Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 86.8%

Time: 17.8s

Alternatives: 7

Speedup: 1.8×

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: 80.7% 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: 86.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m}{d}\\ \mathbf{if}\;M\_m \leq 1.35 \cdot 10^{-239}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(0.125, h \cdot \frac{t\_0 \cdot t\_0}{-\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(\left(M\_m \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \left(\left(M\_m \cdot 0.5\right) \cdot \frac{\frac{D}{d}}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (* D (/ M_m d))))
   (if (<= M_m 1.35e-239)
     (* w0 (fma 0.125 (* h (/ (* t_0 t_0) (- l))) 1.0))
     (*
      w0
      (sqrt
       (-
        1.0
        (* h (* (* (* M_m 0.5) (/ D d)) (* (* M_m 0.5) (/ (/ D d) l))))))))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = D * (M_m / d);
	double tmp;
	if (M_m <= 1.35e-239) {
		tmp = w0 * fma(0.125, (h * ((t_0 * t_0) / -l)), 1.0);
	} else {
		tmp = w0 * sqrt((1.0 - (h * (((M_m * 0.5) * (D / d)) * ((M_m * 0.5) * ((D / d) / l))))));
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(D * Float64(M_m / d))
	tmp = 0.0
	if (M_m <= 1.35e-239)
		tmp = Float64(w0 * fma(0.125, Float64(h * Float64(Float64(t_0 * t_0) / Float64(-l))), 1.0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(M_m * 0.5) * Float64(D / d)) * Float64(Float64(M_m * 0.5) * Float64(Float64(D / d) / l)))))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 1.35e-239], N[(w0 * N[(0.125 * N[(h * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / (-l)), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.35e-239

   1. Initial program 78.8%

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

   3. Simplified 78.8%

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

      1. clear-num 78.8%

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

      2. un-div-inv 80.2%

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

      3. clear-num 80.2%

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

      4. un-div-inv 80.2%

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

      5. associate-/r/ 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. associate-/r/ 86.0%

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

      2. associate-*l/ 86.0%

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

      3. *-commutative 86.0%

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

      4. associate-/l* 86.0%

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

      5. associate-*l/ 86.0%

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

      6. *-commutative 86.0%

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

      7. associate-*l/ 86.0%

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

      8. associate-/l/ 86.7%

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

      9. associate-/r/ 86.7%

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

      10. associate-*l/ 86.7%

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

      11. *-commutative 86.7%

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

      12. associate-*r/ 86.0%

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

      13. *-commutative 86.0%

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

      14. associate-/l/ 86.0%

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

      15. associate-/l* 86.0%

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

   8. Simplified 86.0%

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

      1. add-sqr-sqrt 64.5%

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

      2. pow2 64.5%

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

      3. sqrt-div 46.2%

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

      4. sqrt-pow1 46.8%

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

      5. metadata-eval 46.8%

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

      6. pow1 46.8%

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

      7. associate-/r* 46.8%

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

   10. Applied egg-rr 46.8%

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

   11. Taylor expanded in l around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. fma-define 0.0%

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

   13. Simplified 84.5%

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

       1. unpow2 84.5%

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

   15. Applied egg-rr 84.5%

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

   if 1.35e-239 < M

   1. Initial program 85.6%

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

   3. Simplified 83.9%

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

      1. clear-num 83.9%

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

      2. un-div-inv 84.8%

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

      3. clear-num 84.7%

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

      4. un-div-inv 86.2%

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

      5. associate-/r/ 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. associate-/r/ 88.9%

