Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.7% → 87.7%

Time: 19.0s

Alternatives: 8

Speedup: 0.7×

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: 81.7% 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: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := {\left(M \cdot D\right)}^{2}\\ t_1 := {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot t\_0}{\ell}\right) + -2 \cdot \log d\_m\right)}\right)}^{2}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-39}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{t\_0}{\ell} \cdot \frac{h}{{d\_m}^{2}}\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (let* ((t_0 (pow (* M D) 2.0))
        (t_1 (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l))))
   (if (<= t_1 (- INFINITY))
     (*
      w0
      (pow
       (pow (exp 0.25) (+ (log (* -0.25 (/ (* h t_0) l))) (* -2.0 (log d_m))))
       2.0))
     (if (<= t_1 5e-39)
       (* w0 (sqrt (- 1.0 t_1)))
       (* w0 (sqrt (- 1.0 (* 0.25 (* (/ t_0 l) (/ h (pow d_m 2.0)))))))))))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double t_0 = pow((M * D), 2.0);
	double t_1 = pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = w0 * pow(pow(exp(0.25), (log((-0.25 * ((h * t_0) / l))) + (-2.0 * log(d_m)))), 2.0);
	} else if (t_1 <= 5e-39) {
		tmp = w0 * sqrt((1.0 - t_1));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * ((t_0 / l) * (h / pow(d_m, 2.0))))));
	}
	return tmp;
}

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double t_0 = Math.pow((M * D), 2.0);
	double t_1 = Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.pow(Math.pow(Math.exp(0.25), (Math.log((-0.25 * ((h * t_0) / l))) + (-2.0 * Math.log(d_m)))), 2.0);
	} else if (t_1 <= 5e-39) {
		tmp = w0 * Math.sqrt((1.0 - t_1));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((t_0 / l) * (h / Math.pow(d_m, 2.0))))));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	t_0 = math.pow((M * D), 2.0)
	t_1 = math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = w0 * math.pow(math.pow(math.exp(0.25), (math.log((-0.25 * ((h * t_0) / l))) + (-2.0 * math.log(d_m)))), 2.0)
	elif t_1 <= 5e-39:
		tmp = w0 * math.sqrt((1.0 - t_1))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((t_0 / l) * (h / math.pow(d_m, 2.0))))))
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	t_0 = Float64(M * D) ^ 2.0
	t_1 = Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(w0 * ((exp(0.25) ^ Float64(log(Float64(-0.25 * Float64(Float64(h * t_0) / l))) + Float64(-2.0 * log(d_m)))) ^ 2.0));
	elseif (t_1 <= 5e-39)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_1)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(t_0 / l) * Float64(h / (d_m ^ 2.0)))))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	t_0 = (M * D) ^ 2.0;
	t_1 = (((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = w0 * ((exp(0.25) ^ (log((-0.25 * ((h * t_0) / l))) + (-2.0 * log(d_m)))) ^ 2.0);
	elseif (t_1 <= 5e-39)
		tmp = w0 * sqrt((1.0 - t_1));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * ((t_0 / l) * (h / (d_m ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(w0 * N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[(-0.25 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[d$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-39], N[(w0 * N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(t$95$0 / l), $MachinePrecision] * N[(h / N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 54.3%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in D around 0 49.6%

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

      1. *-un-lft-identity 49.6%

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

      2. add-sqr-sqrt 23.8%

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

      3. times-frac 23.8%

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

      4. sqrt-prod 23.8%

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

      5. unpow2 23.8%

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

      6. sqrt-prod 16.8%

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

      7. add-sqr-sqrt 16.8%

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

      8. associate-*r* 16.8%

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

      9. *-commutative 16.8%

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

      10. pow-prod-down 18.5%

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

      11. sqrt-prod 18.5%

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

      12. unpow2 18.5%

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

      13. sqrt-prod 18.7%

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

      14. add-sqr-sqrt 30.9%

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

   7. Applied egg-rr 30.9%

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

   9. Applied egg-rr 24.1%

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

   10. Taylor expanded in d around 0 35.0%

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

       1. exp-prod 34.7%

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

       2. associate-*r* 34.7%

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

       3. unpow2 34.7%

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

       4. unpow2 34.7%

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

       5. swap-sqr 36.4%

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

       6. unpow2 36.4%

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

       7. *-commutative 36.4%

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

   12. Simplified 36.4%

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

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 4.9999999999999998e-39

   1. Initial program 99.9%

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

   if 4.9999999999999998e-39 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 3.8%

