Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 10.3s

Alternatives: 5

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(1.3333333333333333 / N[(N[(Pi - N[(Pi * 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 98.4%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. metadata-eval 98.4

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

4. Applied egg-rr 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. associate-/r* N/A

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

   2. associate-*r* N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

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

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

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

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

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

6. Applied egg-rr 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. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. neg-mul-1 N/A

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

   5. associate-*l* N/A

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

   6. *-un-lft-identity N/A

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

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

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

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

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

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

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

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

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

   11. PI-lowering-PI.f64 100.0

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

8. Applied egg-rr 100.0%

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

   8. PI-lowering-PI.f64 100.0

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. metadata-eval 98.4

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

4. Applied egg-rr 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. associate-/r* N/A

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

   2. associate-*r* N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

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

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

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

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

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

6. Applied egg-rr 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. Final simplification 100.0%

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

Alternative 3: 98.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in v around 0

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

   1. PI-lowering-PI.f64 99.5

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 98.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in v around 0

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

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

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

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

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

   3. sqrt-lowering-sqrt.f64 99.5

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

7. Simplified 99.5%

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

Alternative 5: 97.3% accurate, 2.1× speedup

Math

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

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(1.3333333333333333 * Float64(sqrt(0.5) / pi))
end

MATLAB

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

Wolfram

code[v_] := N[(1.3333333333333333 * N[(N[Sqrt[0.5], $MachinePrecision] / Pi), $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. 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. *-lowering-*.f64 N/A

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

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

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

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

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

   4. PI-lowering-PI.f64 98.0

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

5. Simplified 98.0%

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

Reproduce

herbie shell --seed 2024196 
(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)))))))