Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 86.4%

Time: 10.7s

Alternatives: 8

Speedup: 0.5×

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: 81.0% 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: 86.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t\_0 \leq 10^{-11}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 1e-11) (* w0 (sqrt (- 1.0 t_0))) w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 1e-11) {
		tmp = w0 * sqrt((1.0 - t_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) :: t_0
    real(8) :: tmp
    t_0 = (((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)
    if (t_0 <= 1d-11) then
        tmp = w0 * sqrt((1.0d0 - t_0))
    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 t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 1e-11) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= 1e-11:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= 1e-11)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (((M * D) / (2.0 * d)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= 1e-11)
		tmp = w0 * sqrt((1.0 - t_0));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-11], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $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)) < 9.99999999999999939e-12

   1. Initial program 91.1%

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

   if 9.99999999999999939e-12 < (*.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%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 72.6%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-11}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+18}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot 0.25\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+18)
   (* w0 (sqrt (- 1.0 (/ (* (* D D) (* (* M (* M h)) 0.25)) (* d (* d l))))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+18) {
		tmp = w0 * sqrt((1.0 - (((D * D) * ((M * (M * h)) * 0.25)) / (d * (d * l)))));
	} 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 (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d+18)) then
        tmp = w0 * sqrt((1.0d0 - (((d * d) * ((m * (m * h)) * 0.25d0)) / (d_1 * (d_1 * l)))))
    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(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+18) {
		tmp = w0 * Math.sqrt((1.0 - (((D * D) * ((M * (M * h)) * 0.25)) / (d * (d * l)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+18:
		tmp = w0 * math.sqrt((1.0 - (((D * D) * ((M * (M * h)) * 0.25)) / (d * (d * l)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+18)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D * D) * Float64(Float64(M * Float64(M * h)) * 0.25)) / Float64(d * Float64(d * l))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e+18)
		tmp = w0 * sqrt((1.0 - (((D * D) * ((M * (M * h)) * 0.25)) / (d * (d * l)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+18], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D * D), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $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)) < -2e18

   1. Initial program 73.4%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 55.4

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

   5. Simplified 55.4%

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

   6. Taylor expanded in h around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 54.3

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

   8. Simplified 54.3%

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

   9. Taylor expanded in M around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. associate-*l* N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 52.4

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

   11. Simplified 52.4%

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

   12. Taylor expanded in D around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 56.3

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

   14. Simplified 56.3%

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

   if -2e18 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.9%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 95.7%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+18}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot 0.25\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(h \cdot \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e-15)
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (* h (* D (* D (* M M))))) (* l (* d d))))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-15) {
		tmp = w0 * sqrt((1.0 - ((0.25 * (h * (D * (D * (M * M))))) / (l * (d * d)))));
	} 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 (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-2d-15)) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * (h * (d * (d * (m * m))))) / (l * (d_1 * d_1)))))
    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(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-15) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * (h * (D * (D * (M * M))))) / (l * (d * d)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e-15:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * (h * (D * (D * (M * M))))) / (l * (d * d)))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e-15)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(h * Float64(D * Float64(D * Float64(M * M))))) / Float64(l * Float64(d * d))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -2e-15)
		tmp = w0 * sqrt((1.0 - ((0.25 * (h * (D * (D * (M * M))))) / (l * (d * d)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e-15], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(h * N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $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)) < -2.0000000000000002e-15

   1. Initial program 74.1%

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

   3. Taylor expanded in M around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 54.9

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

   5. Simplified 54.9%

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

   if -2.0000000000000002e-15 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.8%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 95.8%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(h \cdot \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \left(M \cdot \left(M \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}\right)\right) \cdot 0.125, -1\right) \cdot \left(-w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+207)
   (* (fma (* D D) (* (* M (* M (/ h (* l (* d d))))) 0.125) -1.0) (- w0))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+207) {
		tmp = fma((D * D), ((M * (M * (h / (l * (d * d))))) * 0.125), -1.0) * -w0;
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+207)
		tmp = Float64(fma(Float64(D * D), Float64(Float64(M * Float64(M * Float64(h / Float64(l * Float64(d * d))))) * 0.125), -1.0) * Float64(-w0));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+207], N[(N[(N[(D * D), $MachinePrecision] * N[(N[(M * N[(M * N[(h / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] + -1.0), $MachinePrecision] * (-w0)), $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)) < -2.0000000000000001e207

   1. Initial program 68.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

   5. Simplified 49.3%

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

   6. Taylor expanded in w0 around -inf

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot w0\right) \cdot \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} - 1\right) \cdot \left(-1 \cdot w0\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} - 1\right) \cdot \left(-1 \cdot w0\right)} \]