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

      2. associate-*l/ 88.1%

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

      3. *-commutative 88.1%

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

      4. associate-/l* 88.9%

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

      5. associate-*l/ 89.0%

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

      6. *-commutative 89.0%

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

      7. associate-*l/ 89.0%

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

      8. associate-/l/ 90.1%

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

      9. associate-/r/ 90.1%

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

      10. associate-*l/ 90.1%

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

      11. *-commutative 90.1%

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

      12. associate-*r/ 90.0%

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

      13. *-commutative 90.0%

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

      14. associate-/l/ 90.0%

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

      15. associate-/l* 87.6%

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

   8. Simplified 87.6%

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

   9. Taylor expanded in D around 0 90.0%

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

       1. associate-*l/ 90.1%

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

       2. *-commutative 90.1%

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

   11. Simplified 90.1%

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

       1. unpow2 90.1%

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

       2. *-un-lft-identity 90.1%

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

       3. times-frac 90.9%

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

       4. associate-*r* 90.9%

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

       5. associate-*r* 90.9%

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

   13. Applied egg-rr 90.9%

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

       1. associate-/l* 90.9%

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

   15. Applied egg-rr 90.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.35 \cdot 10^{-239}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(0.125, h \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{-\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \left(\left(M \cdot 0.5\right) \cdot \frac{\frac{D}{d}}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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 - (h * (((d * (m_m / (2.0d0 * d_1))) ** 2.0d0) / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

3. Simplified 81.1%

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

   1. clear-num 81.0%

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

   2. un-div-inv 82.2%

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

   3. clear-num 82.2%

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

   4. un-div-inv 82.8%

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

   5. associate-/r/ 82.8%

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

6. Applied egg-rr 82.8%

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

   1. associate-/r/ 87.3%

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

   2. associate-*l/ 86.9%

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

   3. *-commutative 86.9%

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

   4. associate-/l* 87.3%

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

   5. associate-*l/ 87.3%

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

   6. *-commutative 87.3%

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

   7. associate-*l/ 87.3%

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

   8. associate-/l/ 88.2%

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

   9. associate-/r/ 88.2%

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

   10. associate-*l/ 88.2%

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

   11. *-commutative 88.2%

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

   12. associate-*r/ 87.8%

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

   13. *-commutative 87.8%

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

   14. associate-/l/ 87.8%

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

   15. associate-/l* 86.7%

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

8. Simplified 86.7%

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

9. Final simplification 86.7%

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

Alternative 3: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-313}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \left(t\_0 \cdot t\_0\right) \cdot \frac{h}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (* D (/ M_m d))))
   (if (<= (/ h l) -2e-313)
     (* w0 (fma -0.125 (* (* t_0 t_0) (/ h l)) 1.0))
     w0)))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = D * (M_m / d);
	double tmp;
	if ((h / l) <= -2e-313) {
		tmp = w0 * fma(-0.125, ((t_0 * t_0) * (h / l)), 1.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(D * Float64(M_m / d))
	tmp = 0.0
	if (Float64(h / l) <= -2e-313)
		tmp = Float64(w0 * fma(-0.125, Float64(Float64(t_0 * t_0) * Float64(h / l)), 1.0));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -2e-313], N[(w0 * N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.99999999998e-313

   1. Initial program 79.9%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in D around 0 52.9%

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

      1. associate-/l* 53.6%

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

      2. associate-/l* 52.8%

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

   7. Simplified 52.8%

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

      1. *-un-lft-identity 52.8%

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

      2. cancel-sign-sub-inv 52.8%

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

      3. metadata-eval 52.8%

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

      4. associate-*r* 51.4%

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

      5. pow-prod-down 65.7%

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

      6. associate-/r* 67.1%

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

   9. Applied egg-rr 67.1%

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

       1. *-lft-identity 67.1%

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

   11. Simplified 67.1%

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

   12. Taylor expanded in D around 0 52.3%

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

       1. +-commutative 52.3%

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

       2. fma-define 52.3%

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

   14. Simplified 73.3%

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

       1. unpow2 75.4%

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

   16. Applied egg-rr 73.3%

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

   if -1.99999999998e-313 < (/.f64 h l)

