Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 8.8s

Alternatives: 7

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

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

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(Float64(pi * sqrt(fma(-6.0, Float64(v * v), 2.0))) * Float64(1.0 - Float64(v * v))))
end

Wolfram

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-fma.f64 N/A

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

   7. div-inv N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. frac-times N/A

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

6. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 100.0

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-fma.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(1.3333333333333333 / N[(N[(N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-fma.f64 N/A

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

   7. div-inv N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. frac-times N/A

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

6. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 100.0

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 100.0

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

   11. lift-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-fma.f64 100.0

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

8. Applied rewrites 100.0%

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

Alternative 3: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[(4.0 / N[(3.0 * Pi), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-6.0 * N[(v * v), $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. lower-*.f64 N/A

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

   2. lower-PI.f64 96.9

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

5. Applied rewrites 96.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 98.4

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lift-fma.f64 98.4

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

7. Applied rewrites 98.4%

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

Alternative 4: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in v around 0

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

   1. lower-PI.f64 98.4

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

7. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-fma.f64 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 98.4

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

   10. lift-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-fma.f64 98.4

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

9. Applied rewrites 98.4%

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

Alternative 5: 99.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in v around 0

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

   1. lower-PI.f64 98.4

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

7. Applied rewrites 98.4%

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

Alternative 6: 99.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 98.4%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-PI.f64 98.4

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

7. Applied rewrites 98.4%

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

Alternative 7: 97.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 96.9

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

5. Applied rewrites 96.9%

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

Reproduce

herbie shell --seed 2024235 
(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)))))))