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

   3. Simplified 7.7%

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

   5. Taylor expanded in D around 0 51.5%

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

      1. associate-*r* 59.2%

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

      2. *-commutative 59.2%

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

      3. times-frac 59.2%

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

      4. pow-prod-down 73.8%

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

   7. Applied egg-rr 73.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \left(-0.25 \cdot \frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{2}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-39}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{\left(M \cdot D\right)}^{2}}{\ell} \cdot \frac{h}{{d}^{2}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < +inf.0

   1. Initial program 88.1%

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

   3. Simplified 87.2%

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

   if +inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))))

   1. Initial program 0.0%

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

   3. Simplified 4.0%

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

   5. Taylor expanded in D around 0 68.9%

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

4. Final simplification 85.4%

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

Alternative 3: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{\left(M \cdot D\right)}^{2}}{\ell} \cdot \frac{h}{{d\_m}^{2}}\right)}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* 2.0 d_m)) 2.0) (/ h l)))))
   (if (<= t_0 5e+257)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt
       (- 1.0 (* 0.25 (* (/ (pow (* M D) 2.0) l) (/ h (pow d_m 2.0))))))))))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double t_0 = 1.0 - (pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+257) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * ((pow((M * D), 2.0) / l) * (h / pow(d_m, 2.0))))));
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m * d) / (2.0d0 * d_m)) ** 2.0d0) * (h / l))
    if (t_0 <= 5d+257) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * ((((m * d) ** 2.0d0) / l) * (h / (d_m ** 2.0d0))))))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double t_0 = 1.0 - (Math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+257) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((Math.pow((M * D), 2.0) / l) * (h / Math.pow(d_m, 2.0))))));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d_m)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 5e+257:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((math.pow((M * D), 2.0) / l) * (h / math.pow(d_m, 2.0))))))
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 5e+257)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64((Float64(M * D) ^ 2.0) / l) * Float64(h / (d_m ^ 2.0)))))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	t_0 = 1.0 - ((((M * D) / (2.0 * d_m)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 5e+257)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - (0.25 * ((((M * D) ^ 2.0) / l) * (h / (d_m ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+257], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(h / N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 5.00000000000000028e257

   1. Initial program 99.9%

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

   if 5.00000000000000028e257 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 39.7%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in D around 0 48.5%

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

      1. associate-*r* 50.9%

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

      2. *-commutative 50.9%

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

      3. times-frac 52.1%

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

      4. pow-prod-down 62.5%

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

   7. Applied egg-rr 62.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{{\left(M \cdot D\right)}^{2}}{\ell} \cdot \frac{h}{{d}^{2}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-309}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d\_m}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(D \cdot \left(M \cdot \left(\frac{0.5}{d\_m} \cdot \sqrt{h}\right)\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= h -2e-309)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 d_m))) 2.0)) l))))
   (* w0 (sqrt (- 1.0 (/ (pow (* D (* M (* (/ 0.5 d_m) (sqrt h)))) 2.0) l))))))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (h <= -2e-309) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (D * (0.5 / d_m))), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (pow((D * (M * ((0.5 / d_m) * sqrt(h)))), 2.0) / l)));
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (h <= (-2d-309)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m * (d * (0.5d0 / d_m))) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (((d * (m * ((0.5d0 / d_m) * sqrt(h)))) ** 2.0d0) / l)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (h <= -2e-309) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * (D * (0.5 / d_m))), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow((D * (M * ((0.5 / d_m) * Math.sqrt(h)))), 2.0) / l)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if h <= -2e-309:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M * (D * (0.5 / d_m))), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (math.pow((D * (M * ((0.5 / d_m) * math.sqrt(h)))), 2.0) / l)))
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (h <= -2e-309)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(D * Float64(0.5 / d_m))) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(D * Float64(M * Float64(Float64(0.5 / d_m) * sqrt(h)))) ^ 2.0) / l))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (h <= -2e-309)
		tmp = w0 * sqrt((1.0 - ((h * ((M * (D * (0.5 / d_m))) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (((D * (M * ((0.5 / d_m) * sqrt(h)))) ^ 2.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[h, -2e-309], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(D * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(D * N[(M * N[(N[(0.5 / d$95$m), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.9999999999999988e-309

