Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 88.1%

Time: 19.2s

Alternatives: 7

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: 81.4% 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: 88.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ \begin{array}{l} t_0 := \left(M\_m \cdot 0.5\right) \cdot \frac{D\_m}{d\_m}\\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 10^{+140}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{D\_m \cdot \frac{M\_m}{2}}{d\_m}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{{\left(\sqrt[3]{\log \left(w0\_m \cdot \sqrt{1 - h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)}\right)}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (* (* M_m 0.5) (/ D_m d_m))))
   (*
    w0_s
    (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 1e+140)
      (* w0_m (sqrt (- 1.0 (* h (/ (pow (/ (* D_m (/ M_m 2.0)) d_m) 2.0) l)))))
      (exp
       (pow
        (cbrt (log (* w0_m (sqrt (- 1.0 (* h (* t_0 (/ t_0 l))))))))
        3.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * 0.5) * (D_m / d_m);
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+140) {
		tmp = w0_m * sqrt((1.0 - (h * (pow(((D_m * (M_m / 2.0)) / d_m), 2.0) / l))));
	} else {
		tmp = exp(pow(cbrt(log((w0_m * sqrt((1.0 - (h * (t_0 * (t_0 / l)))))))), 3.0));
	}
	return w0_s * tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * 0.5) * (D_m / d_m);
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 1e+140) {
		tmp = w0_m * Math.sqrt((1.0 - (h * (Math.pow(((D_m * (M_m / 2.0)) / d_m), 2.0) / l))));
	} else {
		tmp = Math.exp(Math.pow(Math.cbrt(Math.log((w0_m * Math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))))))), 3.0));
	}
	return w0_s * tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * 0.5) * Float64(D_m / d_m))
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 1e+140)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(h * Float64((Float64(Float64(D_m * Float64(M_m / 2.0)) / d_m) ^ 2.0) / l)))));
	else
		tmp = exp((cbrt(log(Float64(w0_m * sqrt(Float64(1.0 - Float64(h * Float64(t_0 * Float64(t_0 / l)))))))) ^ 3.0));
	end
	return Float64(w0_s * tmp)
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]}, N[(w0$95$s * If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 1e+140], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(N[(D$95$m * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[Power[N[Power[N[Log[N[(w0$95$m * N[Sqrt[N[(1.0 - N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1.00000000000000006e140

   1. Initial program 84.1%

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

   3. Simplified 83.7%

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

      1. clear-num 83.2%

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

      2. un-div-inv 84.1%

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

      3. clear-num 84.1%

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

      4. un-div-inv 84.1%

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

      5. associate-/r/ 84.1%

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

   6. Applied egg-rr 84.1%

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

      1. associate-/r/ 90.1%

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

      2. associate-*l/ 89.3%

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

      3. *-commutative 89.3%

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

      4. associate-/l* 90.1%

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

      5. associate-*l/ 90.1%

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

      6. *-commutative 90.1%

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

      7. associate-*l/ 90.1%

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

      8. associate-/l/ 89.7%

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

      9. associate-/r/ 89.7%

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

      10. associate-*l/ 89.7%

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

      11. *-commutative 89.7%

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

      12. associate-*r/ 90.6%

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

      13. *-commutative 90.6%

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

      14. associate-/l/ 90.6%

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

      15. associate-/l* 90.1%

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

   8. Simplified 90.1%

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

   9. Taylor expanded in D around 0 90.6%

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

       1. metadata-eval 90.6%

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

       2. metadata-eval 90.6%

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

       3. associate-*l/ 89.7%

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

       4. associate-/r/ 90.1%

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

       5. times-frac 90.1%

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

       6. distribute-lft-neg-in 90.1%

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

       7. neg-mul-1 90.1%

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

       8. times-frac 90.1%

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

       9. metadata-eval 90.1%

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

       10. associate-/l/ 90.1%

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

       11. associate-/r/ 89.7%

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

       12. associate-*l/ 90.6%

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

       13. associate-/l* 90.1%

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

       14. associate-*l/ 90.1%

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

       15. times-frac 90.6%

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

       16. associate-*r/ 90.1%

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

       17. *-lft-identity 90.1%

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

       18. associate-/r* 90.1%

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

       19. associate-*r/ 90.6%

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

   11. Simplified 90.6%

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

   if 1.00000000000000006e140 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 41.3%