   1. Initial program 84.2%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in D around 0 96.6%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-313}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \left(\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)\right) \cdot \frac{h}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \left(M\_m \cdot 0.5\right) \cdot \frac{D}{d}\\ w0 \cdot \sqrt{1 - h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)} \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (* (* M_m 0.5) (/ D d))))
   (* w0 (sqrt (- 1.0 (* h (* t_0 (/ t_0 l))))))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (M_m * 0.5) * (D / d);
	return w0 * sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    t_0 = (m_m * 0.5d0) * (d / d_1)
    code = w0 * sqrt((1.0d0 - (h * (t_0 * (t_0 / l)))))
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (M_m * 0.5) * (D / d);
	return w0 * Math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	t_0 = (M_m * 0.5) * (D / d)
	return w0 * math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))))

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(Float64(M_m * 0.5) * Float64(D / d))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_0 * Float64(t_0 / l))))))
end

MATLAB

M_m = abs(M);
function tmp = code(w0, M_m, D, h, l, d)
	t_0 = (M_m * 0.5) * (D / d);
	tmp = w0 * sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M$95$m * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.8%

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

3. Simplified 81.1%

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

   1. clear-num 81.0%

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

   2. un-div-inv 82.2%

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

   3. clear-num 82.2%

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

   4. un-div-inv 82.8%

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

   5. associate-/r/ 82.8%

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

6. Applied egg-rr 82.8%

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

   1. associate-/r/ 87.3%

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

   2. associate-*l/ 86.9%

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

   3. *-commutative 86.9%

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

   4. associate-/l* 87.3%

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

   5. associate-*l/ 87.3%

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

   6. *-commutative 87.3%

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

   7. associate-*l/ 87.3%

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

   8. associate-/l/ 88.2%

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

   9. associate-/r/ 88.2%

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

   10. associate-*l/ 88.2%

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

   11. *-commutative 88.2%

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

   12. associate-*r/ 87.8%

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

   13. *-commutative 87.8%

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

   14. associate-/l/ 87.8%

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

   15. associate-/l* 86.7%

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

8. Simplified 86.7%

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

9. Taylor expanded in D around 0 87.8%

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

    1. associate-*l/ 88.2%

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

    2. *-commutative 88.2%

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

11. Simplified 88.2%

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

    1. unpow2 88.2%

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

    2. *-un-lft-identity 88.2%

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

    3. times-frac 89.3%

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

    4. associate-*r* 89.3%

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

    5. associate-*r* 89.3%

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

13. Applied egg-rr 89.3%

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

14. Final simplification 89.3%

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

Alternative 5: 80.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := D \cdot \frac{M\_m}{d}\\ w0 \cdot \mathsf{fma}\left(0.125, h \cdot \frac{t\_0 \cdot t\_0}{-\ell}, 1\right) \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (* D (/ M_m d))))
   (* w0 (fma 0.125 (* h (/ (* t_0 t_0) (- l))) 1.0))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = D * (M_m / d);
	return w0 * fma(0.125, (h * ((t_0 * t_0) / -l)), 1.0);
}

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(D * Float64(M_m / d))
	return Float64(w0 * fma(0.125, Float64(h * Float64(Float64(t_0 * t_0) / Float64(-l))), 1.0))
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[(0.125 * N[(h * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / (-l)), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.8%

