Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 9.9s

Alternatives: 4

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, 1.1× speedup

Math

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

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(Float64(pi * Float64(1.0 - Float64(v * v))) * sqrt(fma(v, Float64(v * -6.0), 2.0))))
end

Wolfram

code[v_] := N[(1.3333333333333333 / N[(N[(Pi * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. 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 rewrites 98.5%

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

6. Applied rewrites 100.0%

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

Alternative 2: 99.1% 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

3. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 97.3

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

5. Applied rewrites 97.3%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-/r* N/A

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

   8. div-inv N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. frac-times N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 98.8

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

7. Applied rewrites 98.8%

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

   1. metadata-eval N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. flip-+ N/A

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

   5. flip-+ N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. associate-/l/ N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 N/A

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

9. Applied rewrites 98.8%

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

Alternative 3: 99.1% 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

3. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 97.3

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

5. Applied rewrites 97.3%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-/r* N/A

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

   8. div-inv N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. frac-times N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 98.8

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

7. Applied rewrites 98.8%

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

Alternative 4: 99.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\pi} \cdot \sqrt{0.5} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (/ 1.3333333333333333 PI) (sqrt 0.5)))

C

double code(double v) {
	return (1.3333333333333333 / ((double) M_PI)) * sqrt(0.5);
}

Java

public static double code(double v) {
	return (1.3333333333333333 / Math.PI) * Math.sqrt(0.5);
}

Python

def code(v):
	return (1.3333333333333333 / math.pi) * math.sqrt(0.5)

Julia

function code(v)
	return Float64(Float64(1.3333333333333333 / pi) * sqrt(0.5))
end

MATLAB

function tmp = code(v)
	tmp = (1.3333333333333333 / pi) * sqrt(0.5);
end

Wolfram

code[v_] := N[(N[(1.3333333333333333 / Pi), $MachinePrecision] * N[Sqrt[0.5], $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. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{4}{3} \cdot \frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 97.2

      \[\leadsto \frac{1.3333333333333333 \cdot \sqrt{0.5}}{\color{blue}{\pi}} \]

5. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\frac{1.3333333333333333 \cdot \sqrt{0.5}}{\pi}} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. times-frac N/A

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

   5. metadata-eval N/A

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

   6. associate-/r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. /-rgt-identity N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-/r* N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 98.7

       \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi}} \cdot \sqrt{0.5} \]

7. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi} \cdot \sqrt{0.5}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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)))))))