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

   3. Simplified 45.6%

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

      1. unpow2 45.6%

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

      2. unpow2 45.6%

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

      3. associate-/r* 45.6%

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

      4. associate-*r/ 41.3%

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

      5. *-commutative 41.3%

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

      6. associate-*r/ 41.6%

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

      7. clear-num 41.6%

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

      8. *-commutative 41.6%

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

      9. associate-*l/ 45.9%

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

      10. associate-/r* 45.9%

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

      11. clear-num 45.9%

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

      12. associate-*l/ 45.9%

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

      13. *-un-lft-identity 45.9%

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

      14. associate-/r/ 45.9%

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

   6. Applied egg-rr 45.9%

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

      1. associate-/r/ 45.9%

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

      2. associate-*l/ 45.9%

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

      3. *-commutative 45.9%

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

      4. associate-*l/ 45.9%

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

      5. associate-/l/ 45.9%

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

      6. associate-/r/ 45.9%

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

   8. Simplified 45.9%

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

      1. add-exp-log 19.3%

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

      2. associate-*l/ 19.3%

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

      3. *-un-lft-identity 19.3%

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

      4. *-commutative 19.3%

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

      5. associate-/l/ 19.3%

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

   10. Applied egg-rr 19.3%

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

       1. add-cube-cbrt 19.3%

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

       2. pow3 19.3%

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

       3. associate-/l* 19.3%

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

       4. associate-*r/ 19.3%

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

   12. Applied egg-rr 19.3%

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

       1. unpow2 19.3%

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

       2. *-un-lft-identity 19.3%

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

       3. times-frac 19.3%

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

       4. times-frac 19.3%

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

       5. div-inv 19.3%

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

       6. metadata-eval 19.3%

          \[\leadsto e^{{\left(\sqrt[3]{\log \left(w0 \cdot \sqrt{1 - h \cdot \left(\frac{\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}}{1} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right)}\right)}\right)}^{3}} \]

       7. times-frac 26.5%

          \[\leadsto e^{{\left(\sqrt[3]{\log \left(w0 \cdot \sqrt{1 - h \cdot \left(\frac{\left(M \cdot 0.5\right) \cdot \frac{D}{d}}{1} \cdot \frac{\color{blue}{\frac{M}{2} \cdot \frac{D}{d}}}{\ell}\right)}\right)}\right)}^{3}} \]

       8. div-inv 26.5%

          \[\leadsto e^{{\left(\sqrt[3]{\log \left(w0 \cdot \sqrt{1 - h \cdot \left(\frac{\left(M \cdot 0.5\right) \cdot \frac{D}{d}}{1} \cdot \frac{\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}}{\ell}\right)}\right)}\right)}^{3}} \]

       9. metadata-eval 26.5%

          \[\leadsto e^{{\left(\sqrt[3]{\log \left(w0 \cdot \sqrt{1 - h \cdot \left(\frac{\left(M \cdot 0.5\right) \cdot \frac{D}{d}}{1} \cdot \frac{\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}}{\ell}\right)}\right)}\right)}^{3}} \]