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

3. Simplified 81.1%

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

   1. clear-num 81.0%

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

   2. un-div-inv 82.2%

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

   3. clear-num 82.2%

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

   4. un-div-inv 82.8%

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

   5. associate-/r/ 82.8%

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

6. Applied egg-rr 82.8%

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

   1. associate-/r/ 87.3%

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

   2. associate-*l/ 86.9%

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

   3. *-commutative 86.9%

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

   4. associate-/l* 87.3%

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

   5. associate-*l/ 87.3%

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

   6. *-commutative 87.3%

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

   7. associate-*l/ 87.3%

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

   8. associate-/l/ 88.2%

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

   9. associate-/r/ 88.2%

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

   10. associate-*l/ 88.2%

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

   11. *-commutative 88.2%

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

   12. associate-*r/ 87.8%

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

   13. *-commutative 87.8%

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

   14. associate-/l/ 87.8%

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

   15. associate-/l* 86.7%

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

8. Simplified 86.7%

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

   1. add-sqr-sqrt 67.6%

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

   2. pow2 67.6%

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

   3. sqrt-div 47.3%

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

   4. sqrt-pow1 48.0%

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

   5. metadata-eval 48.0%

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

   6. pow1 48.0%

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

   7. associate-/r* 48.0%

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

10. Applied egg-rr 48.0%

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

11. Taylor expanded in l around -inf 0.0%

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

    1. +-commutative 0.0%

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

    2. fma-define 0.0%

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

13. Simplified 83.8%

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

    1. unpow2 83.8%

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

15. Applied egg-rr 83.8%

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

16. Final simplification 83.8%

    \[\leadsto w0 \cdot \mathsf{fma}\left(0.125, h \cdot \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{-\ell}, 1\right) \]
17. Add Preprocessing

Alternative 6: 62.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 9.8 \cdot 10^{+51}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left({\left(M\_m \cdot \frac{D}{d}\right)}^{2} \cdot \frac{w0 \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 9.8e+51)
   w0
   (* -0.125 (* (pow (* M_m (/ D d)) 2.0) (/ (* w0 h) l)))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 9.8e+51) {
		tmp = w0;
	} else {
		tmp = -0.125 * (pow((M_m * (D / d)), 2.0) * ((w0 * h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m_m <= 9.8d+51) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((m_m * (d / d_1)) ** 2.0d0) * ((w0 * h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 9.8e+51) {
		tmp = w0;
	} else {
		tmp = -0.125 * (Math.pow((M_m * (D / d)), 2.0) * ((w0 * h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 9.8e+51:
		tmp = w0
	else:
		tmp = -0.125 * (math.pow((M_m * (D / d)), 2.0) * ((w0 * h) / l))
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 9.8e+51)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64((Float64(M_m * Float64(D / d)) ^ 2.0) * Float64(Float64(w0 * h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 9.8e+51)
		tmp = w0;
	else
		tmp = -0.125 * (((M_m * (D / d)) ^ 2.0) * ((w0 * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 9.8e+51], w0, N[(-0.125 * N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.79999999999999967e51

   1. Initial program 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in D around 0 75.8%

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

   if 9.79999999999999967e51 < M

   1. Initial program 76.2%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in D around 0 42.4%

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

      1. associate-/l* 42.4%

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

      2. associate-/l* 44.3%

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

   7. Simplified 44.3%

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

      1. *-un-lft-identity 44.3%

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

      2. cancel-sign-sub-inv 44.3%

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

      3. metadata-eval 44.3%

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

      4. associate-*r* 44.3%

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

      5. pow-prod-down 73.7%

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

      6. associate-/r* 73.7%

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

   9. Applied egg-rr 73.7%

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

       1. *-lft-identity 73.7%

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

   11. Simplified 73.7%

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

   12. Taylor expanded in D around 0 42.4%

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

       1. +-commutative 42.4%

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

       2. fma-define 42.4%

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

   14. Simplified 68.4%

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

   15. Taylor expanded in D around inf 28.3%

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

       1. associate-*r* 28.3%

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

       2. times-frac 26.2%

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

       3. associate-/l* 26.2%

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

       4. unpow2 26.2%

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

       5. unpow2 26.2%

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

       6. unpow2 26.2%

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

       7. times-frac 27.5%

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

       8. swap-sqr 28.3%

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

       9. unpow2 28.3%

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

       10. associate-*r/ 28.4%

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

       11. *-commutative 28.4%

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

       12. associate-*r/ 28.4%

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

   17. Simplified 28.4%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9.8 \cdot 10^{+51}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{w0 \cdot h}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

3. Simplified 81.1%

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

5. Taylor expanded in D around 0 70.7%

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

6. Final simplification 70.7%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024072 
(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))))))