Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 9.6s

Alternatives: 5

Speedup: 2.1×

Specification

Math

\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

C

double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
}

Java

public static double code(double v) {
	return 4.0 / (((3.0 * Math.PI) * (1.0 - (v * v))) * Math.sqrt((2.0 - (6.0 * (v * v)))));
}

Python

def code(v):
	return 4.0 / (((3.0 * math.pi) * (1.0 - (v * v))) * math.sqrt((2.0 - (6.0 * (v * v)))))

Julia

function code(v)
	return Float64(4.0 / Float64(Float64(Float64(3.0 * pi) * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v))))))
end

MATLAB

function tmp = code(v)
	tmp = 4.0 / (((3.0 * pi) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
end

Wolfram

code[v_] := N[(4.0 / N[(N[(N[(3.0 * Pi), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

C

double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
}

Java

public static double code(double v) {
	return 4.0 / (((3.0 * Math.PI) * (1.0 - (v * v))) * Math.sqrt((2.0 - (6.0 * (v * v)))));
}

Python

def code(v):
	return 4.0 / (((3.0 * math.pi) * (1.0 - (v * v))) * math.sqrt((2.0 - (6.0 * (v * v)))))

Julia

function code(v)
	return Float64(4.0 / Float64(Float64(Float64(3.0 * pi) * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v))))))
end

MATLAB

function tmp = code(v)
	tmp = 4.0 / (((3.0 * pi) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
end

Wolfram

code[v_] := N[(4.0 / N[(N[(N[(3.0 * Pi), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/
  (/ 1.3333333333333333 (* PI (- 1.0 (* v v))))
  (sqrt (fma -6.0 (* v v) 2.0))))

C

double code(double v) {
	return (1.3333333333333333 / (((double) M_PI) * (1.0 - (v * v)))) / sqrt(fma(-6.0, (v * v), 2.0));
}

Julia

function code(v)
	return Float64(Float64(1.3333333333333333 / Float64(pi * Float64(1.0 - Float64(v * v)))) / sqrt(fma(-6.0, Float64(v * v), 2.0)))
end

Wolfram

code[v_] := N[(N[(1.3333333333333333 / N[(Pi * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{2 + \left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right)}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right) + 2}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(v \cdot v\right) \cdot 6}\right)\right) + 2}} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\left(v \cdot v\right) \cdot \left(\mathsf{neg}\left(6\right)\right)} + 2}} \]

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

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, \mathsf{neg}\left(6\right), 2\right)}}} \]

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

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{v \cdot v}, \mathsf{neg}\left(6\right), 2\right)}} \]

   7. metadata-eval 98.5

      \[\leadsto \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, \color{blue}{-6}, 2\right)}} \]

4. Applied egg-rr 98.5%

   \[\leadsto \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\left(v \cdot v\right) \cdot -6 + 2}}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{v \cdot \left(v \cdot -6\right)} + 2}} \]

   3. flip-+ N/A

      \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{\frac{\left(v \cdot \left(v \cdot -6\right)\right) \cdot \left(v \cdot \left(v \cdot -6\right)\right) - 2 \cdot 2}{v \cdot \left(v \cdot -6\right) - 2}}}} \]

   4. flip-+ N/A

      \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{v \cdot \left(v \cdot -6\right) + 2}}} \]

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

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\frac{4}{\color{blue}{3 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{4}{3}}}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

       \[\leadsto \frac{\frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

       \[\leadsto \frac{\frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(1 - v \cdot v\right)}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

       \[\leadsto \frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(1 - v \cdot v\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

       \[\leadsto \frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(1 - \color{blue}{v \cdot v}\right)}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

       \[\leadsto \frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}{\color{blue}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/
  1.3333333333333333
  (* PI (* (- 1.0 (* v v)) (sqrt (fma v (* v -6.0) 2.0))))))

C

double code(double v) {
	return 1.3333333333333333 / (((double) M_PI) * ((1.0 - (v * v)) * sqrt(fma(v, (v * -6.0), 2.0))));
}

Julia

function code(v)
	return Float64(1.3333333333333333 / Float64(pi * Float64(Float64(1.0 - Float64(v * v)) * sqrt(fma(v, Float64(v * -6.0), 2.0)))))
end

Wolfram

code[v_] := N[(1.3333333333333333 / N[(Pi * N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{2 + \left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right)}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right) + 2}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(v \cdot v\right) \cdot 6}\right)\right) + 2}} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\left(v \cdot v\right) \cdot \left(\mathsf{neg}\left(6\right)\right)} + 2}} \]

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

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, \mathsf{neg}\left(6\right), 2\right)}}} \]

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

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{v \cdot v}, \mathsf{neg}\left(6\right), 2\right)}} \]

   7. metadata-eval 98.5

      \[\leadsto \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, \color{blue}{-6}, 2\right)}} \]

4. Applied egg-rr 98.5%

   \[\leadsto \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\left(v \cdot v\right) \cdot -6 + 2}}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{v \cdot \left(v \cdot -6\right)} + 2}} \]

   3. flip-+ N/A

      \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{\frac{\left(v \cdot \left(v \cdot -6\right)\right) \cdot \left(v \cdot \left(v \cdot -6\right)\right) - 2 \cdot 2}{v \cdot \left(v \cdot -6\right) - 2}}}} \]

   4. flip-+ N/A

      \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{v \cdot \left(v \cdot -6\right) + 2}}} \]

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

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\frac{4}{\color{blue}{3 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{4}{3}}}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

       \[\leadsto \frac{\frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

       \[\leadsto \frac{\frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(1 - v \cdot v\right)}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

       \[\leadsto \frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(1 - v \cdot v\right)}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

       \[\leadsto \frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(1 - \color{blue}{v \cdot v}\right)}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}} \]