   14. Applied egg-rr 26.5%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{+140}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{D \cdot \frac{M}{2}}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{{\left(\sqrt[3]{\log \left(w0 \cdot \sqrt{1 - h \cdot \left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \frac{\left(M \cdot 0.5\right) \cdot \frac{D}{d}}{\ell}\right)}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-289}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - h \cdot \frac{{\left(D\_m \cdot \frac{M\_m}{2 \cdot d\_m}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\_m \cdot \sqrt{1 - h \cdot {\left(D\_m \cdot \left(\frac{M\_m}{\sqrt{\ell}} \cdot \frac{0.5}{d\_m}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= l 4e-289)
    (* w0_m (sqrt (- 1.0 (* h (/ (pow (* D_m (/ M_m (* 2.0 d_m))) 2.0) l)))))
    (*
     w0_m
     (sqrt
      (- 1.0 (* h (pow (* D_m (* (/ M_m (sqrt l)) (/ 0.5 d_m))) 2.0))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (l <= 4e-289) {
		tmp = w0_m * sqrt((1.0 - (h * (pow((D_m * (M_m / (2.0 * d_m))), 2.0) / l))));
	} else {
		tmp = w0_m * sqrt((1.0 - (h * pow((D_m * ((M_m / sqrt(l)) * (0.5 / d_m))), 2.0))));
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (l <= 4d-289) then
        tmp = w0_m * sqrt((1.0d0 - (h * (((d_m * (m_m / (2.0d0 * d_m_1))) ** 2.0d0) / l))))
    else
        tmp = w0_m * sqrt((1.0d0 - (h * ((d_m * ((m_m / sqrt(l)) * (0.5d0 / d_m_1))) ** 2.0d0))))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (l <= 4e-289) {
		tmp = w0_m * Math.sqrt((1.0 - (h * (Math.pow((D_m * (M_m / (2.0 * d_m))), 2.0) / l))));
	} else {
		tmp = w0_m * Math.sqrt((1.0 - (h * Math.pow((D_m * ((M_m / Math.sqrt(l)) * (0.5 / d_m))), 2.0))));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if l <= 4e-289:
		tmp = w0_m * math.sqrt((1.0 - (h * (math.pow((D_m * (M_m / (2.0 * d_m))), 2.0) / l))))
	else:
		tmp = w0_m * math.sqrt((1.0 - (h * math.pow((D_m * ((M_m / math.sqrt(l)) * (0.5 / d_m))), 2.0))))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (l <= 4e-289)
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D_m * Float64(M_m / Float64(2.0 * d_m))) ^ 2.0) / l)))));
	else
		tmp = Float64(w0_m * sqrt(Float64(1.0 - Float64(h * (Float64(D_m * Float64(Float64(M_m / sqrt(l)) * Float64(0.5 / d_m))) ^ 2.0)))));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (l <= 4e-289)
		tmp = w0_m * sqrt((1.0 - (h * (((D_m * (M_m / (2.0 * d_m))) ^ 2.0) / l))));
	else
		tmp = w0_m * sqrt((1.0 - (h * ((D_m * ((M_m / sqrt(l)) * (0.5 / d_m))) ^ 2.0))));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[l, 4e-289], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0$95$m * N[Sqrt[N[(1.0 - N[(h * N[Power[N[(D$95$m * N[(N[(M$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4e-289

   1. Initial program 76.7%

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

   3. Simplified 76.7%

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

      1. clear-num 76.7%

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

      2. un-div-inv 77.6%

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

      3. clear-num 77.5%

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

      4. un-div-inv 77.5%

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

      5. associate-/r/ 77.5%

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

   6. Applied egg-rr 77.5%

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

      1. associate-/r/ 83.5%

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

      2. associate-*l/ 83.5%

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

      3. *-commutative 83.5%

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

      4. associate-/l* 83.5%

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

      5. associate-*l/ 83.5%

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

      6. *-commutative 83.5%

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

      7. associate-*l/ 83.5%

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

      8. associate-/l/ 83.6%

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

      9. associate-/r/ 83.6%

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

      10. associate-*l/ 83.6%

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

      11. *-commutative 83.6%

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

      12. associate-*r/ 83.6%

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

      13. *-commutative 83.6%

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

      14. associate-/l/ 83.6%

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

      15. associate-/l* 83.5%

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

   8. Simplified 83.5%

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

   if 4e-289 < l

   1. Initial program 83.2%

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

   3. Simplified 83.2%

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

      1. clear-num 82.5%

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

      2. un-div-inv 83.2%

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

      3. clear-num 83.2%

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

      4. un-div-inv 83.2%

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

      5. associate-/r/ 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. associate-/r/ 88.4%

