Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 86.1%

Time: 23.4s

Alternatives: 6

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: 81.0% 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: 86.1% accurate, 0.3× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) (- INFINITY))
   (*
    w0
    (exp
     (*
      (-
       (+ (* 2.0 (log D_m)) (log (* -0.25 (/ (/ h (pow d 2.0)) l))))
       (* -2.0 (log M_m)))
      0.5)))
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M_m (* D_m (/ 0.5 d))) 2.0)) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = w0 * exp(((((2.0 * log(D_m)) + log((-0.25 * ((h / pow(d, 2.0)) / l)))) - (-2.0 * log(M_m))) * 0.5));
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((M_m * (D_m * (0.5 / d))), 2.0)) / l)));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.exp(((((2.0 * Math.log(D_m)) + Math.log((-0.25 * ((h / Math.pow(d, 2.0)) / l)))) - (-2.0 * Math.log(M_m))) * 0.5));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M_m * (D_m * (0.5 / d))), 2.0)) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -math.inf:
		tmp = w0 * math.exp(((((2.0 * math.log(D_m)) + math.log((-0.25 * ((h / math.pow(d, 2.0)) / l)))) - (-2.0 * math.log(M_m))) * 0.5))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M_m * (D_m * (0.5 / d))), 2.0)) / l)))
	return tmp

Julia

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

MATLAB

M_m = abs(M);
D_m = abs(D);
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -Inf)
		tmp = w0 * exp(((((2.0 * log(D_m)) + log((-0.25 * ((h / (d ^ 2.0)) / l)))) - (-2.0 * log(M_m))) * 0.5));
	else
		tmp = w0 * sqrt((1.0 - ((h * ((M_m * (D_m * (0.5 / d))) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[Exp[N[(N[(N[(N[(2.0 * N[Log[D$95$m], $MachinePrecision]), $MachinePrecision] + N[Log[N[(-0.25 * N[(N[(h / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[Log[M$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

      1. pow1/2 53.7%

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

      2. *-commutative 53.7%

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

      3. frac-times 53.7%

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

      4. *-commutative 53.7%

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

      5. *-commutative 53.7%

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

      6. pow-to-exp 53.7%

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

   6. Applied egg-rr 53.7%

      \[\leadsto w0 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left(-{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \cdot 0.5}} \]

   7. Taylor expanded in M around inf 29.9%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{M}\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. associate-/r* 30.5%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \color{blue}{\frac{\frac{{D}^{2} \cdot h}{{d}^{2}}}{\ell}}\right) + -2 \cdot \log \left(\frac{1}{M}\right)\right) \cdot 0.5} \]

      2. log-rec 30.5%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{\frac{{D}^{2} \cdot h}{{d}^{2}}}{\ell}\right) + -2 \cdot \color{blue}{\left(-\log M\right)}\right) \cdot 0.5} \]

   9. Simplified 30.5%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{\frac{{D}^{2} \cdot h}{{d}^{2}}}{\ell}\right) + -2 \cdot \left(-\log M\right)\right)} \cdot 0.5} \]

   10. Taylor expanded in D around 0 19.6%

       \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(\log \left(-0.25 \cdot \frac{h}{{d}^{2} \cdot \ell}\right) + 2 \cdot \log D\right)} + -2 \cdot \left(-\log M\right)\right) \cdot 0.5} \]
   11. Step-by-step derivation

       1. +-commutative 19.6%

          \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(2 \cdot \log D + \log \left(-0.25 \cdot \frac{h}{{d}^{2} \cdot \ell}\right)\right)} + -2 \cdot \left(-\log M\right)\right) \cdot 0.5} \]

       2. associate-/r* 19.6%

          \[\leadsto w0 \cdot e^{\left(\left(2 \cdot \log D + \log \left(-0.25 \cdot \color{blue}{\frac{\frac{h}{{d}^{2}}}{\ell}}\right)\right) + -2 \cdot \left(-\log M\right)\right) \cdot 0.5} \]

   12. Simplified 19.6%

       \[\leadsto w0 \cdot e^{\left(\color{blue}{\left(2 \cdot \log D + \log \left(-0.25 \cdot \frac{\frac{h}{{d}^{2}}}{\ell}\right)\right)} + -2 \cdot \left(-\log M\right)\right) \cdot 0.5} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 87.3%

