Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 9.6s

Alternatives: 5

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

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((1.0d0 - (v * v)) * sqrt((((v * v) * (-6.0d0)) + 2.0d0))) / 4.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 4.0), $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. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{4} \cdot \left(1 - v \cdot v\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{4}} \cdot \left(1 - v \cdot v\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)}{\color{blue}{4}}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \color{blue}{\frac{1 - v \cdot v}{4}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right), \color{blue}{\left(\frac{1 - v \cdot v}{4}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - 3 \cdot \left(v \cdot v\right)\right)\right), \left(\frac{\color{blue}{1 - v \cdot v}}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(3 \cdot \left(v \cdot v\right)\right)\right)\right), \left(\frac{\color{blue}{1} - v \cdot v}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \left(v \cdot v\right)\right)\right)\right), \left(\frac{1 - v \cdot v}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1 - v \cdot v}{4}\right)\right)\right) \]

   13. div-sub N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} - \color{blue}{\frac{v \cdot v}{4}}\right)\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} + \color{blue}{\left(\mathsf{neg}\left(\frac{v \cdot v}{4}\right)\right)}\right)\right)\right) \]

   15. distribute-frac-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} + \frac{\mathsf{neg}\left(v \cdot v\right)}{\color{blue}{4}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{4}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(v \cdot v\right)}{4}\right)}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{\color{blue}{\mathsf{neg}\left(v \cdot v\right)}}{4}\right)\right)\right)\right) \]

   18. distribute-frac-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\mathsf{neg}\left(\frac{v \cdot v}{4}\right)\right)\right)\right)\right) \]

   19. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{v \cdot v}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left(v \cdot v\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(0.25 + \frac{v \cdot v}{-4}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. distribute-rgt1-in N/A

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

   8. +-commutative N/A

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

   9. sub-neg N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. associate-/r/ N/A

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

   13. clear-num N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. associate-*l/ N/A

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

6. Applied egg-rr 100.0%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\left(v \cdot v\right) \cdot -3 + 1\right)\right)\right)\right), 4\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\left(v \cdot v\right) \cdot -3\right) + 2 \cdot 1\right)\right)\right), 4\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\left(v \cdot v\right) \cdot -3\right) + 2\right)\right)\right), 4\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)\right), 2\right)\right)\right), 4\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(v \cdot v\right) \cdot -3\right) \cdot 2\right), 2\right)\right)\right), 4\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(v \cdot v\right) \cdot \left(-3 \cdot 2\right)\right), 2\right)\right)\right), 4\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(v \cdot v\right) \cdot -6\right), 2\right)\right)\right), 4\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(v \cdot v\right) \cdot \left(2 \cdot -3\right)\right), 2\right)\right)\right), 4\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(v \cdot v\right), \left(2 \cdot -3\right)\right), 2\right)\right)\right), 4\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(2 \cdot -3\right)\right), 2\right)\right)\right), 4\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right), 2\right)\right)\right), 4\right) \]

8. Applied egg-rr 100.0%

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

Alternative 2: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot \left(0.25 + v \cdot \left(v \cdot \left(-0.625 + \left(v \cdot v\right) \cdot 0.09375\right)\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (sqrt 2.0) (+ 0.25 (* v (* v (+ -0.625 (* (* v v) 0.09375)))))))

C

double code(double v) {
	return sqrt(2.0) * (0.25 + (v * (v * (-0.625 + ((v * v) * 0.09375)))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) * (0.25d0 + (v * (v * ((-0.625d0) + ((v * v) * 0.09375d0)))))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) * (0.25 + (v * (v * (-0.625 + ((v * v) * 0.09375)))));
}

Python

def code(v):
	return math.sqrt(2.0) * (0.25 + (v * (v * (-0.625 + ((v * v) * 0.09375)))))

Julia

function code(v)
	return Float64(sqrt(2.0) * Float64(0.25 + Float64(v * Float64(v * Float64(-0.625 + Float64(Float64(v * v) * 0.09375))))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) * (0.25 + (v * (v * (-0.625 + ((v * v) * 0.09375)))));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 + N[(v * N[(v * N[(-0.625 + N[(N[(v * v), $MachinePrecision] * 0.09375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{4} \cdot \left(1 - v \cdot v\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{4}} \cdot \left(1 - v \cdot v\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)}{\color{blue}{4}}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \color{blue}{\frac{1 - v \cdot v}{4}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right), \color{blue}{\left(\frac{1 - v \cdot v}{4}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - 3 \cdot \left(v \cdot v\right)\right)\right), \left(\frac{\color{blue}{1 - v \cdot v}}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(3 \cdot \left(v \cdot v\right)\right)\right)\right), \left(\frac{\color{blue}{1} - v \cdot v}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \left(v \cdot v\right)\right)\right)\right), \left(\frac{1 - v \cdot v}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1 - v \cdot v}{4}\right)\right)\right) \]

   13. div-sub N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} - \color{blue}{\frac{v \cdot v}{4}}\right)\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} + \color{blue}{\left(\mathsf{neg}\left(\frac{v \cdot v}{4}\right)\right)}\right)\right)\right) \]

   15. distribute-frac-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} + \frac{\mathsf{neg}\left(v \cdot v\right)}{\color{blue}{4}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{4}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(v \cdot v\right)}{4}\right)}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{\color{blue}{\mathsf{neg}\left(v \cdot v\right)}}{4}\right)\right)\right)\right) \]

   18. distribute-frac-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\mathsf{neg}\left(\frac{v \cdot v}{4}\right)\right)\right)\right)\right) \]