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

      2. associate-*l/ 87.0%

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

      3. *-commutative 87.0%

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

      4. associate-/l* 88.4%

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

      5. associate-*l/ 88.4%

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

      6. *-commutative 88.4%

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

      7. associate-*l/ 88.4%

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

      8. associate-/l/ 87.7%

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

      9. associate-/r/ 87.7%

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

      10. associate-*l/ 87.7%

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

      11. *-commutative 87.7%

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

      12. associate-*r/ 88.4%

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

      13. *-commutative 88.4%

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

      14. associate-/l/ 88.4%

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

      15. associate-/l* 88.4%

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

   8. Simplified 88.4%

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

   9. Taylor expanded in D around 0 88.4%

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

       1. metadata-eval 88.4%

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

       2. metadata-eval 88.4%

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

       3. associate-*l/ 87.7%

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

       4. associate-/r/ 88.4%

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

       5. times-frac 88.4%

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

       6. distribute-lft-neg-in 88.4%

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

       7. neg-mul-1 88.4%

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

       8. times-frac 88.4%

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

       9. metadata-eval 88.4%

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

       10. associate-/l/ 88.4%

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

       11. associate-/r/ 87.7%

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

       12. associate-*l/ 88.4%

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

       13. associate-/l* 88.4%

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

       14. associate-*l/ 88.4%

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

       15. times-frac 88.4%

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

       16. associate-*r/ 88.4%

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

       17. *-lft-identity 88.4%

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

       18. associate-/r* 88.4%

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

       19. associate-*r/ 88.4%

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

   11. Simplified 88.4%

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

       1. add-sqr-sqrt 88.4%

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

       2. sqrt-div 88.4%

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

       3. sqrt-pow1 76.6%

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

       4. metadata-eval 76.6%

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

       5. pow1 76.6%

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

       6. associate-/l* 76.6%

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

       7. div-inv 76.6%

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

       8. metadata-eval 76.6%

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

       9. sqrt-div 76.6%

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

       10. sqrt-pow1 90.4%

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

       11. metadata-eval 90.4%

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

       12. pow1 90.4%

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

       13. associate-/l* 91.7%

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

       14. div-inv 91.7%

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

       15. metadata-eval 91.7%

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

   13. Applied egg-rr 91.7%

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

       1. unpow2 91.7%

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

       2. associate-/l* 91.7%

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

   15. Simplified 91.7%

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

       1. associate-/l/ 91.1%

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

       2. times-frac 91.0%

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

   17. Applied egg-rr 91.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-289}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot {\left(D \cdot \left(\frac{M}{\sqrt{\ell}} \cdot \frac{0.5}{d}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (* w0_m (sqrt (- 1.0 (* h (/ (pow (* D_m (/ M_m (* 2.0 d_m))) 2.0) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * (w0_m * sqrt((1.0 - (h * (pow((D_m * (M_m / (2.0 * d_m))), 2.0) / l)))));
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0_s * (w0_m * sqrt((1.0d0 - (h * (((d_m * (m_m / (2.0d0 * d_m_1))) ** 2.0d0) / l)))))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * (w0_m * Math.sqrt((1.0 - (h * (Math.pow((D_m * (M_m / (2.0 * d_m))), 2.0) / l)))));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	return w0_s * (w0_m * math.sqrt((1.0 - (h * (math.pow((D_m * (M_m / (2.0 * d_m))), 2.0) / l)))))

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	return Float64(w0_s * Float64(w0_m * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D_m * Float64(M_m / Float64(2.0 * d_m))) ^ 2.0) / l))))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
function tmp = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = w0_s * (w0_m * sqrt((1.0 - (h * (((D_m * (M_m / (2.0 * d_m))) ^ 2.0) / l)))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * N[(w0$95$m * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D$95$m * N[(M$95$m / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