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

   3. Simplified 87.8%

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

      1. *-commutative 87.8%

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

      2. frac-times 87.3%

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

      3. *-commutative 87.3%

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

      4. associate-*l/ 97.3%

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

      5. div-inv 97.3%

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

      6. associate-*l* 97.2%

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

      7. associate-/r* 97.2%

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

      8. metadata-eval 97.2%

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

   6. Applied egg-rr 97.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

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

Alternative 2: 86.1% accurate, 0.3× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) (- INFINITY))
   (*
    w0
    (exp
     (*
      0.5
      (+
       (log (* -0.25 (/ h (* l (pow d 2.0)))))
       (+ (* 2.0 (log D_m)) (* 2.0 (log M_m)))))))
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M_m (* D_m (/ 0.5 d))) 2.0)) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -((double) INFINITY)) {
		tmp = w0 * exp((0.5 * (log((-0.25 * (h / (l * pow(d, 2.0))))) + ((2.0 * log(D_m)) + (2.0 * log(M_m))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((M_m * (D_m * (0.5 / d))), 2.0)) / l)));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.exp((0.5 * (Math.log((-0.25 * (h / (l * Math.pow(d, 2.0))))) + ((2.0 * Math.log(D_m)) + (2.0 * Math.log(M_m))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M_m * (D_m * (0.5 / d))), 2.0)) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -math.inf:
		tmp = w0 * math.exp((0.5 * (math.log((-0.25 * (h / (l * math.pow(d, 2.0))))) + ((2.0 * math.log(D_m)) + (2.0 * math.log(M_m))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M_m * (D_m * (0.5 / d))), 2.0)) / l)))
	return tmp

Julia

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

MATLAB

M_m = abs(M);
D_m = abs(D);
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -Inf)
		tmp = w0 * exp((0.5 * (log((-0.25 * (h / (l * (d ^ 2.0))))) + ((2.0 * log(D_m)) + (2.0 * log(M_m))))));
	else
		tmp = w0 * sqrt((1.0 - ((h * ((M_m * (D_m * (0.5 / d))) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(w0 * N[Exp[N[(0.5 * N[(N[Log[N[(-0.25 * N[(h / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(2.0 * N[Log[D$95$m], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[M$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

      1. pow1/2 53.7%

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

      2. *-commutative 53.7%

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

      3. frac-times 53.7%

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

      4. *-commutative 53.7%

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

      5. *-commutative 53.7%

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

      6. pow-to-exp 53.7%

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

   6. Applied egg-rr 53.7%

      \[\leadsto w0 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{h}{\ell} \cdot \left(-{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right) \cdot 0.5}} \]

   7. Taylor expanded in M around inf 29.9%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) + -2 \cdot \log \left(\frac{1}{M}\right)\right)} \cdot 0.5} \]
   8. Step-by-step derivation

      1. associate-/r* 30.5%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \color{blue}{\frac{\frac{{D}^{2} \cdot h}{{d}^{2}}}{\ell}}\right) + -2 \cdot \log \left(\frac{1}{M}\right)\right) \cdot 0.5} \]

      2. log-rec 30.5%

         \[\leadsto w0 \cdot e^{\left(\log \left(-0.25 \cdot \frac{\frac{{D}^{2} \cdot h}{{d}^{2}}}{\ell}\right) + -2 \cdot \color{blue}{\left(-\log M\right)}\right) \cdot 0.5} \]

   9. Simplified 30.5%

      \[\leadsto w0 \cdot e^{\color{blue}{\left(\log \left(-0.25 \cdot \frac{\frac{{D}^{2} \cdot h}{{d}^{2}}}{\ell}\right) + -2 \cdot \left(-\log M\right)\right)} \cdot 0.5} \]

   10. Taylor expanded in D around 0 19.6%

       \[\leadsto \color{blue}{w0 \cdot e^{0.5 \cdot \left(\log \left(-0.25 \cdot \frac{h}{{d}^{2} \cdot \ell}\right) + \left(2 \cdot \log D + 2 \cdot \log M\right)\right)}} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 87.3%

