Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 9.1s

Alternatives: 6

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. associate-*l/ N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(1.3333333333333333 / N[(Pi * N[(N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. associate-*l/ N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/r/ N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 4: 98.9% accurate, 1.2× speedup

Math

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

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(v, Float64(v * -6.0), 2.0)))
end

Wolfram

code[v_] := N[(N[(4.0 / N[(Pi * 3.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 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 96.6

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

5. Applied rewrites 96.6%

   \[\leadsto \frac{4}{\color{blue}{\left(\pi \cdot 3\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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 98.1

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. associate-*r* N/A

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

   14. lift-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-fma.f64 98.1

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

7. Applied rewrites 98.1%

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

Alternative 5: 98.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. associate-*l/ N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/r/ N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in v around 0

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

    1. lower-sqrt.f64 98.1

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

11. Applied rewrites 98.1%

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

12. Final simplification 98.1%

    \[\leadsto \frac{1.3333333333333333}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2}\right)} \]
13. Add Preprocessing

Alternative 6: 98.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 96.6

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

5. Applied rewrites 96.6%

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

7. Applied rewrites 98.0%

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

8. Final simplification 98.0%

   \[\leadsto \frac{1.3333333333333333}{\pi} \cdot \sqrt{0.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(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)))))))