   1. Initial program 79.5%

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

   3. Simplified 80.4%

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

      1. *-commutative 80.4%

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

      2. frac-times 79.5%

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

      3. *-commutative 79.5%

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

      4. associate-*l/ 82.9%

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

      5. div-inv 82.9%

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

      6. associate-*l* 84.5%

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

      7. associate-/r* 84.5%

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

      8. metadata-eval 84.5%

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

   6. Applied egg-rr 84.5%

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

   if -1.9999999999999988e-309 < h

   1. Initial program 79.4%

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

   3. Simplified 77.9%

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

      1. *-commutative 77.9%

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

      2. frac-times 79.4%

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

      3. *-commutative 79.4%

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

      4. associate-*l/ 86.5%

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

      5. div-inv 86.5%

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

      6. associate-*l* 85.3%

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

      7. associate-/r* 85.3%

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

      8. metadata-eval 85.3%

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

   6. Applied egg-rr 85.3%

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

      1. add-sqr-sqrt 85.2%

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

      2. pow2 85.2%

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

      3. *-commutative 85.2%

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

      4. sqrt-prod 85.2%

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

      5. unpow2 85.2%

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

      6. sqrt-prod 55.8%

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

      7. add-sqr-sqrt 88.7%

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

      8. *-commutative 88.7%

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

      9. *-commutative 88.7%

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

      10. associate-*l* 90.7%

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

   8. Applied egg-rr 90.7%

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

      1. *-commutative 90.7%

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

      2. associate-*l* 90.0%

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

   10. Simplified 90.0%

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

       1. expm1-log1p-u 76.5%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(D \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot \sqrt{h}\right)\right)\right)\right)}}^{2}}{\ell}} \]

       2. expm1-udef 73.1%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(e^{\mathsf{log1p}\left(\left(D \cdot M\right) \cdot \left(\frac{0.5}{d} \cdot \sqrt{h}\right)\right)} - 1\right)}}^{2}}{\ell}} \]

       3. associate-*l* 73.8%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(e^{\mathsf{log1p}\left(\color{blue}{D \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot \sqrt{h}\right)\right)}\right)} - 1\right)}^{2}}{\ell}} \]

   12. Applied egg-rr 73.8%

       \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(e^{\mathsf{log1p}\left(D \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot \sqrt{h}\right)\right)\right)} - 1\right)}}^{2}}{\ell}} \]
   13. Step-by-step derivation

       1. expm1-def 77.1%

          \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(D \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot \sqrt{h}\right)\right)\right)\right)\right)}}^{2}}{\ell}} \]

       2. expm1-log1p 89.9%

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

       3. *-commutative 89.9%

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

   14. Simplified 89.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-309}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(D \cdot \left(M \cdot \left(\frac{0.5}{d} \cdot \sqrt{h}\right)\right)\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-309}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d\_m}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot D\right) \cdot \left(\frac{0.5}{d\_m} \cdot \sqrt{h}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= h -2e-309)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* M (* D (/ 0.5 d_m))) 2.0)) l))))
   (* w0 (sqrt (- 1.0 (/ (pow (* (* M D) (* (/ 0.5 d_m) (sqrt h))) 2.0) l))))))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (h <= -2e-309) {
		tmp = w0 * sqrt((1.0 - ((h * pow((M * (D * (0.5 / d_m))), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (pow(((M * D) * ((0.5 / d_m) * sqrt(h))), 2.0) / l)));
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (h <= (-2d-309)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((m * (d * (0.5d0 / d_m))) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - ((((m * d) * ((0.5d0 / d_m) * sqrt(h))) ** 2.0d0) / l)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (h <= -2e-309) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((M * (D * (0.5 / d_m))), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(((M * D) * ((0.5 / d_m) * Math.sqrt(h))), 2.0) / l)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if h <= -2e-309:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((M * (D * (0.5 / d_m))), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (math.pow(((M * D) * ((0.5 / d_m) * math.sqrt(h))), 2.0) / l)))
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (h <= -2e-309)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(M * Float64(D * Float64(0.5 / d_m))) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) * Float64(Float64(0.5 / d_m) * sqrt(h))) ^ 2.0) / l))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (h <= -2e-309)
		tmp = w0 * sqrt((1.0 - ((h * ((M * (D * (0.5 / d_m))) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - ((((M * D) * ((0.5 / d_m) * sqrt(h))) ^ 2.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[h, -2e-309], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(M * N[(D * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] * N[(N[(0.5 / d$95$m), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.9999999999999988e-309