   8. Simplified 52.5%

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

   if -2.0000000000000001e207 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.8%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 89.7%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \left(M \cdot \left(M \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}\right)\right) \cdot 0.125, -1\right) \cdot \left(-w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\left(-w0\right) \cdot \mathsf{fma}\left(D \cdot D, 0.125 \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+207)
   (* (- w0) (fma (* D D) (* 0.125 (* M (* M (/ h (* d (* d l)))))) -1.0))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+207) {
		tmp = -w0 * fma((D * D), (0.125 * (M * (M * (h / (d * (d * l)))))), -1.0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+207)
		tmp = Float64(Float64(-w0) * fma(Float64(D * D), Float64(0.125 * Float64(M * Float64(M * Float64(h / Float64(d * Float64(d * l)))))), -1.0));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+207], N[((-w0) * N[(N[(D * D), $MachinePrecision] * N[(0.125 * N[(M * N[(M * N[(h / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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)) < -2.0000000000000001e207

   1. Initial program 68.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

   5. Simplified 49.3%

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

   6. Taylor expanded in w0 around -inf

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot w0\right) \cdot \left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} - 1\right) \cdot \left(-1 \cdot w0\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} - 1\right) \cdot \left(-1 \cdot w0\right)} \]

   8. Simplified 52.5%

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

   9. Taylor expanded in d around 0

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

       1. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \left(M \cdot \left(M \cdot \frac{h}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right)\right) \cdot \frac{1}{8}, -1\right) \cdot \left(\mathsf{neg}\left(w0\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \left(M \cdot \left(M \cdot \frac{h}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}\right)\right) \cdot \frac{1}{8}, -1\right) \cdot \left(\mathsf{neg}\left(w0\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \left(M \cdot \left(M \cdot \frac{h}{d \cdot \color{blue}{\left(\ell \cdot d\right)}}\right)\right) \cdot \frac{1}{8}, -1\right) \cdot \left(\mathsf{neg}\left(w0\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \left(M \cdot \left(M \cdot \frac{h}{\color{blue}{d \cdot \left(\ell \cdot d\right)}}\right)\right) \cdot \frac{1}{8}, -1\right) \cdot \left(\mathsf{neg}\left(w0\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(D \cdot D, \left(M \cdot \left(M \cdot \frac{h}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right)\right) \cdot \frac{1}{8}, -1\right) \cdot \left(\mathsf{neg}\left(w0\right)\right) \]

       6. lower-*.f64 52.5

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

   11. Simplified 52.5%

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

   if -2.0000000000000001e207 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.8%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 89.7%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\left(-w0\right) \cdot \mathsf{fma}\left(D \cdot D, 0.125 \cdot \left(M \cdot \left(M \cdot \frac{h}{d \cdot \left(d \cdot \ell\right)}\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+207)
   (fma (* D D) (/ (* -0.125 (* h (* M (* M w0)))) (* l (* d d))) w0)
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+207) {
		tmp = fma((D * D), ((-0.125 * (h * (M * (M * w0)))) / (l * (d * d))), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+207)
		tmp = fma(Float64(D * D), Float64(Float64(-0.125 * Float64(h * Float64(M * Float64(M * w0)))) / Float64(l * Float64(d * d))), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+207], N[(N[(D * D), $MachinePrecision] * N[(N[(-0.125 * N[(h * N[(M * N[(M * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $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)) < -2.0000000000000001e207

   1. Initial program 68.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}, w0\right)} \]

   5. Simplified 49.3%

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

   if -2.0000000000000001e207 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.8%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 89.7%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(D \cdot D, \frac{-0.125 \cdot \left(h \cdot \left(M \cdot \left(M \cdot w0\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+274}:\\ \;\;\;\;w0 \cdot \frac{-0.125 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -4e+274)
   (* w0 (/ (* -0.125 (* D (* D (* M (* M h))))) (* d (* d l))))
   w0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e+274) {
		tmp = w0 * ((-0.125 * (D * (D * (M * (M * h))))) / (d * (d * l)));
	} 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 (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-4d+274)) then
        tmp = w0 * (((-0.125d0) * (d * (d * (m * (m * h))))) / (d_1 * (d_1 * l)))
    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(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e+274) {
		tmp = w0 * ((-0.125 * (D * (D * (M * (M * h))))) / (d * (d * l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -4e+274:
		tmp = w0 * ((-0.125 * (D * (D * (M * (M * h))))) / (d * (d * l)))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -4e+274)
		tmp = Float64(w0 * Float64(Float64(-0.125 * Float64(D * Float64(D * Float64(M * Float64(M * h))))) / Float64(d * Float64(d * l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -4e+274)
		tmp = w0 * ((-0.125 * (D * (D * (M * (M * h))))) / (d * (d * l)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -4e+274], N[(w0 * N[(N[(-0.125 * N[(D * N[(D * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $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)) < -3.99999999999999969e274

