Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 11.8s

Alternatives: 3

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 - v \cdot v\right) \cdot \left(\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot 0.25\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (- 1.0 (* v v)) (* (sqrt (+ 2.0 (* (* v v) -6.0))) 0.25)))

C

double code(double v) {
	return (1.0 - (v * v)) * (sqrt((2.0 + ((v * v) * -6.0))) * 0.25);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * (sqrt((2.0d0 + ((v * v) * (-6.0d0)))) * 0.25d0)
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * (Math.sqrt((2.0 + ((v * v) * -6.0))) * 0.25);
}

Python

def code(v):
	return (1.0 - (v * v)) * (math.sqrt((2.0 + ((v * v) * -6.0))) * 0.25)

Julia

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) * Float64(sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0))) * 0.25))
end

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * (sqrt((2.0 + ((v * v) * -6.0))) * 0.25);
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

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

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

   5. metadata-eval N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

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

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

   9. metadata-eval 98.5

      \[\leadsto \left(\frac{0.25}{{2}^{\color{blue}{-0.5}}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \left(\color{blue}{\frac{0.25}{{2}^{-0.5}}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
5. Step-by-step derivation

   1. *-commutative N/A

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

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

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

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

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

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

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. pow-flip N/A

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

   8. metadata-eval N/A

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

   9. pow1/2 N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

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

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

   13. sqrt-unprod N/A

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

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

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

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

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

   16. sub-neg N/A

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

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

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

   18. *-commutative N/A

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

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

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

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

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

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

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

   22. metadata-eval 100.0

       \[\leadsto \left(1 - v \cdot v\right) \cdot \left(0.25 \cdot \sqrt{2 \cdot \left(1 + \left(v \cdot v\right) \cdot \color{blue}{-3}\right)}\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(1 - v \cdot v\right) \cdot \left(0.25 \cdot \sqrt{2 \cdot \left(1 + \left(v \cdot v\right) \cdot -3\right)}\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

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

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

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

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

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

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

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

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

   12. metadata-eval 100.0

       \[\leadsto \left(1 - v \cdot v\right) \cdot \left(\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}} \cdot 0.25\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \left(1 - v \cdot v\right) \cdot \color{blue}{\left(\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot 0.25\right)} \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.625\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (sqrt 2.0) (+ 0.25 (* (* v v) -0.625))))

C

double code(double v) {
	return sqrt(2.0) * (0.25 + ((v * v) * -0.625));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) * (0.25d0 + ((v * v) * (-0.625d0)))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) * (0.25 + ((v * v) * -0.625));
}

Python

def code(v):
	return math.sqrt(2.0) * (0.25 + ((v * v) * -0.625))

Julia

function code(v)
	return Float64(sqrt(2.0) * Float64(0.25 + Float64(Float64(v * v) * -0.625)))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) * (0.25 + ((v * v) * -0.625));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 + N[(N[(v * v), $MachinePrecision] * -0.625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0

   \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + {v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) - \frac{3}{2}\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]
4. Step-by-step derivation

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

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + {v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) - \frac{3}{2}\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]

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

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{{v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) - \frac{3}{2}\right)}\right)\right) \cdot \left(1 - v \cdot v\right) \]

   3. unpow2 N/A

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{\left(v \cdot v\right)} \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) - \frac{3}{2}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{\left(v \cdot v\right)} \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) - \frac{3}{2}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \color{blue}{\left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) + \color{blue}{\frac{-3}{2}}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \color{blue}{\left(\frac{-3}{2} + {v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \color{blue}{\left(\frac{-3}{2} + {v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \color{blue}{{v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right)}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   10. unpow2 N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \color{blue}{\left(v \cdot v\right)} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \color{blue}{\left(v \cdot v\right)} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   12. sub-neg N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \color{blue}{\left(\frac{-27}{16} \cdot {v}^{2} + \left(\mathsf{neg}\left(\frac{9}{8}\right)\right)\right)}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-27}{16} \cdot {v}^{2} + \color{blue}{\frac{-9}{8}}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   14. +-commutative N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \color{blue}{\left(\frac{-9}{8} + \frac{-27}{16} \cdot {v}^{2}\right)}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \color{blue}{\left(\frac{-9}{8} + \frac{-27}{16} \cdot {v}^{2}\right)}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   16. *-commutative N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-9}{8} + \color{blue}{{v}^{2} \cdot \frac{-27}{16}}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-9}{8} + \color{blue}{{v}^{2} \cdot \frac{-27}{16}}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   18. unpow2 N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-9}{8} + \color{blue}{\left(v \cdot v\right)} \cdot \frac{-27}{16}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   19. *-lowering-*.f64 99.6

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(-1.5 + \left(v \cdot v\right) \cdot \left(-1.125 + \color{blue}{\left(v \cdot v\right)} \cdot -1.6875\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

5. Simplified 99.6%

   \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + \left(v \cdot v\right) \cdot \left(-1.5 + \left(v \cdot v\right) \cdot \left(-1.125 + \left(v \cdot v\right) \cdot -1.6875\right)\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{-5}{8} \cdot \left({v}^{2} \cdot \sqrt{2}\right) + \frac{1}{4} \cdot \sqrt{2}} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{-5}{8} \cdot {v}^{2}\right) \cdot \sqrt{2}} + \frac{1}{4} \cdot \sqrt{2} \]

   2. distribute-rgt-out N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right)} \]

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

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right)} \]

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

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right) \]

   5. +-commutative N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\frac{1}{4} + \frac{-5}{8} \cdot {v}^{2}\right)} \]

