Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 9.4s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 98.4

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

4. Applied rewrites 98.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. distribute-rgt1-in N/A

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

   5. +-commutative N/A

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

   6. sub-neg N/A

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

   7. lift--.f64 N/A

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

   8. lower-*.f64 100.0

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

10. Applied rewrites 100.0%

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

Alternative 2: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[(4.0 / N[(Pi * 3.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 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. 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 97.6

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

5. Applied rewrites 97.6%

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

   1. lift-/.f64 N/A

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

   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. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.1

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

   6. lift--.f64 N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{2 - \color{blue}{6 \cdot \left(v \cdot v\right)}}} \]

   8. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{\color{blue}{2 + \left(\mathsf{neg}\left(6\right)\right) \cdot \left(v \cdot v\right)}}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{2 + \color{blue}{-6} \cdot \left(v \cdot v\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{2 + \color{blue}{\left(v \cdot v\right) \cdot -6}}} \]

   11. +-commutative N/A

       \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{\color{blue}{\left(v \cdot v\right) \cdot -6 + 2}}} \]

   12. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]

7. Applied rewrites 99.1%

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

8. Final simplification 99.1%

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

Reproduce

herbie shell --seed 2024223 
(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)))))))