Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 2.3s

Alternatives: 5

Speedup: 3.2×

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. mult-flip-rev N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. metadata-eval N/A

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

   14. add-flip N/A

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

   15. lift-*.f64 N/A

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

   16. lift-fma.f64 N/A

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

   17. frac-2neg N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

5. Applied rewrites 100.0%

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

Alternative 3: 99.1% accurate, 1.7× speedup

Math

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

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

Wolfram

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. mult-flip N/A

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

   9. metadata-eval N/A

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. metadata-eval N/A

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

   13. add-flip N/A

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

   14. lift-*.f64 N/A

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

   15. lift-fma.f64 N/A

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

   16. frac-2neg N/A

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

   17. lower-/.f64 N/A

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 99.1%

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

Alternative 4: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 99.0%

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

Alternative 5: 99.0% accurate, 3.2× 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 98.5%

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

2. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\pi \cdot \sqrt{\color{blue}{2}}} \]

   4. lower-sqrt.f64 99.0

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

4. Applied rewrites 99.0%

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

Reproduce

herbie shell --seed 2025162 
(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)))))))