3. Simplified 80.2%

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

   1. clear-num 79.9%

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

   2. un-div-inv 80.6%

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

   3. clear-num 80.6%

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

   4. un-div-inv 80.6%

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

   5. associate-/r/ 80.6%

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

6. Applied egg-rr 80.6%

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

   1. associate-/r/ 86.2%

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

   2. associate-*l/ 85.4%

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

   3. *-commutative 85.4%

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

   4. associate-/l* 86.2%

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

   5. associate-*l/ 86.2%

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

   6. *-commutative 86.2%

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

   7. associate-*l/ 86.2%

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

   8. associate-/l/ 85.8%

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

   9. associate-/r/ 85.8%

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

   10. associate-*l/ 85.8%

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

   11. *-commutative 85.8%

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

   12. associate-*r/ 86.2%

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

   13. *-commutative 86.2%

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

   14. associate-/l/ 86.2%

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

   15. associate-/l* 86.2%

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

8. Simplified 86.2%

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

9. Final simplification 86.2%

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

Alternative 4: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;M\_m \leq 4.2 \cdot 10^{+79}:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0\_m}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (* w0_s (if (<= M_m 4.2e+79) w0_m (log (exp w0_m)))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 4.2e+79) {
		tmp = w0_m;
	} else {
		tmp = log(exp(w0_m));
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 4.2d+79) then
        tmp = w0_m
    else
        tmp = log(exp(w0_m))
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 4.2e+79) {
		tmp = w0_m;
	} else {
		tmp = Math.log(Math.exp(w0_m));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 4.2e+79:
		tmp = w0_m
	else:
		tmp = math.log(math.exp(w0_m))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 4.2e+79)
		tmp = w0_m;
	else
		tmp = log(exp(w0_m));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 4.2e+79)
		tmp = w0_m;
	else
		tmp = log(exp(w0_m));
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[M$95$m, 4.2e+79], w0$95$m, N[Log[N[Exp[w0$95$m], $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if M < 4.20000000000000016e79

   1. Initial program 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in D around 0 74.3%

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

   if 4.20000000000000016e79 < M

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

      1. add-sqr-sqrt 33.8%

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

      2. sqrt-unprod 29.1%

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

      3. *-commutative 29.1%

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

      4. *-commutative 29.1%

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

      5. swap-sqr 28.7%

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

   6. Applied egg-rr 28.7%

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

   7. Taylor expanded in h around 0 23.8%

      \[\leadsto \sqrt{\color{blue}{{w0}^{2}}} \]
   8. Step-by-step derivation

      1. sqrt-pow1 31.4%

         \[\leadsto \color{blue}{{w0}^{\left(\frac{2}{2}\right)}} \]

      2. metadata-eval 31.4%

         \[\leadsto {w0}^{\color{blue}{1}} \]

      3. pow1 31.4%

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

      4. add-log-exp 30.7%

         \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]

   9. Applied egg-rr 30.7%

      \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.2 \cdot 10^{+79}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;D\_m \leq 7.5 \cdot 10^{+171}:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(w0\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (* w0_s (if (<= D_m 7.5e+171) w0_m (log1p (expm1 w0_m)))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 7.5e+171) {
		tmp = w0_m;
	} else {
		tmp = log1p(expm1(w0_m));
	}
	return w0_s * tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (D_m <= 7.5e+171) {
		tmp = w0_m;
	} else {
		tmp = Math.log1p(Math.expm1(w0_m));
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if D_m <= 7.5e+171:
		tmp = w0_m
	else:
		tmp = math.log1p(math.expm1(w0_m))
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (D_m <= 7.5e+171)
		tmp = w0_m;
	else
		tmp = log1p(expm1(w0_m));
	end
	return Float64(w0_s * tmp)
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[D$95$m, 7.5e+171], w0$95$m, N[Log[1 + N[(Exp[w0$95$m] - 1), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if D < 7.4999999999999998e171

   1. Initial program 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in D around 0 71.6%

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

   if 7.4999999999999998e171 < D

   1. Initial program 80.5%

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

   3. Simplified 80.5%

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

      1. unpow2 80.5%

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

      2. unpow2 80.5%

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

      3. associate-/r* 80.5%

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

      4. associate-*r/ 80.5%

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

      5. *-commutative 80.5%

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

      6. associate-*r/ 85.5%

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

      7. clear-num 85.4%

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

      8. *-commutative 85.4%

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

      9. associate-*l/ 85.4%

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

      10. associate-/r* 85.4%

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

      11. clear-num 85.4%

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

      12. associate-*l/ 85.4%

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

      13. *-un-lft-identity 85.4%

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

      14. associate-/r/ 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-/r/ 85.5%

