Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 89.1%

Time: 14.0s

Alternatives: 5

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: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (* (/ (* M (* D 0.5)) d) (/ h (* l (* 2.0 (/ d (* M D))))))))))

C

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

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 * 0.5d0)) / d_1) * (h / (l * (2.0d0 * (d_1 / (m * d))))))))
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 - (((M * (D * 0.5)) / d) * (h / (l * (2.0 * (d / (M * D))))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(M * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(h / N[(l * N[(2.0 * N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

3. Simplified 82.6%

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

   1. unpow2 82.6%

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

   2. associate-*r/ 82.2%

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

   3. clear-num 82.2%

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

   4. un-div-inv 82.2%

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

   5. *-un-lft-identity 82.2%

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

   6. times-frac 82.2%

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

   7. metadata-eval 82.2%

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

   8. times-frac 82.2%

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

6. Applied egg-rr 82.2%

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

   1. frac-times 90.3%

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

   2. associate-*r/ 90.3%

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

8. Applied egg-rr 90.3%

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

   1. associate-/l* 90.7%

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

   2. associate-*r/ 90.4%

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

   3. *-commutative 90.4%

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

   4. *-commutative 90.4%

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

   5. associate-*l/ 90.4%

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

   6. associate-*r/ 90.4%

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

10. Simplified 90.4%

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

11. Taylor expanded in d around 0 92.1%

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

12. Final simplification 92.1%

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

Alternative 2: 74.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 1.6 \cdot 10^{-74}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(M \cdot \frac{D \cdot 0.5}{d}\right) \cdot \left(0.5 \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 1.6e-74)
   w0
   (*
    w0
    (sqrt
     (- 1.0 (* (* M (/ (* D 0.5) d)) (* 0.5 (/ (* D (* M h)) (* d l)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 1.6e-74) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((M * ((D * 0.5) / d)) * (0.5 * ((D * (M * h)) / (d * l))))));
	}
	return tmp;
}

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) :: tmp
    if (d <= 1.6d-74) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((m * ((d * 0.5d0) / d_1)) * (0.5d0 * ((d * (m * h)) / (d_1 * l))))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 1.6e-74) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M * ((D * 0.5) / d)) * (0.5 * ((D * (M * h)) / (d * l))))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 1.6e-74:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((M * ((D * 0.5) / d)) * (0.5 * ((D * (M * h)) / (d * l))))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 1.6e-74)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M * Float64(Float64(D * 0.5) / d)) * Float64(0.5 * Float64(Float64(D * Float64(M * h)) / Float64(d * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 1.6e-74)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((M * ((D * 0.5) / d)) * (0.5 * ((D * (M * h)) / (d * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 1.6e-74], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M * N[(N[(D * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(D * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.5999999999999999e-74

   1. Initial program 82.9%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in M around 0 73.2%

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

   if 1.5999999999999999e-74 < D

   1. Initial program 81.7%

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

   3. Simplified 80.4%

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

      1. unpow2 80.4%

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

      2. associate-*r/ 80.4%

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

      3. clear-num 80.4%

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

      4. un-div-inv 80.4%

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

      5. *-un-lft-identity 80.4%

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

      6. times-frac 80.4%

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

      7. metadata-eval 80.4%

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

      8. times-frac 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. frac-times 91.0%

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

      2. associate-*r/ 91.0%

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

   8. Applied egg-rr 91.0%

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

      1. associate-/l* 91.0%

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

      2. associate-*r/ 91.0%

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

      3. *-commutative 91.0%

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

      4. *-commutative 91.0%

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

      5. associate-*l/ 91.0%

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

      6. associate-*r/ 91.0%

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

   10. Simplified 91.0%

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

       1. associate-/l* 91.0%

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

   12. Applied egg-rr 91.0%

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

   13. Taylor expanded in h around 0 88.3%

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

Alternative 3: 64.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(w0 \cdot h\right)}{\ell \cdot {d}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3.8e+36)
   w0
   (* -0.125 (/ (* (* (* M D) (* M D)) (* w0 h)) (* l (pow d 2.0))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.8e+36) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((M * D) * (M * D)) * (w0 * h)) / (l * pow(d, 2.0)));
	}
	return tmp;
}

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) :: tmp
    if (m <= 3.8d+36) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((((m * d) * (m * d)) * (w0 * h)) / (l * (d_1 ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.8e+36) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((M * D) * (M * D)) * (w0 * h)) / (l * Math.pow(d, 2.0)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3.8e+36:
		tmp = w0
	else:
		tmp = -0.125 * ((((M * D) * (M * D)) * (w0 * h)) / (l * math.pow(d, 2.0)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.8e+36)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(w0 * h)) / Float64(l * (d ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3.8e+36)
		tmp = w0;
	else
		tmp = -0.125 * ((((M * D) * (M * D)) * (w0 * h)) / (l * (d ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3.8e+36], w0, N[(-0.125 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.80000000000000025e36

   1. Initial program 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in M around 0 71.1%

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

   if 3.80000000000000025e36 < M

   1. Initial program 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in M around 0 39.5%

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

      1. +-commutative 39.5%

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

      2. *-commutative 39.5%

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

      3. fma-define 39.5%

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

      4. associate-*r* 50.0%

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

      5. *-commutative 50.0%

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

      6. unpow2 50.0%

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

      7. unpow2 50.0%

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

      8. swap-sqr 71.4%

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

      9. unpow2 71.4%

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

      10. *-commutative 71.4%

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

   7. Simplified 71.4%

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

   8. Taylor expanded in D around inf 19.6%

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

      1. associate-*r* 19.7%

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

      2. unpow2 19.7%

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

      3. unpow2 19.7%

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

      4. swap-sqr 24.2%

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

      5. unpow2 24.2%

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

      6. *-commutative 24.2%

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

   10. Simplified 24.2%

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

       1. *-commutative 24.2%

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

       2. pow2 24.2%

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

   12. Applied egg-rr 24.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \left(w0 \cdot h\right)}{\ell \cdot {d}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (* (* M (/ (* D 0.5) d)) (/ h (* l (/ (/ (* d 2.0) D) M))))))))

C

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

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 * 0.5d0) / d_1)) * (h / (l * (((d_1 * 2.0d0) / d) / m))))))
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 - ((M * ((D * 0.5) / d)) * (h / (l * (((d * 2.0) / D) / M))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M * N[(N[(D * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(h / N[(l * N[(N[(N[(d * 2.0), $MachinePrecision] / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

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

3. Simplified 82.6%

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

   1. unpow2 82.6%

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

   2. associate-*r/ 82.2%

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

   3. clear-num 82.2%

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

   4. un-div-inv 82.2%

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

   5. *-un-lft-identity 82.2%

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

   6. times-frac 82.2%

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

   7. metadata-eval 82.2%

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

   8. times-frac 82.2%

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

6. Applied egg-rr 82.2%

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

   1. frac-times 90.3%

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

   2. associate-*r/ 90.3%

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

8. Applied egg-rr 90.3%

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

   1. associate-/l* 90.7%

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

   2. associate-*r/ 90.4%

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

   3. *-commutative 90.4%

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

   4. *-commutative 90.4%

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

   5. associate-*l/ 90.4%

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

   6. associate-*r/ 90.4%

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

10. Simplified 90.4%

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

    1. associate-/l* 90.7%

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

12. Applied egg-rr 90.7%

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

13. Final simplification 90.7%

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

Alternative 5: 68.2% 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 82.6%

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

3. Simplified 82.6%

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

5. Taylor expanded in M around 0 68.4%

   \[\leadsto \color{blue}{w0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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))))))