)}^{2} \]

      4. *-commutative 30.3%

         \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(-\color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell} \cdot 0.25}\right)\right)\right)}\right)}^{2} \]

      5. distribute-rgt-neg-in 30.3%

         \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \color{blue}{\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell} \cdot \left(-0.25\right)\right)}\right)\right)}\right)}^{2} \]

      6. associate-*r* 30.3%

         \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{\ell} \cdot \left(-0.25\right)\right)\right)\right)}\right)}^{2} \]

      7. unpow2 30.3%

         \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(\frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{\ell} \cdot \left(-0.25\right)\right)\right)\right)}\right)}^{2} \]

      8. unpow2 30.3%

         \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(\frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{\ell} \cdot \left(-0.25\right)\right)\right)\right)}\right)}^{2} \]

      9. swap-sqr 32.0%

         \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(\frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{\ell} \cdot \left(-0.25\right)\right)\right)\right)}\right)}^{2} \]

      10. unpow2 32.0%

          \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{\ell} \cdot \left(-0.25\right)\right)\right)\right)}\right)}^{2} \]

      11. metadata-eval 32.0%

          \[\leadsto w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell} \cdot \color{blue}{-0.25}\right)\right)\right)}\right)}^{2} \]

   10. Simplified 32.0%

       \[\leadsto w0 \cdot {\color{blue}{\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell} \cdot -0.25\right)\right)\right)}\right)}}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+296}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\mathsf{fma}\left(-2, \log d, \log \left(\frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell} \cdot -0.25\right)\right)\right)}\right)}^{2}\\ \end{array} \]

Alternative 2: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} d = |d|\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.0001:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) 0.0001)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D 2.0) (/ M d)) 2.0)))))
   w0))

C

d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= 0.0001) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: d should be positive before calling this function
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
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= 0.0001d0) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d / 2.0d0) * (m / d_1)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

d = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= 0.0001) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

d = abs(d)
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= 0.0001:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D / 2.0) * (M / d)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

d = abs(d)
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 0.0001)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / 2.0) * Float64(M / d)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

d = abs(d)
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= 0.0001)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D / 2.0) * (M / d)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 0.0001], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 1.00000000000000005e-4

   1. Initial program 88.2%

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

   3. Simplified 89.0%

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

   if 1.00000000000000005e-4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.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%

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

   4. Taylor expanded in D around 0 73.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.0001:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} d = |d|\\ \\ w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}}} \end{array} \]

FPCore

NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (+ 1.0 (/ -1.0 (/ l (* (pow (* 0.5 (/ (* M D) d)) 2.0) h)))))))

C

d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 + (-1.0 / (l / (pow((0.5 * ((M * D) / d)), 2.0) * h)))));
}

Fortran

NOTE: d should be positive before calling this function
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 + ((-1.0d0) / (l / (((0.5d0 * ((m * d) / d_1)) ** 2.0d0) * h)))))
end function

Java

d = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 + (-1.0 / (l / (Math.pow((0.5 * ((M * D) / d)), 2.0) * h)))));
}

Python

d = abs(d)
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 + (-1.0 / (l / (math.pow((0.5 * ((M * D) / d)), 2.0) * h)))))

Julia

d = abs(d)
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 + Float64(-1.0 / Float64(l / Float64((Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0) * h))))))
end

MATLAB

d = abs(d)
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 + (-1.0 / (l / (((0.5 * ((M * D) / d)) ^ 2.0) * h)))));
end

Wolfram

NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 + N[(-1.0 / N[(l / N[(N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Simplified 81.7%

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

   1. associate-*r/ 87.6%

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

   2. frac-times 86.9%

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

   3. *-commutative 86.9%

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

   4. clear-num 86.9%

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

   5. unpow2 86.9%

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

   6. unpow2 86.9%

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

   7. *-un-lft-identity 86.9%

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

   8. times-frac 86.9%

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

   9. metadata-eval 86.9%

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

   10. *-commutative 86.9%

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

5. Applied egg-rr 86.9%

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

6. Final simplification 86.9%

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

Alternative 4: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} d = |d|\\ \\ w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}{\ell}} \end{array} \]

FPCore

NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (/ (* (pow (* 0.5 (/ (* M D) d)) 2.0) h) l)))))

C

d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((pow((0.5 * ((M * D) / d)), 2.0) * h) / l)));
}

Fortran

NOTE: d should be positive before calling this function
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 - ((((0.5d0 * ((m * d) / d_1)) ** 2.0d0) * h) / l)))
end function

