Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 87.6%

Time: 9.8s

Alternatives: 11

Speedup: 2.0×

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: 81.3% 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: 87.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 84.7%

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

6. Applied rewrites 88.4%

   \[\leadsto w0 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\frac{-0.5 \cdot \left(D \cdot M\right)}{d} \cdot h}{\ell} \cdot \frac{0.5}{d}, D \cdot M, 1\right)}} \]
7. Add Preprocessing

Alternative 2: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+47}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{\left(M \cdot h\right) \cdot D}{d} \cdot -0.25}{\ell \cdot d}, D \cdot M, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(\frac{h \cdot M}{d} \cdot \frac{\frac{M}{\ell}}{d}\right) \cdot D, -0.125 \cdot D, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+47)
   (* w0 (sqrt (fma (/ (* (/ (* (* M h) D) d) -0.25) (* l d)) (* D M) 1.0)))
   (* w0 (fma (* (* (/ (* h M) d) (/ (/ M l) d)) D) (* -0.125 D) 1.0))))

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+47) {
		tmp = w0 * sqrt(fma((((((M * h) * D) / d) * -0.25) / (l * d)), (D * M), 1.0));
	} else {
		tmp = w0 * fma(((((h * M) / d) * ((M / l) / d)) * D), (-0.125 * D), 1.0);
	}
	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+47)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(Float64(Float64(M * h) * D) / d) * -0.25) / Float64(l * d)), Float64(D * M), 1.0)));
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(h * M) / d) * Float64(Float64(M / l) / d)) * D), Float64(-0.125 * D), 1.0));
	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+47], N[(w0 * N[Sqrt[N[(N[(N[(N[(N[(N[(M * h), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * -0.25), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(D * M), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * N[(-0.125 * D), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

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)) < -2.0000000000000001e47

   1. Initial program 62.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 58.1%

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

   6. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 62.0

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

   8. Applied rewrites 62.0%

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

   9. Taylor expanded in M around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 54.3

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

   11. Applied rewrites 54.3%

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

   if -2.0000000000000001e47 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.6%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 70.9%

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

   9. Applied rewrites 89.2%

      \[\leadsto w0 \cdot \mathsf{fma}\left(\left(\frac{h \cdot M}{d} \cdot \frac{\frac{M}{\ell}}{d}\right) \cdot D, \color{blue}{-0.125 \cdot D}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+47}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\left(-0.25 \cdot \frac{D}{d \cdot d}\right) \cdot \frac{M \cdot h}{\ell}, D \cdot M, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(\frac{h \cdot M}{d} \cdot \frac{\frac{M}{\ell}}{d}\right) \cdot D, -0.125 \cdot D, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -5e+47)
   (* w0 (sqrt (fma (* (* -0.25 (/ D (* d d))) (/ (* M h) l)) (* D M) 1.0)))
   (* w0 (fma (* (* (/ (* h M) d) (/ (/ M l) d)) D) (* -0.125 D) 1.0))))

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+47) {
		tmp = w0 * sqrt(fma(((-0.25 * (D / (d * d))) * ((M * h) / l)), (D * M), 1.0));
	} else {
		tmp = w0 * fma(((((h * M) / d) * ((M / l) / d)) * D), (-0.125 * D), 1.0);
	}
	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+47)
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(-0.25 * Float64(D / Float64(d * d))) * Float64(Float64(M * h) / l)), Float64(D * M), 1.0)));
	else
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(h * M) / d) * Float64(Float64(M / l) / d)) * D), Float64(-0.125 * D), 1.0));
	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+47], N[(w0 * N[Sqrt[N[(N[(N[(-0.25 * N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(D * M), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * N[(-0.125 * D), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

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)) < -5.00000000000000022e47

   1. Initial program 61.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 58.9%

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

   6. Applied rewrites 64.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 61.5

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

   8. Applied rewrites 61.5%

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

   9. Taylor expanded in M around 0

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

       1. times-frac N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 50.0

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

   11. Applied rewrites 50.0%

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

   if -5.00000000000000022e47 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 64.3%

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

   7. Applied rewrites 70.5%

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

   9. Applied rewrites 88.7%

      \[\leadsto w0 \cdot \mathsf{fma}\left(\left(\frac{h \cdot M}{d} \cdot \frac{\frac{M}{\ell}}{d}\right) \cdot D, \color{blue}{-0.125 \cdot D}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+76}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, M \cdot \left(M \cdot \frac{h}{\left(d \cdot d\right) \cdot \ell}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e+76)
   (* w0 (fma (* (* D D) -0.125) (* M (* M (/ h (* (* d d) l)))) 1.0))
   (* w0 1.0)))

