Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.7% → 89.3%

Time: 17.0s

Alternatives: 16

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ w0 \cdot \sqrt{1 + \frac{t\_0}{\ell} \cdot \frac{t\_0}{\frac{-1}{h}}} \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (* w0 (sqrt (+ 1.0 (* (/ t_0 l) (/ t_0 (/ -1.0 h))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (2.0 * d);
	return w0 * sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    t_0 = (m * d) / (2.0d0 * d_1)
    code = w0 * sqrt((1.0d0 + ((t_0 / l) * (t_0 / ((-1.0d0) / h)))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (2.0 * d);
	return w0 * Math.sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (M * D) / (2.0 * d)
	return w0 * math.sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))))

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d))
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(t_0 / l) * Float64(t_0 / Float64(-1.0 / h))))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	t_0 = (M * D) / (2.0 * d);
	tmp = w0 * sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
end

Wolfram

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

Derivation

1. Initial program 81.5%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. unpow2 N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. /-lowering-/.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. /-lowering-/.f64 93.2

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

4. Applied egg-rr 93.2%

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

5. Final simplification 93.2%

   \[\leadsto w0 \cdot \sqrt{1 + \frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{-1}{h}}} \]
6. Add Preprocessing

Alternative 2: 84.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 2.0) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt(fma(-0.25, (((M * D) * ((M * D) * h)) / (d * (d * l))), 1.0));
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 99.8%

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

   if 2 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 51.6%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 51.7

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

   5. Simplified 51.7%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 67.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 73.4

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

   9. Applied egg-rr 73.4%

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

4. Final simplification 89.8%

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

Alternative 3: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -20:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \left(M \cdot \left(D \cdot h\right)\right) \cdot \frac{M \cdot D}{d \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -20.0)
   (* w0 (sqrt (fma -0.25 (* (* M (* D h)) (/ (* M D) (* 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)) <= -20.0) {
		tmp = w0 * sqrt(fma(-0.25, ((M * (D * h)) * ((M * D) / (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)) <= -20.0)
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(M * Float64(D * h)) * Float64(Float64(M * D) / 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], -20.0], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(M * N[(D * h), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -20

   1. Initial program 69.5%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 52.8

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

   5. Simplified 52.8%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 68.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 71.1

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

   9. Applied egg-rr 71.1%

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

   if -20 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.3%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 96.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -20:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \left(M \cdot \left(D \cdot h\right)\right) \cdot \frac{M \cdot D}{d \cdot \left(d \cdot \ell\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -20:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \left(M \cdot D\right) \cdot \left(h \cdot \frac{M \cdot D}{\ell \cdot \left(d \cdot d\right)}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -20.0)
   (* w0 (sqrt (fma -0.25 (* (* M D) (* h (/ (* M D) (* l (* d d))))) 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)) <= -20.0) {
		tmp = w0 * sqrt(fma(-0.25, ((M * D) * (h * ((M * D) / (l * (d * d))))), 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)) <= -20.0)
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(M * D) * Float64(h * Float64(Float64(M * D) / Float64(l * Float64(d * d))))), 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], -20.0], N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(M * D), $MachinePrecision] * N[(h * N[(N[(M * D), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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)) < -20

   1. Initial program 69.5%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 52.8

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

   5. Simplified 52.8%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 68.6%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 69.2

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

   9. Applied egg-rr 69.2%

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

       1. associate-*r/ N/A

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

       2. associate-/l/ N/A

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

       3. associate-*l/ N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. associate-*l* N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. /-lowering-/.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 N/A