Java

d = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - ((Math.pow((0.5 * ((M * D) / d)), 2.0) * h) / l)));
}

Python

d = abs(d)
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((math.pow((0.5 * ((M * D) / d)), 2.0) * h) / l)))

Julia

d = abs(d)
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0) * h) / l))))
end

MATLAB

d = abs(d)
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((0.5 * ((M * D) / d)) ^ 2.0) * h) / l)));
end

Wolfram

NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Simplified 81.7%

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

   1. associate-*r/ 87.6%

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

   2. frac-times 86.9%

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

   3. *-commutative 86.9%

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

   4. unpow2 86.9%

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

   5. unpow2 86.9%

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

   6. *-un-lft-identity 86.9%

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

   7. times-frac 86.9%

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

   8. metadata-eval 86.9%

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

   9. *-commutative 86.9%

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

5. Applied egg-rr 86.9%

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

6. Final simplification 86.9%

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

Alternative 5: 81.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} d = |d|\\ \\ w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \end{array} \]

FPCore

NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (+ 1.0 (* -0.125 (/ (* h (pow (* D (/ M d)) 2.0)) l)))))

C

d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((h * pow((D * (M / d)), 2.0)) / l)));
}

Fortran

NOTE: d should be positive before calling this function
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 * (1.0d0 + ((-0.125d0) * ((h * ((d * (m / d_1)) ** 2.0d0)) / l)))
end function

Java

d = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * (1.0 + (-0.125 * ((h * Math.pow((D * (M / d)), 2.0)) / l)));
}

Python

d = abs(d)
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + (-0.125 * ((h * math.pow((D * (M / d)), 2.0)) / l)))

Julia

d = abs(d)
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h * (Float64(D * Float64(M / d)) ^ 2.0)) / l))))
end

MATLAB

d = abs(d)
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + (-0.125 * ((h * ((D * (M / d)) ^ 2.0)) / l)));
end

Wolfram

NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Simplified 81.7%

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

4. Taylor expanded in D around 0 53.6%

   \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 42.7%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)\right)}\right) \]

   2. expm1-udef 42.7%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} - 1\right)}\right) \]

   3. associate-*r* 46.3%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)} - 1\right)\right) \]

   4. unpow-prod-down 53.9%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{{d}^{2} \cdot \ell}\right)} - 1\right)\right) \]

   5. *-commutative 53.9%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{\ell \cdot {d}^{2}}}\right)} - 1\right)\right) \]

6. Applied egg-rr 53.9%

   \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell \cdot {d}^{2}}\right)} - 1\right)}\right) \]
7. Step-by-step derivation

   1. expm1-def 53.9%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell \cdot {d}^{2}}\right)\right)}\right) \]

   2. expm1-log1p 66.8%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell \cdot {d}^{2}}}\right) \]

   3. *-commutative 66.8%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{h \cdot {\left(D \cdot M\right)}^{2}}}{\ell \cdot {d}^{2}}\right) \]

   4. times-frac 64.3%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}}\right)}\right) \]

   5. unpow2 64.3%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{{\left(D \cdot M\right)}^{2}}{\color{blue}{d \cdot d}}\right)\right) \]

   6. unpow2 64.3%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{d \cdot d}\right)\right) \]

   7. times-frac 73.7%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right)}\right)\right) \]

   8. associate-/l* 73.7%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\color{blue}{\frac{D}{\frac{d}{M}}} \cdot \frac{D \cdot M}{d}\right)\right)\right) \]

   9. associate-/l* 74.5%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \color{blue}{\frac{D}{\frac{d}{M}}}\right)\right)\right) \]

   10. unpow2 74.5%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}\right)\right) \]

   11. associate-/l* 73.7%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

   12. *-commutative 73.7%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{\color{blue}{M \cdot D}}{d}\right)}^{2}\right)\right) \]

   13. associate-/l* 73.7%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}\right)\right) \]

8. Simplified 73.7%

   \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}\right) \]
9. Step-by-step derivation

   1. associate-*l/ 79.3%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{h \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}}\right) \]

   2. associate-/r/ 80.1%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}}{\ell}\right) \]

10. Applied egg-rr 80.1%

    \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{h \cdot {\left(\frac{M}{d} \cdot D\right)}^{2}}{\ell}}\right) \]

11. Final simplification 80.1%

    \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right) \]

Alternative 6: 73.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} d = |d|\\ \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ \mathbf{if}\;M \cdot D \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(t_0 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) d)))
   (if (<= (* M D) 2.4e-22)
     w0
     (* w0 (+ 1.0 (* -0.125 (* t_0 (* (/ h l) t_0))))))))

