Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 10.1s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(4.0 / N[(N[(N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision] * N[(v * N[(v * -3.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * Pi), $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 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. Taylor expanded in v around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 98.5

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

7. Simplified 98.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 100.0

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

   12. lift-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 100.0

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

9. Applied egg-rr 100.0%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

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

Wolfram

code[v_] := N[(4.0 / N[(Pi * N[(N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision] * 3.0), $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 97.5

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

5. Simplified 97.5%

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

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. +-commutative N/A

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

   15. lift-fma.f64 N/A

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

   16. lower-*.f64 99.0

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

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

Reproduce

herbie shell --seed 2024218 
(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)))))))