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

       15. *-lowering-*.f64 64.1

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

   11. Applied egg-rr 64.1%

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

   if -20 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.3%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 96.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -20:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \left(M \cdot D\right) \cdot \left(h \cdot \frac{M \cdot D}{\ell \cdot \left(d \cdot d\right)}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -20:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}, h \cdot -0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -20.0)
   (* w0 (sqrt (fma (/ (* D (* D (* M M))) (* l (* d d))) (* h -0.25) 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)) <= -20.0) {
		tmp = w0 * sqrt(fma(((D * (D * (M * M))) / (l * (d * d))), (h * -0.25), 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)) <= -20.0)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(D * Float64(D * Float64(M * M))) / Float64(l * Float64(d * d))), Float64(h * -0.25), 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], -20.0], N[(w0 * N[Sqrt[N[(N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * -0.25), $MachinePrecision] + 1.0), $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)) < -20

   1. Initial program 69.5%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. unpow2 N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 81.4

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

   4. Applied egg-rr 81.4%

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

   5. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 54.3%

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

   if -20 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.3%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 96.3%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -20:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}, h \cdot -0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+54}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell \cdot \left(d \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+54)
   (* w0 (sqrt (/ (* (* -0.25 (* D D)) (* M (* M h))) (* 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+54) {
		tmp = w0 * sqrt((((-0.25 * (D * D)) * (M * (M * h))) / (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+54)) then
        tmp = w0 * sqrt(((((-0.25d0) * (d * d)) * (m * (m * h))) / (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+54) {
		tmp = w0 * Math.sqrt((((-0.25 * (D * D)) * (M * (M * h))) / (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+54:
		tmp = w0 * math.sqrt((((-0.25 * (D * D)) * (M * (M * h))) / (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+54)
		tmp = Float64(w0 * sqrt(Float64(Float64(Float64(-0.25 * Float64(D * D)) * Float64(M * Float64(M * h))) / 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+54)
		tmp = w0 * sqrt((((-0.25 * (D * D)) * (M * (M * h))) / (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+54], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $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)) < -2.0000000000000002e54

   1. Initial program 68.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. unpow2 N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 80.6

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

   4. Applied egg-rr 80.6%

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

   5. Taylor expanded in M around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 50.1

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

   7. Simplified 50.1%

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

   if -2.0000000000000002e54 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   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. Taylor expanded in M around 0

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

   5. Simplified 94.9%

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

Alternative 7: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \left(h \cdot -0.125\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+156)
   (fma (/ (* (* M (* D D)) (* M (* h -0.125))) (* d (* d l))) w0 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)) <= -5e+156) {
		tmp = fma((((M * (D * D)) * (M * (h * -0.125))) / (d * (d * l))), w0, 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)) <= -5e+156)
		tmp = fma(Float64(Float64(Float64(M * Float64(D * D)) * Float64(M * Float64(h * -0.125))) / Float64(d * Float64(d * l))), w0, 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], -5e+156], N[(N[(N[(N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(M * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w0 + 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)) < -4.99999999999999992e156

   1. Initial program 64.7%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 54.7

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

   5. Simplified 54.7%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 49.5%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. /-lowering-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 54.8

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

   10. Applied egg-rr 54.8%

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

   if -4.99999999999999992e156 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 86.9%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 91.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \left(h \cdot -0.125\right)\right)}{d \cdot \left(d \cdot \ell\right)}, w0, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(D, w0 \cdot \left(D \cdot \left(\left(M \cdot h\right) \cdot \left(M \cdot \frac{-0.125}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+264)
   (fma D (* w0 (* D (* (* M h) (* M (/ -0.125 (* d (* d l))))))) 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+264) {
		tmp = fma(D, (w0 * (D * ((M * h) * (M * (-0.125 / (d * (d * l))))))), 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+264)
		tmp = fma(D, Float64(w0 * Float64(D * Float64(Float64(M * h) * Float64(M * Float64(-0.125 / Float64(d * Float64(d * l))))))), 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+264], N[(D * N[(w0 * N[(D * N[(N[(M * h), $MachinePrecision] * N[(M * N[(-0.125 / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -2.00000000000000009e264

   1. Initial program 62.9%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 55.8

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

   5. Simplified 55.8%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 52.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   10. Applied egg-rr 58.0%