   1. Initial program 79.5%

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

   3. Simplified 80.4%

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

      1. *-commutative 80.4%

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

      2. frac-times 79.5%

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

      3. *-commutative 79.5%

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

      4. associate-*l/ 82.9%

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

      5. div-inv 82.9%

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

      6. associate-*l* 84.5%

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

      7. associate-/r* 84.5%

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

      8. metadata-eval 84.5%

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

   6. Applied egg-rr 84.5%

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

   if -1.9999999999999988e-309 < h

   1. Initial program 79.4%

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

   3. Simplified 77.9%

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

      1. *-commutative 77.9%

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

      2. frac-times 79.4%

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

      3. *-commutative 79.4%

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

      4. associate-*l/ 86.5%

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

      5. div-inv 86.5%

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

      6. associate-*l* 85.3%

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

      7. associate-/r* 85.3%

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

      8. metadata-eval 85.3%

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

   6. Applied egg-rr 85.3%

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

      1. add-sqr-sqrt 85.2%

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

      2. pow2 85.2%

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

      3. *-commutative 85.2%

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

      4. sqrt-prod 85.2%

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

      5. unpow2 85.2%

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

      6. sqrt-prod 55.8%

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

      7. add-sqr-sqrt 88.7%

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

      8. *-commutative 88.7%

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

      9. *-commutative 88.7%

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

      10. associate-*l* 90.7%

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

   8. Applied egg-rr 90.7%

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

      1. *-commutative 90.7%

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

      2. associate-*l* 90.0%

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

   10. Simplified 90.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-309}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot \sqrt{h}\right)\right)}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;D \leq 8 \cdot 10^{+16} \lor \neg \left(D \leq 8 \cdot 10^{+114}\right) \land D \leq 2.4 \cdot 10^{+203}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \left(h \cdot {d\_m}^{-2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (or (<= D 8e+16) (and (not (<= D 8e+114)) (<= D 2.4e+203)))
   w0
   (+
    w0
    (* w0 (* -0.125 (* (/ (* (* M D) (* M D)) l) (* h (pow d_m -2.0))))))))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((D <= 8e+16) || (!(D <= 8e+114) && (D <= 2.4e+203))) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (-0.125 * ((((M * D) * (M * D)) / l) * (h * pow(d_m, -2.0)))));
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if ((d <= 8d+16) .or. (.not. (d <= 8d+114)) .and. (d <= 2.4d+203)) then
        tmp = w0
    else
        tmp = w0 + (w0 * ((-0.125d0) * ((((m * d) * (m * d)) / l) * (h * (d_m ** (-2.0d0))))))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if ((D <= 8e+16) || (!(D <= 8e+114) && (D <= 2.4e+203))) {
		tmp = w0;
	} else {
		tmp = w0 + (w0 * (-0.125 * ((((M * D) * (M * D)) / l) * (h * Math.pow(d_m, -2.0)))));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if (D <= 8e+16) or (not (D <= 8e+114) and (D <= 2.4e+203)):
		tmp = w0
	else:
		tmp = w0 + (w0 * (-0.125 * ((((M * D) * (M * D)) / l) * (h * math.pow(d_m, -2.0)))))
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if ((D <= 8e+16) || (!(D <= 8e+114) && (D <= 2.4e+203)))
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(w0 * Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) / l) * Float64(h * (d_m ^ -2.0))))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if ((D <= 8e+16) || (~((D <= 8e+114)) && (D <= 2.4e+203)))
		tmp = w0;
	else
		tmp = w0 + (w0 * (-0.125 * ((((M * D) * (M * D)) / l) * (h * (d_m ^ -2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[Or[LessEqual[D, 8e+16], And[N[Not[LessEqual[D, 8e+114]], $MachinePrecision], LessEqual[D, 2.4e+203]]], w0, N[(w0 + N[(w0 * N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(h * N[Power[d$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 8e16 or 8e114 < D < 2.4000000000000001e203

   1. Initial program 79.6%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in D around 0 72.0%

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

   if 8e16 < D < 8e114 or 2.4000000000000001e203 < D

   1. Initial program 78.6%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in D around 0 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*r* 55.7%