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)) <= -1e+76) {
		tmp = w0 * fma(((D * D) * -0.125), (M * (M * (h / ((d * d) * l)))), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	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)) <= -1e+76)
		tmp = Float64(w0 * fma(Float64(Float64(D * D) * -0.125), Float64(M * Float64(M * Float64(h / Float64(Float64(d * d) * l)))), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	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], -1e+76], N[(w0 * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] * N[(M * N[(M * N[(h / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

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)) < -1e76

   1. Initial program 59.9%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 38.1%

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

   7. Applied rewrites 42.9%

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

   9. Applied rewrites 44.3%

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

   if -1e76 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 94.5%

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

Alternative 5: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+47}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, D \cdot \left(\left(-0.125 \cdot \frac{M \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot M\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+47)
   (* w0 (fma D (* D (* (* -0.125 (/ (* M h) (* (* d d) l))) M)) 1.0))
   (* w0 1.0)))

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+47) {
		tmp = w0 * fma(D, (D * ((-0.125 * ((M * h) / ((d * d) * l))) * M)), 1.0);
	} else {
		tmp = w0 * 1.0;
	}
	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+47)
		tmp = Float64(w0 * fma(D, Float64(D * Float64(Float64(-0.125 * Float64(Float64(M * h) / Float64(Float64(d * d) * l))) * M)), 1.0));
	else
		tmp = Float64(w0 * 1.0);
	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+47], N[(w0 * N[(D * N[(D * N[(N[(-0.125 * N[(N[(M * h), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * 1.0), $MachinePrecision]]

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)) < -2.0000000000000001e47

   1. Initial program 62.0%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 36.2%

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

   7. Applied rewrites 38.4%

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

   9. Applied rewrites 44.3%

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

   11. Applied rewrites 43.0%

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

   if -2.0000000000000001e47 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))

   1. Initial program 85.6%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 96.3%

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

Alternative 6: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{-291}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, D \cdot \left(\left(-0.125 \cdot \frac{\frac{M}{d} \cdot h}{\ell \cdot d}\right) \cdot M\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 1e-291)
   (* w0 1.0)
   (* w0 (fma D (* D (* (* -0.125 (/ (* (/ M d) h) (* l d))) M)) 1.0))))

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) <= 1e-291) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma(D, (D * ((-0.125 * (((M / d) * h) / (l * d))) * M)), 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)) ^ 2.0) <= 1e-291)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(D, Float64(D * Float64(Float64(-0.125 * Float64(Float64(Float64(M / d) * h) / Float64(l * d))) * M)), 1.0));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1e-291], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(D * N[(D * N[(N[(-0.125 * N[(N[(N[(M / d), $MachinePrecision] * h), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 9.99999999999999962e-292

   1. Initial program 87.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 100.0%

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

   if 9.99999999999999962e-292 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 70.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 37.3%

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

   7. Applied rewrites 44.8%

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

   9. Applied rewrites 50.5%

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

   11. Applied rewrites 62.6%

       \[\leadsto w0 \cdot \mathsf{fma}\left(D, D \cdot \left(\left(-0.125 \cdot \frac{\frac{M}{d} \cdot h}{\ell \cdot d}\right) \cdot M\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{-291}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, D \cdot \left(\left(-0.125 \cdot \left(\frac{M}{d} \cdot \frac{h}{\ell \cdot d}\right)\right) \cdot M\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 1e-291)
   (* w0 1.0)
   (* w0 (fma D (* D (* (* -0.125 (* (/ M d) (/ h (* l d)))) M)) 1.0))))

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) <= 1e-291) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma(D, (D * ((-0.125 * ((M / d) * (h / (l * d)))) * M)), 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)) ^ 2.0) <= 1e-291)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(D, Float64(D * Float64(Float64(-0.125 * Float64(Float64(M / d) * Float64(h / Float64(l * d)))) * M)), 1.0));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1e-291], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(D * N[(D * N[(N[(-0.125 * N[(N[(M / d), $MachinePrecision] * N[(h / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) < 9.99999999999999962e-292

   1. Initial program 87.9%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 100.0%

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

   if 9.99999999999999962e-292 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))