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

   if -2.00000000000000009e264 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.1%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 90.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(D, w0 \cdot \left(D \cdot \left(\left(M \cdot h\right) \cdot \left(M \cdot \frac{-0.125}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+264}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot \left(\left(M \cdot h\right) \cdot \left(M \cdot \frac{-0.125}{d \cdot \left(d \cdot \ell\right)}\right)\right), D, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+264)
   (* w0 (fma (* D (* (* M h) (* M (/ -0.125 (* d (* d l)))))) D 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+264) {
		tmp = w0 * fma((D * ((M * h) * (M * (-0.125 / (d * (d * l)))))), D, 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+264)
		tmp = Float64(w0 * fma(Float64(D * Float64(Float64(M * h) * Float64(M * Float64(-0.125 / Float64(d * Float64(d * l)))))), D, 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+264], N[(w0 * N[(N[(D * N[(N[(M * h), $MachinePrecision] * N[(M * N[(-0.125 / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * D + 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)) < -2.00000000000000009e264

   1. Initial program 62.9%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 55.8

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

   5. Simplified 55.8%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 52.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 58.0

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

   10. Applied egg-rr 58.0%

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

   if -2.00000000000000009e264 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 87.1%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 90.5%

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

Alternative 10: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D \cdot D, \left(M \cdot M\right) \cdot \left(h \cdot \frac{-0.125}{d \cdot \left(d \cdot \ell\right)}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+302)
   (* w0 (fma (* D D) (* (* M M) (* h (/ -0.125 (* 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+302) {
		tmp = w0 * fma((D * D), ((M * M) * (h * (-0.125 / (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+302)
		tmp = Float64(w0 * fma(Float64(D * D), Float64(Float64(M * M) * Float64(h * Float64(-0.125 / 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+302], N[(w0 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h * N[(-0.125 / N[(d * N[(d * l), $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)) < -2.0000000000000002e302

   1. Initial program 62.2%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 56.7

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

   5. Simplified 56.7%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 52.9%

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 52.9

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

   10. Applied egg-rr 52.9%

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

   if -2.0000000000000002e302 < (*.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

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

   5. Simplified 90.1%

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

Alternative 11: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+302)
   (* w0 (* (* D D) (/ (* -0.125 (* h (* M M))) (* 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+302) {
		tmp = w0 * ((D * D) * ((-0.125 * (h * (M * M))) / (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+302)) then
        tmp = w0 * ((d * d) * (((-0.125d0) * (h * (m * m))) / (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+302) {
		tmp = w0 * ((D * D) * ((-0.125 * (h * (M * M))) / (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+302:
		tmp = w0 * ((D * D) * ((-0.125 * (h * (M * M))) / (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+302)
		tmp = Float64(w0 * Float64(Float64(D * D) * Float64(Float64(-0.125 * Float64(h * Float64(M * M))) / 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+302)
		tmp = w0 * ((D * D) * ((-0.125 * (h * (M * M))) / (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+302], N[(w0 * N[(N[(D * D), $MachinePrecision] * N[(N[(-0.125 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $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)) < -2.0000000000000002e302

   1. Initial program 62.2%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 56.7

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

   5. Simplified 56.7%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 52.9%

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

   9. Taylor expanded in D around inf

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. associate-*r/ N/A

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

       9. /-lowering-/.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 50.9

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

   11. Simplified 50.9%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 54.5

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

   13. Applied egg-rr 54.5%

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

   if -2.0000000000000002e302 < (*.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

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

   5. Simplified 90.1%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+302)
   (* w0 (* (* D D) (/ (* -0.125 (* h (* 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+302) {
		tmp = w0 * ((D * D) * ((-0.125 * (h * (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+302)) then
        tmp = w0 * ((d * d) * (((-0.125d0) * (h * (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+302) {
		tmp = w0 * ((D * D) * ((-0.125 * (h * (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+302:
		tmp = w0 * ((D * D) * ((-0.125 * (h * (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+302)
		tmp = Float64(w0 * Float64(Float64(D * D) * Float64(Float64(-0.125 * Float64(h * 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+302)
		tmp = w0 * ((D * D) * ((-0.125 * (h * (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+302], N[(w0 * N[(N[(D * D), $MachinePrecision] * N[(N[(-0.125 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $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)) < -2.0000000000000002e302

