Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 86.6%

Time: 17.4s

Alternatives: 9

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ h l)) 0.01)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (* D M) (/ 0.5 d)) 2.0)))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((D * M) / (2.0 * d)), 2.0) * (h / l)) <= 0.01) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D * M) * (0.5 / d)), 2.0))));
	} else {
		tmp = w0;
	}
	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 * m) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= 0.01d0) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d * m) * (0.5d0 / d_1)) ** 2.0d0))))
    else
        tmp = w0
    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 ((Math.pow(((D * M) / (2.0 * d)), 2.0) * (h / l)) <= 0.01) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((D * M) * (0.5 / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((D * M) / (2.0 * d)), 2.0) * (h / l)) <= 0.01:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((D * M) * (0.5 / d)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= 0.01)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D * M) * Float64(0.5 / d)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((D * M) / (2.0 * d)) ^ 2.0) * (h / l)) <= 0.01)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((D * M) * (0.5 / d)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 0.01], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * M), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)) < 0.0100000000000000002

   1. Initial program 86.5%

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

      1. clear-num 86.5%

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

      2. associate-/r/ 86.5%

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

      3. associate-/r* 86.5%

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

      4. metadata-eval 86.5%

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

      5. *-commutative 86.5%

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

   4. Applied egg-rr 86.5%

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

   if 0.0100000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.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 0.0%

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

   5. Taylor expanded in D around 0 81.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.01:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\left(D \cdot M\right) \cdot \frac{0.5}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 1.0× speedup

Math

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

FPCore

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

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - ((math.pow((((D * M) / 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(Float64(Float64(D * M) / 2.0) / d) ^ 2.0) * h) / l))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((((D * M) / 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[(N[Power[N[(N[(N[(D * M), $MachinePrecision] / 2.0), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

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

3. Simplified 74.9%

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

   1. associate-*r/ 84.3%

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

   2. associate-/r* 84.3%

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

6. Applied egg-rr 84.3%

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

   1. associate-*r/ 85.0%

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

8. Applied egg-rr 85.0%

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

   1. associate-/r* 85.0%

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

10. Simplified 85.0%

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

Alternative 3: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((h * pow((D * (M / (2.0 * d))), 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 * sqrt((1.0d0 - ((h * ((d * (m / (2.0d0 * d_1))) ** 2.0d0)) / 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 - ((h * Math.pow((D * (M / (2.0 * d))), 2.0)) / l)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Simplified 74.9%

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

   1. associate-*r/ 84.3%

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

   2. associate-/r* 84.3%

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

6. Applied egg-rr 84.3%

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

7. Final simplification 84.3%

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

Alternative 4: 85.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := D \cdot \frac{M}{2 \cdot d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{w0 \cdot h}{\ell \cdot {d}^{2}}\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-306}:\\ \;\;\;\;w0 \cdot \sqrt{\left(2 - \frac{h}{\ell} \cdot \left(t\_0 \cdot t\_0\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* D (/ M (* 2.0 d)))))
   (if (<= (/ h l) (- INFINITY))
     (+ w0 (* -0.125 (* (* D (* M (* D M))) (/ (* w0 h) (* l (pow d 2.0))))))
     (if (<= (/ h l) -2e-306)
       (* w0 (sqrt (+ (- 2.0 (* (/ h l) (* t_0 t_0))) -1.0)))
       w0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D * (M / (2.0 * d));
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 + (-0.125 * ((D * (M * (D * M))) * ((w0 * h) / (l * pow(d, 2.0)))));
	} else if ((h / l) <= -2e-306) {
		tmp = w0 * sqrt(((2.0 - ((h / l) * (t_0 * t_0))) + -1.0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D * (M / (2.0 * d));
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 + (-0.125 * ((D * (M * (D * M))) * ((w0 * h) / (l * Math.pow(d, 2.0)))));
	} else if ((h / l) <= -2e-306) {
		tmp = w0 * Math.sqrt(((2.0 - ((h / l) * (t_0 * t_0))) + -1.0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = D * (M / (2.0 * d))
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 + (-0.125 * ((D * (M * (D * M))) * ((w0 * h) / (l * math.pow(d, 2.0)))))
	elif (h / l) <= -2e-306:
		tmp = w0 * math.sqrt(((2.0 - ((h / l) * (t_0 * t_0))) + -1.0))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(D * Float64(M / Float64(2.0 * d)))
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(D * Float64(M * Float64(D * M))) * Float64(Float64(w0 * h) / Float64(l * (d ^ 2.0))))));
	elseif (Float64(h / l) <= -2e-306)
		tmp = Float64(w0 * sqrt(Float64(Float64(2.0 - Float64(Float64(h / l) * Float64(t_0 * t_0))) + -1.0)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = D * (M / (2.0 * d));
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0 + (-0.125 * ((D * (M * (D * M))) * ((w0 * h) / (l * (d ^ 2.0)))));
	elseif ((h / l) <= -2e-306)
		tmp = w0 * sqrt(((2.0 - ((h / l) * (t_0 * t_0))) + -1.0));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 + N[(-0.125 * N[(N[(D * N[(M * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -2e-306], N[(w0 * N[Sqrt[N[(N[(2.0 - N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 22.9%