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

      2. associate-*l/ 85.5%

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

      3. *-commutative 85.5%

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

      4. associate-*l/ 85.4%

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

      5. associate-/l/ 85.5%

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

      6. associate-/r/ 85.5%

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

   8. Simplified 85.5%

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

      1. add-exp-log 44.5%

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

      2. associate-*l/ 44.5%

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

      3. *-un-lft-identity 44.5%

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

      4. *-commutative 44.5%

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

      5. associate-/l/ 44.5%

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

   10. Applied egg-rr 44.5%

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

   11. Taylor expanded in h around 0 21.0%

       \[\leadsto e^{\color{blue}{\log w0}} \]
   12. Step-by-step derivation

       1. rem-exp-log 35.4%

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

       2. log1p-expm1-u 52.3%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)} \]

   13. Applied egg-rr 52.3%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 7.5 \cdot 10^{+171}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(w0\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot \begin{array}{l} \mathbf{if}\;M\_m \leq 2.4 \cdot 10^{+91}:\\ \;\;\;\;w0\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(w0\_m + 1\right)}^{2}}{-2 - w0\_m}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M_m D_m h l d_m)
 :precision binary64
 (*
  w0_s
  (if (<= M_m 2.4e+91) w0_m (/ (- 1.0 (pow (+ w0_m 1.0) 2.0)) (- -2.0 w0_m)))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2.4e+91) {
		tmp = w0_m;
	} else {
		tmp = (1.0 - pow((w0_m + 1.0), 2.0)) / (-2.0 - w0_m);
	}
	return w0_s * tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 2.4d+91) then
        tmp = w0_m
    else
        tmp = (1.0d0 - ((w0_m + 1.0d0) ** 2.0d0)) / ((-2.0d0) - w0_m)
    end if
    code = w0_s * tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2.4e+91) {
		tmp = w0_m;
	} else {
		tmp = (1.0 - Math.pow((w0_m + 1.0), 2.0)) / (-2.0 - w0_m);
	}
	return w0_s * tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 2.4e+91:
		tmp = w0_m
	else:
		tmp = (1.0 - math.pow((w0_m + 1.0), 2.0)) / (-2.0 - w0_m)
	return w0_s * tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 2.4e+91)
		tmp = w0_m;
	else
		tmp = Float64(Float64(1.0 - (Float64(w0_m + 1.0) ^ 2.0)) / Float64(-2.0 - w0_m));
	end
	return Float64(w0_s * tmp)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
function tmp_2 = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 2.4e+91)
		tmp = w0_m;
	else
		tmp = (1.0 - ((w0_m + 1.0) ^ 2.0)) / (-2.0 - w0_m);
	end
	tmp_2 = w0_s * tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * If[LessEqual[M$95$m, 2.4e+91], w0$95$m, N[(N[(1.0 - N[Power[N[(w0$95$m + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(-2.0 - w0$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if M < 2.39999999999999983e91