   1. Initial program 62.2%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 56.7

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

   5. Simplified 56.7%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 52.9%

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

   9. Taylor expanded in D around inf

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. associate-*r/ N/A

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

       9. /-lowering-/.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 50.9

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

   11. Simplified 50.9%

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

   if -2.0000000000000002e302 < (*.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

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

   5. Simplified 90.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot D\right) \cdot \frac{-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+302)
   (* w0 (* (* D D) (* (* (* M M) -0.125) (/ h (* 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+302) {
		tmp = w0 * ((D * D) * (((M * M) * -0.125) * (h / (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+302)) then
        tmp = w0 * ((d * d) * (((m * m) * (-0.125d0)) * (h / (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+302) {
		tmp = w0 * ((D * D) * (((M * M) * -0.125) * (h / (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+302:
		tmp = w0 * ((D * D) * (((M * M) * -0.125) * (h / (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+302)
		tmp = Float64(w0 * Float64(Float64(D * D) * Float64(Float64(Float64(M * M) * -0.125) * Float64(h / 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+302)
		tmp = w0 * ((D * D) * (((M * M) * -0.125) * (h / (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+302], N[(w0 * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(h / 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)) < -2.0000000000000002e302

   1. Initial program 62.2%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 56.7

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

   5. Simplified 56.7%

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

   6. Taylor expanded in D around 0

      \[\leadsto w0 \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. 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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 52.9%

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

   9. Taylor expanded in D around inf

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. associate-*r/ N/A

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

       9. /-lowering-/.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 50.9

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

   11. Simplified 50.9%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. associate-/l* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. /-lowering-/.f64 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 49.3

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

   13. Applied egg-rr 49.3%

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

   if -2.0000000000000002e302 < (*.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

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

   5. Simplified 90.1%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{h}{\ell \cdot \left(d \cdot d\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{-179}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \frac{M \cdot \left(D \cdot h\right)}{d \cdot \ell}}{d}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 1e-179)
   w0
   (* w0 (sqrt (fma -0.25 (/ (* (* M D) (/ (* M (* D h)) (* d l))) d) 1.0)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (((M * D) / (2.0 * d)) <= 1e-179) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt(fma(-0.25, (((M * D) * ((M * (D * h)) / (d * l))) / d), 1.0));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(Float64(M * D) / Float64(2.0 * d)) <= 1e-179)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(fma(-0.25, Float64(Float64(Float64(M * D) * Float64(Float64(M * Float64(D * h)) / Float64(d * l))) / d), 1.0)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 1e-179], w0, N[(w0 * N[Sqrt[N[(-0.25 * N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * N[(D * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 1e-179

   1. Initial program 85.7%

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

   3. Taylor expanded in M around 0

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

   5. Simplified 80.6%

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

   if 1e-179 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 72.6%

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

   3. Taylor expanded in w0 around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 48.5

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

   5. Simplified 48.5%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 72.9%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 79.7

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

   9. Applied egg-rr 79.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{-179}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(-0.25, \frac{\left(M \cdot D\right) \cdot \frac{M \cdot \left(D \cdot h\right)}{d \cdot \ell}}{d}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 87.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. unpow2 N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. /-lowering-/.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. /-lowering-/.f64 93.2

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

4. Applied egg-rr 93.2%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}}} \]
5. Step-by-step derivation

   1. div-inv N/A

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

   2. remove-double-div N/A

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

   3. associate-*l/ N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 91.6

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

6. Applied egg-rr 91.6%

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

Alternative 16: 68.6% 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 81.5%

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

3. Taylor expanded in M around 0

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

5. Simplified 70.7%

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

Reproduce

herbie shell --seed 2024199 
(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))))))