C

d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / d;
	double tmp;
	if ((M * D) <= 2.4e-22) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (t_0 * ((h / l) * t_0))));
	}
	return tmp;
}

Fortran

NOTE: d should be positive before calling this function
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m * d) / d_1
    if ((m * d) <= 2.4d-22) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (t_0 * ((h / l) * t_0))))
    end if
    code = tmp
end function

Java

d = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / d;
	double tmp;
	if ((M * D) <= 2.4e-22) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (t_0 * ((h / l) * t_0))));
	}
	return tmp;
}

Python

d = abs(d)
def code(w0, M, D, h, l, d):
	t_0 = (M * D) / d
	tmp = 0
	if (M * D) <= 2.4e-22:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (t_0 * ((h / l) * t_0))))
	return tmp

Julia

d = abs(d)
function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / d)
	tmp = 0.0
	if (Float64(M * D) <= 2.4e-22)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(t_0 * Float64(Float64(h / l) * t_0)))));
	end
	return tmp
end

MATLAB

d = abs(d)
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (M * D) / d;
	tmp = 0.0;
	if ((M * D) <= 2.4e-22)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (t_0 * ((h / l) * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 2.4e-22], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.40000000000000002e-22

   1. Initial program 83.9%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in D around 0 74.0%

      \[\leadsto \color{blue}{w0} \]

   if 2.40000000000000002e-22 < (*.f64 M D)

   1. Initial program 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in D around 0 41.4%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 14.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)\right)}\right) \]

      2. expm1-udef 14.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} - 1\right)}\right) \]

      3. associate-*r* 12.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)} - 1\right)\right) \]

      4. unpow-prod-down 24.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{{d}^{2} \cdot \ell}\right)} - 1\right)\right) \]

      5. *-commutative 24.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{\ell \cdot {d}^{2}}}\right)} - 1\right)\right) \]

   6. Applied egg-rr 24.0%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell \cdot {d}^{2}}\right)} - 1\right)}\right) \]
   7. Step-by-step derivation

      1. expm1-def 24.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell \cdot {d}^{2}}\right)\right)}\right) \]

      2. expm1-log1p 55.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell \cdot {d}^{2}}}\right) \]

      3. *-commutative 55.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{h \cdot {\left(D \cdot M\right)}^{2}}}{\ell \cdot {d}^{2}}\right) \]

      4. times-frac 57.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}}\right)}\right) \]

      5. unpow2 57.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{{\left(D \cdot M\right)}^{2}}{\color{blue}{d \cdot d}}\right)\right) \]

      6. unpow2 57.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{d \cdot d}\right)\right) \]

      7. times-frac 63.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right)}\right)\right) \]

      8. associate-/l* 63.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\color{blue}{\frac{D}{\frac{d}{M}}} \cdot \frac{D \cdot M}{d}\right)\right)\right) \]

      9. associate-/l* 63.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \color{blue}{\frac{D}{\frac{d}{M}}}\right)\right)\right) \]

      10. unpow2 63.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}\right)\right) \]

      11. associate-/l* 63.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

      12. *-commutative 63.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{\color{blue}{M \cdot D}}{d}\right)}^{2}\right)\right) \]

      13. associate-/l* 63.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}\right)\right) \]

   8. Simplified 63.5%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*l/ 65.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{h \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}}{\ell}}\right) \]

      2. associate-/r/ 65.6%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}}{\ell}\right) \]

   10. Applied egg-rr 65.6%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{h \cdot {\left(\frac{M}{d} \cdot D\right)}^{2}}{\ell}}\right) \]
   11. Step-by-step derivation

       1. pow2 65.6%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{h \cdot \color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)}}{\ell}\right) \]

       2. associate-*l/ 63.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)\right)}\right) \]

       3. associate-*r* 67.8%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)}\right) \]

       4. associate-*l/ 65.8%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{h}{\ell} \cdot \color{blue}{\frac{M \cdot D}{d}}\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)\right) \]

       5. associate-*l/ 65.8%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d}\right) \cdot \color{blue}{\frac{M \cdot D}{d}}\right)\right) \]

   12. Applied egg-rr 65.8%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d}\right) \cdot \frac{M \cdot D}{d}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{M \cdot D}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d}\right)\right)\right)\\ \end{array} \]

Alternative 7: 73.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} d = |d|\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-22}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(t_0 \cdot t_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* D (/ M d))))
   (if (<= (* M D) 2e-22)
     w0
     (* w0 (+ 1.0 (* -0.125 (* (/ h l) (* t_0 t_0))))))))