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

   7. Simplified 55.7%

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

      1. expm1-log1p-u 55.7%

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

      2. expm1-udef 53.2%

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

      3. pow-prod-down 63.3%

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

   9. Applied egg-rr 63.3%

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

       1. expm1-def 65.9%

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

       2. expm1-log1p 65.9%

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

   11. Simplified 65.9%

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

       1. distribute-rgt-in 65.9%

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

       2. *-un-lft-identity 65.9%

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

       3. *-commutative 65.9%

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

       4. *-commutative 65.9%

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

       5. frac-times 68.7%

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

       6. div-inv 68.7%

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

       7. pow-flip 68.7%

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

       8. metadata-eval 68.7%

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

   13. Applied egg-rr 68.7%

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

       1. unpow2 31.9%

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

   15. Applied egg-rr 68.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 8 \cdot 10^{+16} \lor \neg \left(D \leq 8 \cdot 10^{+114}\right) \land D \leq 2.4 \cdot 10^{+203}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + w0 \cdot \left(-0.125 \cdot \left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\ell} \cdot \left(h \cdot {d}^{-2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{+75}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{\ell \cdot {d\_m}^{2}}{h \cdot w0}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (w0 M D h l d_m)
 :precision binary64
 (if (<= M 1.05e+75)
   w0
   (* -0.125 (/ (* (* M D) (* M D)) (/ (* l (pow d_m 2.0)) (* h w0))))))

C

d_m = fabs(d);
double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (M <= 1.05e+75) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((M * D) * (M * D)) / ((l * pow(d_m, 2.0)) / (h * w0)));
	}
	return tmp;
}

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    real(8) :: tmp
    if (m <= 1.05d+75) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((m * d) * (m * d)) / ((l * (d_m ** 2.0d0)) / (h * w0)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double w0, double M, double D, double h, double l, double d_m) {
	double tmp;
	if (M <= 1.05e+75) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((M * D) * (M * D)) / ((l * Math.pow(d_m, 2.0)) / (h * w0)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(w0, M, D, h, l, d_m):
	tmp = 0
	if M <= 1.05e+75:
		tmp = w0
	else:
		tmp = -0.125 * (((M * D) * (M * D)) / ((l * math.pow(d_m, 2.0)) / (h * w0)))
	return tmp

Julia

d_m = abs(d)
function code(w0, M, D, h, l, d_m)
	tmp = 0.0
	if (M <= 1.05e+75)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(M * D) * Float64(M * D)) / Float64(Float64(l * (d_m ^ 2.0)) / Float64(h * w0))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(w0, M, D, h, l, d_m)
	tmp = 0.0;
	if (M <= 1.05e+75)
		tmp = w0;
	else
		tmp = -0.125 * (((M * D) * (M * D)) / ((l * (d_m ^ 2.0)) / (h * w0)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := If[LessEqual[M, 1.05e+75], w0, N[(-0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(N[(l * N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.04999999999999999e75

   1. Initial program 79.5%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in D around 0 70.5%

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

   if 1.04999999999999999e75 < M

   1. Initial program 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in D around 0 32.7%

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

      1. *-commutative 32.7%

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

      2. associate-*r* 36.3%

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

   7. Simplified 36.3%

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

      1. expm1-log1p-u 36.3%

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

      2. expm1-udef 32.7%

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

      3. pow-prod-down 61.4%

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

   9. Applied egg-rr 61.4%

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

       1. expm1-def 65.0%

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

       2. expm1-log1p 65.0%

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

   11. Simplified 65.0%

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

   12. Taylor expanded in D around inf 22.6%

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

       1. associate-*r* 22.4%

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

       2. unpow2 22.4%

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

       3. unpow2 22.4%

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

       4. swap-sqr 27.1%

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

       5. unpow2 27.1%

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

       6. *-commutative 27.1%

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

       7. associate-/l* 27.1%

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

       8. *-commutative 27.1%

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

   14. Simplified 27.1%

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

       1. unpow2 27.1%

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

   16. Applied egg-rr 27.1%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.05 \cdot 10^{+75}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{\ell \cdot {d}^{2}}{h \cdot w0}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

d_m = abs(d)
real(8) function code(w0, m, d, h, l, d_m)
    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_m
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[w0_, M_, D_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 79.5%

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

3. Simplified 79.1%

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

5. Taylor expanded in D around 0 67.8%

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

6. Final simplification 67.8%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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))))))