   1. Initial program 64.8%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left({D}^{2}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right)} \]

      7. unpow2 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\color{blue}{D \cdot D}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right) \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(\color{blue}{D \cdot D}, \frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, 1\right) \]

      9. associate-*r/ N/A

         \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}, 1\right) \]

      10. lower-/.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \color{blue}{\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\color{blue}{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell}, 1\right) \]

      13. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}, 1\right) \]

      14. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot h\right)}{{d}^{2} \cdot \ell}, 1\right) \]

      15. lower-*.f64 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}, 1\right) \]

      16. unpow2 N/A

          \[\leadsto w0 \cdot \mathsf{fma}\left(D \cdot D, \frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}, 1\right) \]

      17. lower-*.f64 52.5

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

   5. Simplified 52.5%

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

   6. Taylor expanded in D around inf

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

      1. associate-*r/ N/A

         \[\leadsto w0 \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto w0 \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}} \]

      3. *-commutative N/A

         \[\leadsto w0 \cdot \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{-1}{8}}}{{d}^{2} \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto w0 \cdot \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{-1}{8}}}{{d}^{2} \cdot \ell} \]

      5. *-commutative N/A

         \[\leadsto w0 \cdot \frac{\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)} \cdot \frac{-1}{8}}{{d}^{2} \cdot \ell} \]

      6. unpow2 N/A

         \[\leadsto w0 \cdot \frac{\left(\left({M}^{2} \cdot h\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{-1}{8}}{{d}^{2} \cdot \ell} \]

      7. associate-*r* N/A

         \[\leadsto w0 \cdot \frac{\color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)} \cdot \frac{-1}{8}}{{d}^{2} \cdot \ell} \]

      8. lower-*.f64 N/A

         \[\leadsto w0 \cdot \frac{\color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot D\right)} \cdot \frac{-1}{8}}{{d}^{2} \cdot \ell} \]

      9. lower-*.f64 N/A

         \[\leadsto w0 \cdot \frac{\left(\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)} \cdot D\right) \cdot \frac{-1}{8}}{{d}^{2} \cdot \ell} \]

      10. lower-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\color{blue}{\left({M}^{2} \cdot h\right)} \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{{d}^{2} \cdot \ell} \]

      11. unpow2 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{{d}^{2} \cdot \ell} \]

      12. lower-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{{d}^{2} \cdot \ell} \]

      13. lower-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{\color{blue}{{d}^{2} \cdot \ell}} \]

      14. unpow2 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

      15. lower-*.f64 54.6

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

   8. Simplified 54.6%

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

   9. Taylor expanded in d around 0

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

       1. unpow2 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{\color{blue}{\left(d \cdot d\right)} \cdot \ell} \]

       2. associate-*l* N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{\color{blue}{d \cdot \left(d \cdot \ell\right)}} \]

       3. *-commutative N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{d \cdot \color{blue}{\left(\ell \cdot d\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{\color{blue}{d \cdot \left(\ell \cdot d\right)}} \]

       5. *-commutative N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

       6. lower-*.f64 54.6

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

   11. Simplified 54.6%

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

   12. Taylor expanded in M around 0

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

       1. unpow2 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\left(\color{blue}{\left(M \cdot M\right)} \cdot h\right) \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{d \cdot \left(d \cdot \ell\right)} \]

       2. associate-*l* N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\color{blue}{\left(M \cdot \left(M \cdot h\right)\right)} \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{d \cdot \left(d \cdot \ell\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto w0 \cdot \frac{\left(\left(\color{blue}{\left(M \cdot \left(M \cdot h\right)\right)} \cdot D\right) \cdot D\right) \cdot \frac{-1}{8}}{d \cdot \left(d \cdot \ell\right)} \]

       4. lower-*.f64 58.1

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

   14. Simplified 58.1%

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

   if -3.99999999999999969e274 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.2%

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

   3. Taylor expanded in M around 0

      \[\leadsto w0 \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 87.2%

      \[\leadsto w0 \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -4 \cdot 10^{+274}:\\ \;\;\;\;w0 \cdot \frac{-0.125 \cdot \left(D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.2% accurate, 157.0× 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 82.2%

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

3. Taylor expanded in M around 0

   \[\leadsto w0 \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 69.2%

   \[\leadsto w0 \cdot \color{blue}{1} \]

6. Final simplification 69.2%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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))))))