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

   3. Simplified 22.9%

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

   5. Taylor expanded in D around 0 48.1%

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

      1. +-commutative 48.1%

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

      2. fma-define 48.1%

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

      3. associate-*r* 56.8%

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

      4. unpow2 56.8%

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

      5. unpow2 56.8%

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

      6. swap-sqr 62.5%

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

      7. unpow2 62.5%

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

      8. *-commutative 62.5%

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

   7. Simplified 62.5%

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

      1. fma-undefine 62.5%

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

      2. associate-/l* 62.5%

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

      3. *-commutative 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. add-exp-log 62.5%

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

       2. log-pow 26.9%

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

   11. Applied egg-rr 26.9%

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

       1. *-commutative 26.9%

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

       2. pow-to-exp 62.5%

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

       3. pow2 62.5%

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

       4. associate-*l* 62.5%

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

   13. Applied egg-rr 62.5%

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

   if -inf.0 < (/.f64 h l) < -2.00000000000000006e-306

   1. Initial program 81.0%

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

   3. Simplified 78.8%

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

      1. associate-*r/ 78.0%

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

      2. associate-/r* 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. associate-*r/ 79.6%

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

   8. Applied egg-rr 79.6%

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

      1. associate-/r* 79.6%

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

   10. Simplified 79.6%

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

       1. expm1-log1p-u 78.5%

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

       2. *-commutative 78.5%

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

       3. associate-/l* 78.5%

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

       4. associate-*r/ 77.2%

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

       5. associate-*l/ 77.8%

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

       6. associate-/l/ 77.8%

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

   12. Applied egg-rr 77.8%

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

       1. expm1-undefine 77.8%

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

       2. sub-neg 77.8%

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

       3. log1p-undefine 77.8%

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

       4. rem-exp-log 78.8%

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

       5. associate-+r- 78.8%

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

       6. metadata-eval 78.8%

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

       7. associate-*r/ 81.0%

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

       8. *-commutative 81.0%

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

       9. metadata-eval 81.0%

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

   14. Simplified 81.0%

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

       1. unpow2 81.0%

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

       2. associate-/l* 78.1%

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

       3. associate-/l* 78.8%

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

   16. Applied egg-rr 78.8%

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

   if -2.00000000000000006e-306 < (/.f64 h l)

   1. Initial program 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in D around 0 95.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{w0 \cdot h}{\ell \cdot {d}^{2}}\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-306}:\\ \;\;\;\;w0 \cdot \sqrt{\left(2 - \frac{h}{\ell} \cdot \left(\left(D \cdot \frac{M}{2 \cdot d}\right) \cdot \left(D \cdot \frac{M}{2 \cdot d}\right)\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 6.8 \cdot 10^{+145}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{w0 \cdot h}{\ell \cdot {d}^{2}} \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 6.8e+145)
   w0
   (+ w0 (* -0.125 (* (/ (* w0 h) (* l (pow d 2.0))) (* (* D M) (* D M)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 6.8e+145) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((w0 * h) / (l * pow(d, 2.0))) * ((D * M) * (D * M))));
	}
	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 <= 6.8d+145) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((w0 * h) / (l * (d_1 ** 2.0d0))) * ((d * m) * (d * m))))
    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 <= 6.8e+145) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((w0 * h) / (l * Math.pow(d, 2.0))) * ((D * M) * (D * M))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 6.8e+145:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((w0 * h) / (l * math.pow(d, 2.0))) * ((D * M) * (D * M))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 6.8e+145)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(w0 * h) / Float64(l * (d ^ 2.0))) * Float64(Float64(D * M) * Float64(D * M)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 6.8e+145)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((w0 * h) / (l * (d ^ 2.0))) * ((D * M) * (D * M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 6.8e+145], w0, N[(w0 + N[(-0.125 * N[(N[(N[(w0 * h), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 6.7999999999999998e145