   1. Initial program 70.1%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 37.3%

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

   7. Applied rewrites 44.8%

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

   9. Applied rewrites 50.5%

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

   11. Applied rewrites 62.6%

       \[\leadsto w0 \cdot \mathsf{fma}\left(D, D \cdot \left(\left(-0.125 \cdot \left(\frac{M}{d} \cdot \frac{h}{\ell \cdot d}\right)\right) \cdot M\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(\left(\frac{h \cdot M}{d} \cdot \frac{\frac{M}{\ell}}{d}\right) \cdot D, -0.125 \cdot D, 1\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\mathsf{fma}\left(\frac{h \cdot \left(\left(M \cdot D\right) \cdot -0.5\right)}{\ell \cdot d} \cdot \frac{0.5}{d}, D \cdot M, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ (* M D) (* 2.0 d)) 5e-80)
   (* w0 (fma (* (* (/ (* h M) d) (/ (/ M l) d)) D) (* -0.125 D) 1.0))
   (*
    w0
    (sqrt
     (fma (* (/ (* h (* (* M D) -0.5)) (* l d)) (/ 0.5 d)) (* D M) 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)) <= 5e-80) {
		tmp = w0 * fma(((((h * M) / d) * ((M / l) / d)) * D), (-0.125 * D), 1.0);
	} else {
		tmp = w0 * sqrt(fma((((h * ((M * D) * -0.5)) / (l * d)) * (0.5 / d)), (D * M), 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)) <= 5e-80)
		tmp = Float64(w0 * fma(Float64(Float64(Float64(Float64(h * M) / d) * Float64(Float64(M / l) / d)) * D), Float64(-0.125 * D), 1.0));
	else
		tmp = Float64(w0 * sqrt(fma(Float64(Float64(Float64(h * Float64(Float64(M * D) * -0.5)) / Float64(l * d)) * Float64(0.5 / d)), Float64(D * M), 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], 5e-80], N[(w0 * N[(N[(N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * N[(-0.125 * D), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(N[(N[(h * N[(N[(M * D), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * N[(D * M), $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)) < 5e-80

   1. Initial program 81.3%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 61.5%

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

   7. Applied rewrites 66.1%

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

   9. Applied rewrites 80.2%

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

   if 5e-80 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 68.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 69.0%

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

   6. Applied rewrites 72.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 68.8

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

   8. Applied rewrites 68.8%

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

Alternative 9: 75.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \mathsf{fma}\left(\left(\frac{h \cdot M}{d} \cdot \frac{\frac{M}{\ell}}{d}\right) \cdot D, -0.125 \cdot D, 1\right) \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (fma (* (* (/ (* h M) d) (/ (/ M l) d)) D) (* -0.125 D) 1.0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * fma(((((h * M) / d) * ((M / l) / d)) * D), (-0.125 * D), 1.0);
}

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * fma(Float64(Float64(Float64(Float64(h * M) / d) * Float64(Float64(M / l) / d)) * D), Float64(-0.125 * D), 1.0))
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(N[(N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * N[(-0.125 * D), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

3. Taylor expanded in M around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 56.3%

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

7. Applied rewrites 62.0%

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

9. Applied rewrites 75.4%

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

Alternative 10: 70.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-36}:\\ \;\;\;\;w0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(D, D \cdot \left(h \cdot \frac{-0.125 \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* M D) 1e-36)
   (* w0 1.0)
   (* w0 (fma D (* D (* h (/ (* -0.125 (* M M)) (* (* d d) l)))) 1.0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M * D) <= 1e-36) {
		tmp = w0 * 1.0;
	} else {
		tmp = w0 * fma(D, (D * (h * ((-0.125 * (M * M)) / ((d * d) * l)))), 1.0);
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 1e-36)
		tmp = Float64(w0 * 1.0);
	else
		tmp = Float64(w0 * fma(D, Float64(D * Float64(h * Float64(Float64(-0.125 * Float64(M * M)) / Float64(Float64(d * d) * l)))), 1.0));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(M * D), $MachinePrecision], 1e-36], N[(w0 * 1.0), $MachinePrecision], N[(w0 * N[(D * N[(D * N[(h * N[(N[(-0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 9.9999999999999994e-37

   1. Initial program 82.4%

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

   3. Taylor expanded in M around 0

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

   5. Applied rewrites 78.6%

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

   if 9.9999999999999994e-37 < (*.f64 M D)

   1. Initial program 61.3%

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

   3. Taylor expanded in M around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 43.0%

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

   7. Applied rewrites 48.0%

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

   9. Applied rewrites 45.6%

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

   11. Applied rewrites 52.5%

       \[\leadsto w0 \cdot \mathsf{fma}\left(D, D \cdot \left(h \cdot \color{blue}{\frac{-0.125 \cdot \left(M \cdot M\right)}{\left(d \cdot d\right) \cdot \ell}}\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 68.4% accurate, 26.2× speedup

Math

\[\begin{array}{l} \\ w0 \cdot 1 \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 (* w0 1.0))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * 1.0;
}

Fortran

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

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * 1.0;
}

Python

def code(w0, M, D, h, l, d):
	return w0 * 1.0

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * 1.0)
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * 1.0;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * 1.0), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

3. Taylor expanded in M around 0

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

5. Applied rewrites 69.7%

   \[\leadsto w0 \cdot \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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))))))