Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 88.7%

Time: 16.9s

Alternatives: 4

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: 81.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

C

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

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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
    real(8) :: t_0
    t_0 = (m * d) * (0.5d0 / d_1)
    code = w0 * sqrt((1.0d0 - (t_0 * (h / (l / t_0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 79.2%

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

   1. frac-times 79.6%

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

   2. *-commutative 79.6%

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

   3. expm1-log1p-u 63.3%

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

   4. expm1-udef 63.3%

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

5. Applied egg-rr 79.6%

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

   1. add-exp-log 63.3%

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

   2. log1p-udef 63.3%

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

   3. expm1-udef 63.3%

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

   4. expm1-log1p-u 79.6%

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

   5. associate-*l/ 84.4%

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

   6. associate-/l* 84.8%

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

   7. associate-*l* 84.4%

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

7. Applied egg-rr 84.4%

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

   1. *-un-lft-identity 84.4%

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

   2. unpow2 84.4%

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

   3. times-frac 85.2%

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

   4. *-commutative 85.2%

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

   5. associate-*r* 83.4%

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

   6. *-commutative 83.4%

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

   7. *-commutative 83.4%

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

   8. associate-*r* 84.8%

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

   9. *-commutative 84.8%

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

9. Applied egg-rr 84.8%

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

    1. associate-*l/ 84.8%

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

    2. *-lft-identity 84.8%

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

11. Simplified 84.8%

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

    1. associate-/r/ 86.3%

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

    2. associate-*r* 84.8%

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

    3. associate-*r* 87.0%

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

13. Applied egg-rr 87.0%

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

14. Final simplification 87.0%

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

Alternative 2: 86.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \left(D \cdot \frac{0.5}{d}\right)\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty \lor \neg \left(\frac{h}{\ell} \leq -2 \cdot 10^{-238}\right):\\ \;\;\;\;w0 + -0.125 \cdot \left(w0 \cdot \frac{h \cdot {\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(\frac{h}{\ell} \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* M (* D (/ 0.5 d)))))
   (if (or (<= (/ h l) (- INFINITY)) (not (<= (/ h l) -2e-238)))
     (+ w0 (* -0.125 (* w0 (/ (* h (pow (/ D (/ d M)) 2.0)) l))))
     (* w0 (sqrt (- 1.0 (* t_0 (* (/ h l) t_0))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (D * (0.5 / d));
	double tmp;
	if (((h / l) <= -((double) INFINITY)) || !((h / l) <= -2e-238)) {
		tmp = w0 + (-0.125 * (w0 * ((h * pow((D / (d / M)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	}
	return tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (D * (0.5 / d));
	double tmp;
	if (((h / l) <= -Double.POSITIVE_INFINITY) || !((h / l) <= -2e-238)) {
		tmp = w0 + (-0.125 * (w0 * ((h * Math.pow((D / (d / M)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = M * (D * (0.5 / d))
	tmp = 0
	if ((h / l) <= -math.inf) or not ((h / l) <= -2e-238):
		tmp = w0 + (-0.125 * (w0 * ((h * math.pow((D / (d / M)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * ((h / l) * t_0))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(M * Float64(D * Float64(0.5 / d)))
	tmp = 0.0
	if ((Float64(h / l) <= Float64(-Inf)) || !(Float64(h / l) <= -2e-238))
		tmp = Float64(w0 + Float64(-0.125 * Float64(w0 * Float64(Float64(h * (Float64(D / Float64(d / M)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(h / l) * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = M * (D * (0.5 / d));
	tmp = 0.0;
	if (((h / l) <= -Inf) || ~(((h / l) <= -2e-238)))
		tmp = w0 + (-0.125 * (w0 * ((h * ((D / (d / M)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (t_0 * ((h / l) * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(h / l), $MachinePrecision], -2e-238]], $MachinePrecision]], N[(w0 + N[(-0.125 * N[(w0 * N[(N[(h * N[Power[N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -inf.0 or -2e-238 < (/.f64 h l)

   1. Initial program 80.3%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in D around 0 55.7%

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

      1. expm1-log1p-u 51.8%

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

      2. expm1-udef 51.8%

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

      3. associate-*r* 52.6%

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

      4. pow-prod-down 66.3%

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

      5. *-commutative 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. expm1-def 66.3%

