Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 9.8s

Alternatives: 5

Speedup: 1.1×

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

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

FPCore

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

C

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

Java

public static double code(double v) {
	return 1.3333333333333333 / (((1.0 - (v * v)) * Math.sqrt((2.0 + ((v * v) * -6.0)))) * Math.PI);
}

Python

def code(v):
	return 1.3333333333333333 / (((1.0 - (v * v)) * math.sqrt((2.0 + ((v * v) * -6.0)))) * math.pi)

Julia

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

MATLAB

function tmp = code(v)
	tmp = 1.3333333333333333 / (((1.0 - (v * v)) * sqrt((2.0 + ((v * v) * -6.0)))) * pi);
end

Wolfram

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

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. associate-/l/ N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

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

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

   10. metadata-eval N/A

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

   11. associate-*r* N/A

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

   12. associate-/l/ N/A

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

   13. associate-/l/ N/A

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

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

       \[\leadsto \mathsf{/.f64}\left(\frac{4}{3}, \color{blue}{\left(\left(\sqrt{2 + v \cdot \left(v \cdot -6\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \]

4. Applied egg-rr 100.0%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(N[(4.0 / Pi), $MachinePrecision] / N[Sqrt[N[(2.0 + N[(v * N[(v * -6.0), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-/r* N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}\right), \color{blue}{\left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{4}{3 \cdot \mathsf{PI}\left(\right)}}{1 - v \cdot v}\right), \left(\sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{4}{3 \cdot \mathsf{PI}\left(\right)}\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{\color{blue}{2} - 6 \cdot \left(v \cdot v\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{4}{3}\right), \mathsf{PI}\left(\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{\color{blue}{2} - 6 \cdot \left(v \cdot v\right)}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI}\left(\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \left(v \cdot v\right)\right)\right), \left(\sqrt{2 - \color{blue}{6 \cdot \left(v \cdot v\right)}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \left(\sqrt{2 - 6 \cdot \color{blue}{\left(v \cdot v\right)}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 - 6 \cdot \left(v \cdot v\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 + \left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right)\right)\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\left(6 \cdot v\right) \cdot v\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(v \cdot \left(6 \cdot v\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(v \cdot \left(\mathsf{neg}\left(6 \cdot v\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(6 \cdot v\right)\right)\right)\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(v \cdot 6\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \left(v \cdot \left(\mathsf{neg}\left(6\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(6\right)\right)\right)\right)\right)\right)\right) \]

   21. metadata-eval 100.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -6\right)\right)\right)\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}\right)}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -6\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

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

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

   2. PI-lowering-PI.f64 98.6%

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

7. Simplified 98.6%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \left(\sqrt{2 + v \cdot \left(v \cdot -6\right)}\right)\right), \mathsf{PI}\left(\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{sqrt.f64}\left(\left(2 + v \cdot \left(v \cdot -6\right)\right)\right)\right), \mathsf{PI}\left(\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(v \cdot \left(v \cdot -6\right)\right)\right)\right)\right), \mathsf{PI}\left(\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \left(v \cdot -6\right)\right)\right)\right)\right), \mathsf{PI}\left(\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -6\right)\right)\right)\right)\right), \mathsf{PI}\left(\right)\right) \]

   9. PI-lowering-PI.f64 98.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -6\right)\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

9. Applied egg-rr 98.6%

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

    1. associate-/l/ N/A

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

    2. metadata-eval N/A

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

    3. metadata-eval N/A

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

    4. frac-times N/A

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

    5. metadata-eval N/A

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

    6. metadata-eval N/A

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

    7. associate-/r* N/A

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

    8. div-inv N/A

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

    9. *-commutative N/A

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

    10. associate-/r* N/A

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

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

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

11. Applied egg-rr 98.6%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double v) {
	return 1.3333333333333333 / (Math.PI * Math.sqrt((2.0 + (v * (v * -6.0)))));
}

Python

def code(v):
	return 1.3333333333333333 / (math.pi * math.sqrt((2.0 + (v * (v * -6.0)))))

Julia

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

MATLAB

function tmp = code(v)
	tmp = 1.3333333333333333 / (pi * sqrt((2.0 + (v * (v * -6.0)))));
end

Wolfram

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

   1. associate-/r* N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}\right), \color{blue}{\left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{4}{3 \cdot \mathsf{PI}\left(\right)}}{1 - v \cdot v}\right), \left(\sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{4}{3 \cdot \mathsf{PI}\left(\right)}\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{\color{blue}{2} - 6 \cdot \left(v \cdot v\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{4}{3}\right), \mathsf{PI}\left(\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{\color{blue}{2} - 6 \cdot \left(v \cdot v\right)}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI}\left(\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \left(v \cdot v\right)\right)\right), \left(\sqrt{2 - \color{blue}{6 \cdot \left(v \cdot v\right)}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \left(\sqrt{2 - 6 \cdot \color{blue}{\left(v \cdot v\right)}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 - 6 \cdot \left(v \cdot v\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 + \left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right)\right)\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\left(6 \cdot v\right) \cdot v\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(v \cdot \left(6 \cdot v\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(v \cdot \left(\mathsf{neg}\left(6 \cdot v\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(6 \cdot v\right)\right)\right)\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(v \cdot 6\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \left(v \cdot \left(\mathsf{neg}\left(6\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(6\right)\right)\right)\right)\right)\right)\right) \]

   21. metadata-eval 100.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -6\right)\right)\right)\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}\right)}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -6\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

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

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

   2. PI-lowering-PI.f64 98.6%

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

7. Simplified 98.6%

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

   1. associate-/l/ N/A

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

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

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

   3. *-commutative N/A

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

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

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

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

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

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

      \[\leadsto \mathsf{/.f64}\left(\frac{4}{3}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{sqrt.f64}\left(\left(2 + v \cdot \left(v \cdot -6\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\frac{4}{3}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(v \cdot \left(v \cdot -6\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\frac{4}{3}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \left(v \cdot -6\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 98.6%

      \[\leadsto \mathsf{/.f64}\left(\frac{4}{3}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -6\right)\right)\right)\right)\right)\right) \]

9. Applied egg-rr 98.6%

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

Alternative 4: 99.0% accurate, 1.1× speedup

Math

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

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(Float64(1.3333333333333333 / pi) / sqrt(2.0))
end

MATLAB

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

Wolfram

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

   \[\leadsto \mathsf{/.f64}\left(4, \color{blue}{\left(3 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(4, \left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{2}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(4, \left(\left(\mathsf{PI}\left(\right) \cdot 3\right) \cdot \sqrt{\color{blue}{2}}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(4, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(3 \cdot \sqrt{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(4, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(3 \cdot \sqrt{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(4, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{3} \cdot \sqrt{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(4, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(3, \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 98.5%

      \[\leadsto \mathsf{/.f64}\left(4, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 98.5%

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

   1. associate-*r* N/A

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

   2. associate-/r* N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{4}{3}}{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI}\left(\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \left(\sqrt{2}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 98.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{4}{3}, \mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

7. Applied egg-rr 98.5%

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

Alternative 5: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(1.3333333333333333 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / Pi), $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

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{4}{3} \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\mathsf{PI}\left(\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{4}{3}, \left(\sqrt{\frac{1}{2}}\right)\right), \mathsf{PI}\left(\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{4}{3}, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \mathsf{PI}\left(\right)\right) \]

   5. PI-lowering-PI.f64 97.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{4}{3}, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

5. Simplified 97.0%

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

Reproduce

herbie shell --seed 2024144 
(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)))))))