   1. Initial program 77.1%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in D around 0 68.0%

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

   if 6.7999999999999998e145 < D

   1. Initial program 67.6%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in D around 0 33.8%

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

      1. +-commutative 33.8%

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

      2. fma-define 33.8%

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

      3. associate-*r* 33.8%

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

      4. unpow2 33.8%

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

      5. unpow2 33.8%

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

      6. swap-sqr 54.4%

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

      7. unpow2 54.4%

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

      8. *-commutative 54.4%

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

   7. Simplified 54.4%

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

      1. fma-undefine 54.4%

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

      2. associate-/l* 57.4%

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

      3. *-commutative 57.4%

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

   9. Applied egg-rr 57.4%

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

       1. unpow2 57.4%

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

   11. Applied egg-rr 57.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 6.8 \cdot 10^{+145}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{w0 \cdot h}{\ell \cdot {d}^{2}} \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 4.6 \cdot 10^{+146}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{w0 \cdot h}{\ell \cdot {d}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 4.6e+146)
   w0
   (+ w0 (* -0.125 (* (* D (* M (* D M))) (/ (* 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 (D <= 4.6e+146) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((D * (M * (D * M))) * ((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 (d <= 4.6d+146) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((d * (m * (d * m))) * ((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 (D <= 4.6e+146) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((D * (M * (D * M))) * ((w0 * h) / (l * Math.pow(d, 2.0)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 4.6e+146:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((D * (M * (D * M))) * ((w0 * h) / (l * math.pow(d, 2.0)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 4.6e+146)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(D * Float64(M * Float64(D * M))) * Float64(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 (D <= 4.6e+146)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((D * (M * (D * M))) * ((w0 * h) / (l * (d ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 4.6e+146], w0, N[(w0 + N[(-0.125 * N[(N[(D * N[(M * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] / N[(l * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.60000000000000001e146

   1. Initial program 77.2%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in D around 0 67.7%

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

   if 4.60000000000000001e146 < D

   1. Initial program 66.5%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in D around 0 31.5%

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

      1. +-commutative 31.5%

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

      2. fma-define 31.5%

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

      3. associate-*r* 31.6%

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

      4. unpow2 31.6%

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

      5. unpow2 31.6%

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

      6. swap-sqr 52.8%

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

      7. unpow2 52.8%

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

      8. *-commutative 52.8%

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

   7. Simplified 52.8%

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

      1. fma-undefine 52.8%

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

      2. associate-/l* 55.9%

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

      3. *-commutative 55.9%

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

   9. Applied egg-rr 55.9%

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

       1. add-exp-log 55.9%

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

       2. log-pow 21.2%

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

   11. Applied egg-rr 21.2%

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

       1. *-commutative 21.2%

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

       2. pow-to-exp 55.9%

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

       3. pow2 55.9%

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

       4. associate-*l* 55.9%

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

   13. Applied egg-rr 55.9%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 4.6 \cdot 10^{+146}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(D \cdot \left(M \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{w0 \cdot h}{\ell \cdot {d}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{+101}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{w0 \cdot \left(h \cdot {d}^{-2}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.6e+101)
   w0
   (* -0.125 (* (* (* D M) (* D M)) (/ (* w0 (* h (pow d -2.0))) l)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.6e+101) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * M) * (D * M)) * ((w0 * (h * pow(d, -2.0))) / 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 (m <= 2.6d+101) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * m) * (d * m)) * ((w0 * (h * (d_1 ** (-2.0d0)))) / 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 (M <= 2.6e+101) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * M) * (D * M)) * ((w0 * (h * Math.pow(d, -2.0))) / l));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.6e+101:
		tmp = w0
	else:
		tmp = -0.125 * (((D * M) * (D * M)) * ((w0 * (h * math.pow(d, -2.0))) / l))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.6e+101)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * M) * Float64(D * M)) * Float64(Float64(w0 * Float64(h * (d ^ -2.0))) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.6e+101)
		tmp = w0;
	else
		tmp = -0.125 * (((D * M) * (D * M)) * ((w0 * (h * (d ^ -2.0))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.6e+101], w0, N[(-0.125 * N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(h * N[Power[d, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.6e101