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

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\frac{1}{4} + \frac{-5}{8} \cdot {v}^{2}\right)} \]

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

      \[\leadsto \sqrt{2} \cdot \left(\frac{1}{4} + \color{blue}{\frac{-5}{8} \cdot {v}^{2}}\right) \]

   8. unpow2 N/A

      \[\leadsto \sqrt{2} \cdot \left(\frac{1}{4} + \frac{-5}{8} \cdot \color{blue}{\left(v \cdot v\right)}\right) \]

   9. *-lowering-*.f64 98.9

      \[\leadsto \sqrt{2} \cdot \left(0.25 + -0.625 \cdot \color{blue}{\left(v \cdot v\right)}\right) \]

8. Simplified 98.9%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(0.25 + -0.625 \cdot \left(v \cdot v\right)\right)} \]

9. Final simplification 98.9%

   \[\leadsto \sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.625\right) \]
10. Add Preprocessing

Alternative 3: 99.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {0.015625}^{0.25} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (pow 0.015625 0.25))

C

double code(double v) {
	return pow(0.015625, 0.25);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = 0.015625d0 ** 0.25d0
end function

Java

public static double code(double v) {
	return Math.pow(0.015625, 0.25);
}

Python

def code(v):
	return math.pow(0.015625, 0.25)

Julia

function code(v)
	return 0.015625 ^ 0.25
end

MATLAB

function tmp = code(v)
	tmp = 0.015625 ^ 0.25;
end

Wolfram

code[v_] := N[Power[0.015625, 0.25], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0

   \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + {v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) - \frac{3}{2}\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]
4. Step-by-step derivation

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

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + {v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) - \frac{3}{2}\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]

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

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{{v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) - \frac{3}{2}\right)}\right)\right) \cdot \left(1 - v \cdot v\right) \]

   3. unpow2 N/A

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{\left(v \cdot v\right)} \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) - \frac{3}{2}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{\left(v \cdot v\right)} \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) - \frac{3}{2}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \color{blue}{\left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left({v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right) + \color{blue}{\frac{-3}{2}}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \color{blue}{\left(\frac{-3}{2} + {v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \color{blue}{\left(\frac{-3}{2} + {v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \color{blue}{{v}^{2} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right)}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   10. unpow2 N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \color{blue}{\left(v \cdot v\right)} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \color{blue}{\left(v \cdot v\right)} \cdot \left(\frac{-27}{16} \cdot {v}^{2} - \frac{9}{8}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   12. sub-neg N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \color{blue}{\left(\frac{-27}{16} \cdot {v}^{2} + \left(\mathsf{neg}\left(\frac{9}{8}\right)\right)\right)}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-27}{16} \cdot {v}^{2} + \color{blue}{\frac{-9}{8}}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   14. +-commutative N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \color{blue}{\left(\frac{-9}{8} + \frac{-27}{16} \cdot {v}^{2}\right)}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \color{blue}{\left(\frac{-9}{8} + \frac{-27}{16} \cdot {v}^{2}\right)}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   16. *-commutative N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-9}{8} + \color{blue}{{v}^{2} \cdot \frac{-27}{16}}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

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

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-9}{8} + \color{blue}{{v}^{2} \cdot \frac{-27}{16}}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   18. unpow2 N/A

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-9}{8} + \color{blue}{\left(v \cdot v\right)} \cdot \frac{-27}{16}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   19. *-lowering-*.f64 99.6

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot \left(-1.5 + \left(v \cdot v\right) \cdot \left(-1.125 + \color{blue}{\left(v \cdot v\right)} \cdot -1.6875\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

5. Simplified 99.6%

   \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + \left(v \cdot v\right) \cdot \left(-1.5 + \left(v \cdot v\right) \cdot \left(-1.125 + \left(v \cdot v\right) \cdot -1.6875\right)\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{4} \cdot \left(\left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-9}{8} + \left(v \cdot v\right) \cdot \frac{-27}{16}\right)\right)\right) \cdot \left(1 - v \cdot v\right)\right)} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{4}{\sqrt{2}}}} \cdot \left(\left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-9}{8} + \left(v \cdot v\right) \cdot \frac{-27}{16}\right)\right)\right) \cdot \left(1 - v \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \left(\left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-9}{8} + \left(v \cdot v\right) \cdot \frac{-27}{16}\right)\right)\right) \cdot \left(1 - v \cdot v\right)\right)}{\frac{4}{\sqrt{2}}}} \]

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

      \[\leadsto \color{blue}{\frac{1 \cdot \left(\left(1 + \left(v \cdot v\right) \cdot \left(\frac{-3}{2} + \left(v \cdot v\right) \cdot \left(\frac{-9}{8} + \left(v \cdot v\right) \cdot \frac{-27}{16}\right)\right)\right) \cdot \left(1 - v \cdot v\right)\right)}{\frac{4}{\sqrt{2}}}} \]

7. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\frac{1 \cdot \left(\left(1 + v \cdot \left(v \cdot \left(-1.5 + v \cdot \left(v \cdot \left(-1.125 + \left(v \cdot v\right) \cdot -1.6875\right)\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right)\right)}{{4}^{0.75}}} \]

8. Taylor expanded in v around 0

   \[\leadsto \color{blue}{{\frac{1}{64}}^{\frac{1}{4}}} \]
9. Step-by-step derivation

   1. pow-lowering-pow.f64 98.4

      \[\leadsto \color{blue}{{0.015625}^{0.25}} \]

10. Simplified 98.4%

    \[\leadsto \color{blue}{{0.015625}^{0.25}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))