   1. Initial program 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in D around 0 74.2%

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

   if 2.39999999999999983e91 < M

   1. Initial program 70.8%

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

   3. Simplified 70.7%

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

      1. unpow2 70.7%

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

      2. unpow2 70.7%

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

      3. associate-/r* 70.7%

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

      4. associate-*r/ 70.8%

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

      5. *-commutative 70.8%

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

      6. associate-*r/ 71.4%

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

      7. clear-num 71.4%

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

      8. *-commutative 71.4%

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

      9. associate-*l/ 71.4%

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

      10. associate-/r* 71.4%

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

      11. clear-num 71.4%

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

      12. associate-*l/ 71.4%

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

      13. *-un-lft-identity 71.4%

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

      14. associate-/r/ 71.4%

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

   6. Applied egg-rr 71.4%

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

      1. associate-/r/ 71.4%

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

      2. associate-*l/ 71.4%

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

      3. *-commutative 71.4%

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

      4. associate-*l/ 71.3%

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

      5. associate-/l/ 71.3%

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

      6. associate-/r/ 71.4%

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

   8. Simplified 71.4%

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

      1. add-exp-log 33.4%

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

      2. associate-*l/ 33.4%

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

      3. *-un-lft-identity 33.4%

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

      4. *-commutative 33.4%

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

      5. associate-/l/ 33.4%

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

   10. Applied egg-rr 33.4%

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

   11. Taylor expanded in h around 0 8.2%

       \[\leadsto e^{\color{blue}{\log w0}} \]
   12. Step-by-step derivation

       1. rem-exp-log 27.7%

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

       2. expm1-log1p-u 22.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0\right)\right)} \]

       3. expm1-define 9.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(w0\right)} - 1} \]

       4. flip-- 25.7%

          \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(w0\right)} \cdot e^{\mathsf{log1p}\left(w0\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(w0\right)} + 1}} \]

       5. frac-2neg 25.7%

          \[\leadsto \color{blue}{\frac{-\left(e^{\mathsf{log1p}\left(w0\right)} \cdot e^{\mathsf{log1p}\left(w0\right)} - 1 \cdot 1\right)}{-\left(e^{\mathsf{log1p}\left(w0\right)} + 1\right)}} \]

       6. metadata-eval 25.7%

          \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left(w0\right)} \cdot e^{\mathsf{log1p}\left(w0\right)} - \color{blue}{1}\right)}{-\left(e^{\mathsf{log1p}\left(w0\right)} + 1\right)} \]

       7. sub-neg 25.7%

          \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left(w0\right)} \cdot e^{\mathsf{log1p}\left(w0\right)} + \left(-1\right)\right)}}{-\left(e^{\mathsf{log1p}\left(w0\right)} + 1\right)} \]

       8. pow2 25.7%

          \[\leadsto \frac{-\left(\color{blue}{{\left(e^{\mathsf{log1p}\left(w0\right)}\right)}^{2}} + \left(-1\right)\right)}{-\left(e^{\mathsf{log1p}\left(w0\right)} + 1\right)} \]

       9. log1p-undefine 25.7%

          \[\leadsto \frac{-\left({\left(e^{\color{blue}{\log \left(1 + w0\right)}}\right)}^{2} + \left(-1\right)\right)}{-\left(e^{\mathsf{log1p}\left(w0\right)} + 1\right)} \]

       10. rem-exp-log 25.7%

           \[\leadsto \frac{-\left({\color{blue}{\left(1 + w0\right)}}^{2} + \left(-1\right)\right)}{-\left(e^{\mathsf{log1p}\left(w0\right)} + 1\right)} \]

       11. metadata-eval 25.7%

           \[\leadsto \frac{-\left({\left(1 + w0\right)}^{2} + \color{blue}{-1}\right)}{-\left(e^{\mathsf{log1p}\left(w0\right)} + 1\right)} \]

       12. log1p-undefine 25.7%

           \[\leadsto \frac{-\left({\left(1 + w0\right)}^{2} + -1\right)}{-\left(e^{\color{blue}{\log \left(1 + w0\right)}} + 1\right)} \]

       13. rem-exp-log 33.1%

           \[\leadsto \frac{-\left({\left(1 + w0\right)}^{2} + -1\right)}{-\left(\color{blue}{\left(1 + w0\right)} + 1\right)} \]

   13. Applied egg-rr 33.1%

       \[\leadsto \color{blue}{\frac{-\left({\left(1 + w0\right)}^{2} + -1\right)}{-\left(\left(1 + w0\right) + 1\right)}} \]
   14. Step-by-step derivation

       1. neg-sub0 33.1%

          \[\leadsto \frac{\color{blue}{0 - \left({\left(1 + w0\right)}^{2} + -1\right)}}{-\left(\left(1 + w0\right) + 1\right)} \]