C

d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D * (M / d);
	double tmp;
	if ((M * D) <= 2e-22) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))));
	}
	return tmp;
}

Fortran

NOTE: d should be positive before calling this function
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d * (m / d_1)
    if ((m * d) <= 2d-22) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((h / l) * (t_0 * t_0))))
    end if
    code = tmp
end function

Java

d = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D * (M / d);
	double tmp;
	if ((M * D) <= 2e-22) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))));
	}
	return tmp;
}

Python

d = abs(d)
def code(w0, M, D, h, l, d):
	t_0 = D * (M / d)
	tmp = 0
	if (M * D) <= 2e-22:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))))
	return tmp

Julia

d = abs(d)
function code(w0, M, D, h, l, d)
	t_0 = Float64(D * Float64(M / d))
	tmp = 0.0
	if (Float64(M * D) <= 2e-22)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * Float64(t_0 * t_0)))));
	end
	return tmp
end

MATLAB

d = abs(d)
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = D * (M / d);
	tmp = 0.0;
	if ((M * D) <= 2e-22)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 2e-22], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.0000000000000001e-22

   1. Initial program 83.9%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in D around 0 74.0%

      \[\leadsto \color{blue}{w0} \]

   if 2.0000000000000001e-22 < (*.f64 M D)

   1. Initial program 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in D around 0 41.4%

      \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 14.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)\right)}\right) \]

      2. expm1-udef 14.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} - 1\right)}\right) \]

      3. associate-*r* 12.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}\right)} - 1\right)\right) \]

      4. unpow-prod-down 24.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h}{{d}^{2} \cdot \ell}\right)} - 1\right)\right) \]

      5. *-commutative 24.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\color{blue}{\ell \cdot {d}^{2}}}\right)} - 1\right)\right) \]

   6. Applied egg-rr 24.0%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell \cdot {d}^{2}}\right)} - 1\right)}\right) \]
   7. Step-by-step derivation

      1. expm1-def 24.0%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell \cdot {d}^{2}}\right)\right)}\right) \]

      2. expm1-log1p 55.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\frac{{\left(D \cdot M\right)}^{2} \cdot h}{\ell \cdot {d}^{2}}}\right) \]

      3. *-commutative 55.4%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \frac{\color{blue}{h \cdot {\left(D \cdot M\right)}^{2}}}{\ell \cdot {d}^{2}}\right) \]

      4. times-frac 57.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{{\left(D \cdot M\right)}^{2}}{{d}^{2}}\right)}\right) \]

      5. unpow2 57.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{{\left(D \cdot M\right)}^{2}}{\color{blue}{d \cdot d}}\right)\right) \]

      6. unpow2 57.2%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{d \cdot d}\right)\right) \]

      7. times-frac 63.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right)}\right)\right) \]

      8. associate-/l* 63.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\color{blue}{\frac{D}{\frac{d}{M}}} \cdot \frac{D \cdot M}{d}\right)\right)\right) \]

      9. associate-/l* 63.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \color{blue}{\frac{D}{\frac{d}{M}}}\right)\right)\right) \]

      10. unpow2 63.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}\right)\right) \]

      11. associate-/l* 63.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)\right) \]

      12. *-commutative 63.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{\color{blue}{M \cdot D}}{d}\right)}^{2}\right)\right) \]

      13. associate-/l* 63.5%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\color{blue}{\left(\frac{M}{\frac{d}{D}}\right)}}^{2}\right)\right) \]

   8. Simplified 63.5%

      \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(\frac{M}{\frac{d}{D}}\right)}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 63.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}\right)}\right)\right) \]

      2. associate-/r/ 63.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \frac{M}{\frac{d}{D}}\right)\right)\right) \]

      3. associate-/r/ 63.5%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(\frac{M}{d} \cdot D\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right)\right) \]

   10. Applied egg-rr 63.5%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-22}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 68.4% accurate, 216.0× speedup

Math

\[\begin{array}{l} d = |d|\\ \\ w0 \end{array} \]

FPCore

NOTE: d should be positive before calling this function
(FPCore (w0 M D h l d) :precision binary64 w0)

C

d = abs(d);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Fortran

NOTE: d should be positive before calling this function
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
end function

Java

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

Python

d = abs(d)
def code(w0, M, D, h, l, d):
	return w0

Julia

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

MATLAB

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

Wolfram

NOTE: d should be positive before calling this function
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 81.0%

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

3. Simplified 81.7%

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

4. Taylor expanded in D around 0 67.4%

   \[\leadsto \color{blue}{w0} \]

5. Final simplification 67.4%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023301 
(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))))))