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

      2. expm1-log1p 71.0%

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

      3. associate-*r* 74.0%

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

      4. *-commutative 74.0%

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

      5. times-frac 71.1%

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

      6. associate-/l* 71.7%

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

   8. Simplified 71.7%

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

      1. associate-*r/ 76.8%

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

      2. add-sqr-sqrt 66.5%

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

      3. pow2 66.5%

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

      4. sqrt-div 32.0%

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

      5. unpow2 32.0%

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

      6. sqrt-prod 20.3%

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

      7. add-sqr-sqrt 32.8%

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

      8. sqrt-div 32.9%

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

      9. unpow2 32.9%

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

      10. sqrt-prod 19.2%

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

      11. add-sqr-sqrt 38.0%

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

   10. Applied egg-rr 38.0%

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

       1. expm1-log1p-u 35.4%

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

       2. expm1-udef 35.4%

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

       3. associate-*r/ 32.5%

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

       4. *-commutative 32.5%

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

       5. associate-/r/ 32.5%

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

       6. unpow-prod-down 32.5%

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

       7. *-commutative 32.5%

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

       8. pow2 32.5%

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

       9. add-sqr-sqrt 79.3%

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

   12. Applied egg-rr 79.3%

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

       1. expm1-def 79.3%

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

       2. expm1-log1p 84.2%

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

       3. associate-*l/ 88.7%

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

       4. *-rgt-identity 88.7%

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

       5. associate-*r/ 88.7%

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

       6. associate-*l* 89.4%

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

       7. associate-*r/ 89.4%

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

       8. *-rgt-identity 89.4%

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

       9. *-commutative 89.4%

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

       10. *-commutative 89.4%

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

       11. associate-/l* 88.7%

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

   14. Simplified 88.7%

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

   if -inf.0 < (/.f64 h l) < -2e-238

   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%

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

      1. frac-times 78.8%

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

      2. *-commutative 78.8%

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

      3. expm1-log1p-u 50.4%

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

      4. expm1-udef 50.4%

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

   5. Applied egg-rr 78.8%

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

      1. add-exp-log 50.4%

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

      2. log1p-udef 50.4%

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

      3. expm1-udef 50.4%

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

      4. expm1-log1p-u 78.8%

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

      5. associate-*l/ 77.1%

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

      6. associate-/l* 78.0%

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

      7. associate-*l* 78.0%

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

   7. Applied egg-rr 78.0%

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

      1. *-un-lft-identity 78.0%

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

      2. unpow2 78.0%

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

      3. times-frac 78.0%

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

      4. *-commutative 78.0%

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

      5. associate-*r* 77.4%

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

      6. *-commutative 77.4%

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

      7. *-commutative 77.4%

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

      8. associate-*r* 79.0%

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

      9. *-commutative 79.0%

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

   9. Applied egg-rr 79.0%

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

       1. associate-*l/ 79.0%

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

       2. *-lft-identity 79.0%

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

   11. Simplified 79.0%

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

       1. associate-/r/ 81.4%

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

       2. associate-*r* 79.0%

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

       3. associate-*r* 79.6%

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

   13. Applied egg-rr 79.6%

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

       1. *-commutative 79.6%

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

       2. associate-*l* 79.0%

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

       3. associate-/r/ 79.8%

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

       4. associate-*l* 82.3%

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

   15. Simplified 82.3%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty \lor \neg \left(\frac{h}{\ell} \leq -2 \cdot 10^{-238}\right):\\ \;\;\;\;w0 + -0.125 \cdot \left(w0 \cdot \frac{h \cdot {\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot \left(\frac{h}{\ell} \cdot \left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)\right)}\\ \end{array} \]

Alternative 3: 80.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (+ w0 (* -0.125 (* w0 (/ (* h (pow (/ D (/ d M)) 2.0)) l)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 + (-0.125 * (w0 * ((h * pow((D / (d / M)), 2.0)) / 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 + ((-0.125d0) * (w0 * ((h * ((d / (d_1 / m)) ** 2.0d0)) / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 + N[(-0.125 * N[(w0 * N[(N[(h * N[Power[N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 79.2%

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

4. Taylor expanded in D around 0 49.4%

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

   1. expm1-log1p-u 44.0%

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

   2. expm1-udef 44.0%

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

   3. associate-*r* 45.1%

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

   4. pow-prod-down 55.4%

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

   5. *-commutative 55.4%

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

6. Applied egg-rr 55.4%

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

   1. expm1-def 55.4%

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

   2. expm1-log1p 62.5%

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

   3. associate-*r* 66.2%

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

   4. *-commutative 66.2%

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

   5. times-frac 62.4%

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

   6. associate-/l* 63.9%

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

8. Simplified 63.9%

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

   1. associate-*r/ 69.5%

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

   2. add-sqr-sqrt 58.5%

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

   3. pow2 58.5%

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

   4. sqrt-div 33.0%

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

   5. unpow2 33.0%

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

   6. sqrt-prod 20.6%

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

   7. add-sqr-sqrt 33.5%

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

   8. sqrt-div 33.6%

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

   9. unpow2 33.6%

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

   10. sqrt-prod 19.4%

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

   11. add-sqr-sqrt 38.4%

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

10. Applied egg-rr 38.4%

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

    1. expm1-log1p-u 35.2%

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

    2. expm1-udef 35.2%

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

    3. associate-*r/ 32.0%

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

    4. *-commutative 32.0%

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

    5. associate-/r/ 32.4%

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

    6. unpow-prod-down 32.3%

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

    7. *-commutative 32.3%

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

    8. pow2 32.3%

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

    9. add-sqr-sqrt 67.4%

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

12. Applied egg-rr 67.4%

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

    1. expm1-def 67.6%

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

    2. expm1-log1p 75.3%

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

    3. associate-*l/ 80.1%

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

    4. *-rgt-identity 80.1%

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

    5. associate-*r/ 80.1%

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

    6. associate-*l* 80.8%

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

    7. associate-*r/ 80.8%

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

    8. *-rgt-identity 80.8%

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

    9. *-commutative 80.8%

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

    10. *-commutative 80.8%

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

    11. associate-/l* 80.2%

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

14. Simplified 80.2%

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

15. Final simplification 80.2%

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

Alternative 4: 67.7% accurate, 216.0× speedup

Math

\[\begin{array}{l} \\ w0 \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 w0)

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

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 79.2%

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

4. Taylor expanded in D around 0 67.9%

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

5. Final simplification 67.9%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023332 
(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))))))