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

       \[\leadsto \frac{\frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)}}{\color{blue}{\sqrt{v \cdot \left(v \cdot -6\right) + 2}}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]
7. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{3}}{\sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)\right)}} \]

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

      \[\leadsto \color{blue}{\frac{\frac{4}{3}}{\sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - v \cdot v\right)\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\left(\sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{-6 \cdot \left(v \cdot v\right) + 2}\right)} \cdot \left(1 - v \cdot v\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \left(1 - v \cdot v\right)\right)}} \]

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

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \left(1 - v \cdot v\right)\right)}} \]

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

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \left(1 - v \cdot v\right)\right)} \]

   8. flip3-- N/A

      \[\leadsto \frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \color{blue}{\frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\right)} \]

   9. clear-num N/A

      \[\leadsto \frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}{{1}^{3} - {\left(v \cdot v\right)}^{3}}}}\right)} \]

   10. clear-num N/A

       \[\leadsto \frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \frac{1}{\color{blue}{\frac{1}{\frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}}\right)} \]

   11. flip3-- N/A

       \[\leadsto \frac{\frac{4}{3}}{\mathsf{PI}\left(\right) \cdot \left(\sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \frac{1}{\frac{1}{\color{blue}{1 - v \cdot v}}}\right)} \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right)}} \]
9. Add Preprocessing

Alternative 3: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{4}{\pi \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{3} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 4.0 (* PI (sqrt (fma v (* v -6.0) 2.0)))) 3.0))

C

double code(double v) {
	return (4.0 / (((double) M_PI) * sqrt(fma(v, (v * -6.0), 2.0)))) / 3.0;
}

Julia

function code(v)
	return Float64(Float64(4.0 / Float64(pi * sqrt(fma(v, Float64(v * -6.0), 2.0)))) / 3.0)
end

Wolfram

code[v_] := N[(N[(4.0 / N[(Pi * N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

4. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\pi \cdot \mathsf{fma}\left(v, v, 1\right)\right)}} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{4}{3}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]
6. Step-by-step derivation

   1. PI-lowering-PI.f64 99.3

      \[\leadsto \frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \color{blue}{\pi}} \]

7. Simplified 99.3%

   \[\leadsto \frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \color{blue}{\pi}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{4}{3}}}{\sqrt{v \cdot \left(v \cdot -6\right) + 2} \cdot \mathsf{PI}\left(\right)} \]

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{4}{3 \cdot \left(\sqrt{v \cdot \left(v \cdot -6\right) + 2} \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\sqrt{v \cdot \left(v \cdot -6\right) + 2} \cdot \mathsf{PI}\left(\right)}}{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{4}{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{v \cdot \left(v \cdot -6\right) + 2}}}}{3} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\frac{4}{\mathsf{PI}\left(\right) \cdot \sqrt{\color{blue}{\left(v \cdot v\right) \cdot -6} + 2}}}{3} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{4}{\mathsf{PI}\left(\right) \cdot \sqrt{\color{blue}{-6 \cdot \left(v \cdot v\right)} + 2}}}{3} \]

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

      \[\leadsto \color{blue}{\frac{\frac{4}{\mathsf{PI}\left(\right) \cdot \sqrt{-6 \cdot \left(v \cdot v\right) + 2}}}{3}} \]

9. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{\frac{4}{\pi \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{3}} \]
10. Add Preprocessing

Alternative 4: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\pi \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 1.3333333333333333 (* PI (sqrt (fma v (* v -6.0) 2.0)))))

C

double code(double v) {
	return 1.3333333333333333 / (((double) M_PI) * sqrt(fma(v, (v * -6.0), 2.0)));
}

Julia

function code(v)
	return Float64(1.3333333333333333 / Float64(pi * sqrt(fma(v, Float64(v * -6.0), 2.0))))
end

Wolfram

code[v_] := N[(1.3333333333333333 / N[(Pi * N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

4. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\pi \cdot \mathsf{fma}\left(v, v, 1\right)\right)}} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{4}{3}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]
6. Step-by-step derivation

   1. PI-lowering-PI.f64 99.3

      \[\leadsto \frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \color{blue}{\pi}} \]

7. Simplified 99.3%

   \[\leadsto \frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \color{blue}{\pi}} \]

8. Final simplification 99.3%

   \[\leadsto \frac{1.3333333333333333}{\pi \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}} \]
9. Add Preprocessing

Alternative 5: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\pi \cdot \sqrt{2}} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (/ 1.3333333333333333 (* PI (sqrt 2.0))))

C

double code(double v) {
	return 1.3333333333333333 / (((double) M_PI) * sqrt(2.0));
}

Java

public static double code(double v) {
	return 1.3333333333333333 / (Math.PI * Math.sqrt(2.0));
}

Python

def code(v):
	return 1.3333333333333333 / (math.pi * math.sqrt(2.0))

Julia

function code(v)
	return Float64(1.3333333333333333 / Float64(pi * sqrt(2.0)))
end

MATLAB

function tmp = code(v)
	tmp = 1.3333333333333333 / (pi * sqrt(2.0));
end

Wolfram

code[v_] := N[(1.3333333333333333 / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

4. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\pi \cdot \mathsf{fma}\left(v, v, 1\right)\right)}} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}} \]
6. Step-by-step derivation

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

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}} \]

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

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{2}} \]

   3. sqrt-lowering-sqrt.f64 99.2

      \[\leadsto \frac{1.3333333333333333}{\pi \cdot \color{blue}{\sqrt{2}}} \]

7. Simplified 99.2%

   \[\leadsto \frac{1.3333333333333333}{\color{blue}{\pi \cdot \sqrt{2}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))