   1. Initial program 77.3%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in D around 0 68.4%

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

   if 2.6e101 < M

   1. Initial program 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in D around 0 30.0%

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

      1. +-commutative 30.0%

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

      2. fma-define 30.0%

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

      3. associate-*r* 32.4%

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

      4. unpow2 32.4%

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

      5. unpow2 32.4%

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

      6. swap-sqr 50.0%

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

      7. unpow2 50.0%

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

      8. *-commutative 50.0%

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

   7. Simplified 50.0%

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

   8. Taylor expanded in D around inf 20.5%

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

      1. associate-*r* 20.6%

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

      2. unpow2 20.6%

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

      3. unpow2 20.6%

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

      4. swap-sqr 21.7%

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

      5. unpow2 21.7%

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

      6. *-commutative 21.7%

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

      7. *-commutative 21.7%

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

      8. associate-*r/ 21.5%

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

      9. times-frac 24.1%

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

   10. Simplified 24.1%

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

       1. unpow2 50.9%

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

   12. Applied egg-rr 24.1%

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

       1. associate-*l/ 24.3%

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

       2. div-inv 24.3%

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

       3. pow-flip 26.7%

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

       4. metadata-eval 26.7%

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

   14. Applied egg-rr 26.7%

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

Alternative 8: 66.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 7.8 \cdot 10^{+98}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \left({d}^{-2} \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 7.8e+98)
   w0
   (* -0.125 (* (* (* D M) (* D M)) (* (pow d -2.0) (* w0 (/ h l)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 7.8e+98) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * M) * (D * M)) * (pow(d, -2.0) * (w0 * (h / 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 (m <= 7.8d+98) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * m) * (d * m)) * ((d_1 ** (-2.0d0)) * (w0 * (h / 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 (M <= 7.8e+98) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * M) * (D * M)) * (Math.pow(d, -2.0) * (w0 * (h / l))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 7.8e+98:
		tmp = w0
	else:
		tmp = -0.125 * (((D * M) * (D * M)) * (math.pow(d, -2.0) * (w0 * (h / l))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 7.8e+98)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * M) * Float64(D * M)) * Float64((d ^ -2.0) * Float64(w0 * Float64(h / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 7.8e+98)
		tmp = w0;
	else
		tmp = -0.125 * (((D * M) * (D * M)) * ((d ^ -2.0) * (w0 * (h / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 7.8e+98], w0, N[(-0.125 * N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(N[Power[d, -2.0], $MachinePrecision] * N[(w0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.7999999999999999e98

   1. Initial program 77.3%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in D around 0 68.4%

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

   if 7.7999999999999999e98 < M

   1. Initial program 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in D around 0 30.0%

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

      1. +-commutative 30.0%

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

      2. fma-define 30.0%

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

      3. associate-*r* 32.4%

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

      4. unpow2 32.4%

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

      5. unpow2 32.4%

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

      6. swap-sqr 50.0%

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

      7. unpow2 50.0%

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

      8. *-commutative 50.0%

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

   7. Simplified 50.0%

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

   8. Taylor expanded in D around inf 20.5%

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

      1. associate-*r* 20.6%

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

      2. unpow2 20.6%

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

      3. unpow2 20.6%

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

      4. swap-sqr 21.7%

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

      5. unpow2 21.7%

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

      6. *-commutative 21.7%

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

      7. *-commutative 21.7%

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

      8. associate-*r/ 21.5%

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

      9. times-frac 24.1%

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

   10. Simplified 24.1%

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

       1. unpow2 50.9%

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

   12. Applied egg-rr 24.1%

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

       1. associate-*l/ 24.3%

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

       2. div-inv 24.3%

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

       3. pow-flip 26.7%

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

       4. metadata-eval 26.7%

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

   14. Applied egg-rr 26.7%

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

       1. associate-*r* 26.4%

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

       2. associate-*l/ 26.3%

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

       3. associate-/l* 26.2%

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

   16. Simplified 26.2%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.8 \cdot 10^{+98}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot \left({d}^{-2} \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.3% 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 76.0%

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

3. Simplified 74.9%

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

5. Taylor expanded in D around 0 64.3%

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

Reproduce

herbie shell --seed 2024150 
(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))))))