       2. +-commutative 33.1%

          \[\leadsto \frac{0 - \color{blue}{\left(-1 + {\left(1 + w0\right)}^{2}\right)}}{-\left(\left(1 + w0\right) + 1\right)} \]

       3. associate--r+ 33.1%

          \[\leadsto \frac{\color{blue}{\left(0 - -1\right) - {\left(1 + w0\right)}^{2}}}{-\left(\left(1 + w0\right) + 1\right)} \]

       4. metadata-eval 33.1%

          \[\leadsto \frac{\color{blue}{1} - {\left(1 + w0\right)}^{2}}{-\left(\left(1 + w0\right) + 1\right)} \]

       5. +-commutative 33.1%

          \[\leadsto \frac{1 - {\color{blue}{\left(w0 + 1\right)}}^{2}}{-\left(\left(1 + w0\right) + 1\right)} \]

       6. neg-sub0 33.1%

          \[\leadsto \frac{1 - {\left(w0 + 1\right)}^{2}}{\color{blue}{0 - \left(\left(1 + w0\right) + 1\right)}} \]

       7. +-commutative 33.1%

          \[\leadsto \frac{1 - {\left(w0 + 1\right)}^{2}}{0 - \color{blue}{\left(1 + \left(1 + w0\right)\right)}} \]

       8. associate-+r+ 33.1%

          \[\leadsto \frac{1 - {\left(w0 + 1\right)}^{2}}{0 - \color{blue}{\left(\left(1 + 1\right) + w0\right)}} \]

       9. metadata-eval 33.1%

          \[\leadsto \frac{1 - {\left(w0 + 1\right)}^{2}}{0 - \left(\color{blue}{2} + w0\right)} \]

       10. associate--r+ 33.1%

           \[\leadsto \frac{1 - {\left(w0 + 1\right)}^{2}}{\color{blue}{\left(0 - 2\right) - w0}} \]

       11. metadata-eval 33.1%

           \[\leadsto \frac{1 - {\left(w0 + 1\right)}^{2}}{\color{blue}{-2} - w0} \]

   15. Simplified 33.1%

       \[\leadsto \color{blue}{\frac{1 - {\left(w0 + 1\right)}^{2}}{-2 - w0}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.4 \cdot 10^{+91}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(w0 + 1\right)}^{2}}{-2 - w0}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ w0\_m = \left|w0\right| \\ w0\_s = \mathsf{copysign}\left(1, w0\right) \\ w0\_s \cdot w0\_m \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
w0\_m = (fabs.f64 w0)
w0\_s = (copysign.f64 #s(literal 1 binary64) w0)
(FPCore (w0_s w0_m M_m D_m h l d_m) :precision binary64 (* w0_s w0_m))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
w0\_m = fabs(w0);
w0\_s = copysign(1.0, w0);
double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * w0_m;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0d0, w0)
real(8) function code(w0_s, w0_m, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0_s
    real(8), intent (in) :: w0_m
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0_s * w0_m
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
w0\_m = Math.abs(w0);
w0\_s = Math.copySign(1.0, w0);
public static double code(double w0_s, double w0_m, double M_m, double D_m, double h, double l, double d_m) {
	return w0_s * w0_m;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
w0\_m = math.fabs(w0)
w0\_s = math.copysign(1.0, w0)
def code(w0_s, w0_m, M_m, D_m, h, l, d_m):
	return w0_s * w0_m

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0\_m = abs(w0)
w0\_s = copysign(1.0, w0)
function code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	return Float64(w0_s * w0_m)
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0\_m = abs(w0);
w0\_s = sign(w0) * abs(1.0);
function tmp = code(w0_s, w0_m, M_m, D_m, h, l, d_m)
	tmp = w0_s * w0_m;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
w0\_m = N[Abs[w0], $MachinePrecision]
w0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[w0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[w0$95$s_, w0$95$m_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0$95$s * w0$95$m), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

3. Simplified 80.2%

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

5. Taylor expanded in D around 0 68.8%

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

6. Final simplification 68.8%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(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))))))