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

   3. Simplified 87.8%

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

      1. *-commutative 87.8%

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

      2. frac-times 87.3%

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

      3. *-commutative 87.3%

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

      4. associate-*l/ 97.3%

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

      5. div-inv 97.3%

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

      6. associate-*l* 97.2%

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

      7. associate-/r* 97.2%

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

      8. metadata-eval 97.2%

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

   6. Applied egg-rr 97.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

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

Alternative 3: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M_m \cdot D_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D_m}{2} \cdot \frac{M_m}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l)) -2e-11)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D_m 2.0) (/ M_m d)) 2.0)))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e-11) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D_m / 2.0) * (M_m / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l)) <= (-2d-11)) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d_m / 2.0d0) * (m_m / d)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e-11) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D_m / 2.0) * (M_m / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l)) <= -2e-11:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D_m / 2.0) * (M_m / d)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-11)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D_m / 2.0) * Float64(M_m / d)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e-11)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D_m / 2.0) * (M_m / d)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-11], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D$95$m / 2.0), $MachinePrecision] * N[(M$95$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)) < -1.99999999999999988e-11

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   if -1.99999999999999988e-11 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 86.4%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in D around 0 97.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-11}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (/ (* h (pow (* M_m (* D_m (/ 0.5 d))) 2.0)) l)))))

C

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

Fortran

M_m = abs(M)
D_m = abs(D)
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    code = w0 * sqrt((1.0d0 - ((h * ((m_m * (d_m * (0.5d0 / d))) ** 2.0d0)) / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

3. Simplified 79.9%

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

   1. *-commutative 79.9%

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

   2. frac-times 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l/ 87.6%

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

   5. div-inv 87.6%

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

   6. associate-*l* 87.6%

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

   7. associate-/r* 87.6%

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

   8. metadata-eval 87.6%

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

6. Applied egg-rr 87.6%

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

7. Final simplification 87.6%

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

Alternative 5: 70.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ \begin{array}{l} \mathbf{if}\;M_m \leq 1.65 \cdot 10^{-35}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{\left(M_m \cdot D_m\right) \cdot \left(M_m \cdot D_m\right)}{\frac{{d}^{2}}{h}} \cdot \frac{w0}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 1.65e-35)
   w0
   (+
    w0
    (*
     -0.125
     (* (/ (* (* M_m D_m) (* M_m D_m)) (/ (pow d 2.0) h)) (/ w0 l))))))

C

M_m = fabs(M);
D_m = fabs(D);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.65e-35) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((((M_m * D_m) * (M_m * D_m)) / (pow(d, 2.0) / h)) * (w0 / l)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (m_m <= 1.65d-35) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((((m_m * d_m) * (m_m * d_m)) / ((d ** 2.0d0) / h)) * (w0 / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.65e-35) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((((M_m * D_m) * (M_m * D_m)) / (Math.pow(d, 2.0) / h)) * (w0 / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 1.65e-35:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((((M_m * D_m) * (M_m * D_m)) / (math.pow(d, 2.0) / h)) * (w0 / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 1.65e-35)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) / Float64((d ^ 2.0) / h)) * Float64(w0 / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 1.65e-35)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((((M_m * D_m) * (M_m * D_m)) / ((d ^ 2.0) / h)) * (w0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 1.65e-35], w0, N[(w0 + N[(-0.125 * N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Power[d, 2.0], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision] * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.65e-35

   1. Initial program 79.5%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in D around 0 76.2%

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

   if 1.65e-35 < M

   1. Initial program 79.7%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in D around 0 48.5%

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

      1. expm1-log1p-u 39.8%

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

      2. expm1-udef 38.0%

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

      3. associate-*r* 39.7%

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

      4. pow-prod-down 44.7%

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

      5. *-commutative 44.7%

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

   7. Applied egg-rr 44.7%

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

      1. expm1-def 46.4%

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

      2. expm1-log1p 56.9%

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

      3. associate-*r* 60.2%

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

      4. *-commutative 60.2%

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

   9. Simplified 60.2%

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

       1. times-frac 56.9%

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

   11. Applied egg-rr 56.9%

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

       1. associate-/l* 58.5%

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

   13. Simplified 58.5%

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

       1. unpow2 58.5%

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

   15. Applied egg-rr 58.5%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.65 \cdot 10^{-35}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{{d}^{2}}{h}} \cdot \frac{w0}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(M)
D_m = abs(D)
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.9%

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

5. Taylor expanded in D around 0 71.8%

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

6. Final simplification 71.8%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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))))))