   19. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{v \cdot v}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left(v \cdot v\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(0.25 + \frac{v \cdot v}{-4}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \color{blue}{\left(\frac{1}{4} + {v}^{2} \cdot \left(\frac{3}{32} \cdot {v}^{2} - \frac{5}{8}\right)\right)}\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left({v}^{2} \cdot \left(\frac{3}{32} \cdot {v}^{2} - \frac{5}{8}\right)\right)}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(v \cdot v\right) \cdot \left(\color{blue}{\frac{3}{32} \cdot {v}^{2}} - \frac{5}{8}\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(v \cdot \color{blue}{\left(v \cdot \left(\frac{3}{32} \cdot {v}^{2} - \frac{5}{8}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \color{blue}{\left(v \cdot \left(\frac{3}{32} \cdot {v}^{2} - \frac{5}{8}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \color{blue}{\left(\frac{3}{32} \cdot {v}^{2} - \frac{5}{8}\right)}\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(\frac{3}{32} \cdot {v}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{5}{8}\right)\right)}\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(\frac{3}{32} \cdot {v}^{2} + \frac{-5}{8}\right)\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(\frac{-5}{8} + \color{blue}{\frac{3}{32} \cdot {v}^{2}}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(\frac{-5}{8}, \color{blue}{\left(\frac{3}{32} \cdot {v}^{2}\right)}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(\frac{-5}{8}, \left({v}^{2} \cdot \color{blue}{\frac{3}{32}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(\frac{-5}{8}, \mathsf{*.f64}\left(\left({v}^{2}\right), \color{blue}{\frac{3}{32}}\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(\frac{-5}{8}, \mathsf{*.f64}\left(\left(v \cdot v\right), \frac{3}{32}\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(\frac{-5}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \frac{3}{32}\right)\right)\right)\right)\right)\right) \]

7. Simplified 99.3%

   \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0.25 + v \cdot \left(v \cdot \left(-0.625 + \left(v \cdot v\right) \cdot 0.09375\right)\right)\right)} \]
8. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2}}{\frac{1}{0.25 + \left(v \cdot v\right) \cdot -0.625}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (sqrt 2.0) (/ 1.0 (+ 0.25 (* (* v v) -0.625)))))

C

double code(double v) {
	return sqrt(2.0) / (1.0 / (0.25 + ((v * v) * -0.625)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) / (1.0d0 / (0.25d0 + ((v * v) * (-0.625d0))))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) / (1.0 / (0.25 + ((v * v) * -0.625)));
}

Python

def code(v):
	return math.sqrt(2.0) / (1.0 / (0.25 + ((v * v) * -0.625)))

Julia

function code(v)
	return Float64(sqrt(2.0) / Float64(1.0 / Float64(0.25 + Float64(Float64(v * v) * -0.625))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) / (1.0 / (0.25 + ((v * v) * -0.625)));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] / N[(1.0 / N[(0.25 + N[(N[(v * v), $MachinePrecision] * -0.625), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{4} \cdot \left(1 - v \cdot v\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{4}} \cdot \left(1 - v \cdot v\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)}{\color{blue}{4}}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \color{blue}{\frac{1 - v \cdot v}{4}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right), \color{blue}{\left(\frac{1 - v \cdot v}{4}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - 3 \cdot \left(v \cdot v\right)\right)\right), \left(\frac{\color{blue}{1 - v \cdot v}}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(3 \cdot \left(v \cdot v\right)\right)\right)\right), \left(\frac{\color{blue}{1} - v \cdot v}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \left(v \cdot v\right)\right)\right)\right), \left(\frac{1 - v \cdot v}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1 - v \cdot v}{4}\right)\right)\right) \]

   13. div-sub N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} - \color{blue}{\frac{v \cdot v}{4}}\right)\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} + \color{blue}{\left(\mathsf{neg}\left(\frac{v \cdot v}{4}\right)\right)}\right)\right)\right) \]

   15. distribute-frac-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} + \frac{\mathsf{neg}\left(v \cdot v\right)}{\color{blue}{4}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{4}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(v \cdot v\right)}{4}\right)}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{\color{blue}{\mathsf{neg}\left(v \cdot v\right)}}{4}\right)\right)\right)\right) \]

   18. distribute-frac-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\mathsf{neg}\left(\frac{v \cdot v}{4}\right)\right)\right)\right)\right) \]

   19. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{v \cdot v}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left(v \cdot v\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(0.25 + \frac{v \cdot v}{-4}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-out N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\frac{1}{4} + \frac{-5}{8} \cdot {v}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{\frac{1}{4}} + \frac{-5}{8} \cdot {v}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-5}{8} \cdot {v}^{2}\right)}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left({v}^{2} \cdot \color{blue}{\frac{-5}{8}}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(v \cdot \color{blue}{\left(v \cdot \frac{-5}{8}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \color{blue}{\left(v \cdot \frac{-5}{8}\right)}\right)\right)\right) \]

   14. *-lowering-*.f64 99.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \color{blue}{\frac{-5}{8}}\right)\right)\right)\right) \]

7. Simplified 99.2%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(0.25 + v \cdot \left(v \cdot -0.625\right)\right)} \]
8. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{\frac{1}{4} \cdot \frac{1}{4} - \left(v \cdot \left(v \cdot \frac{-5}{8}\right)\right) \cdot \left(v \cdot \left(v \cdot \frac{-5}{8}\right)\right)}{\color{blue}{\frac{1}{4} - v \cdot \left(v \cdot \frac{-5}{8}\right)}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{1}{\color{blue}{\frac{\frac{1}{4} - v \cdot \left(v \cdot \frac{-5}{8}\right)}{\frac{1}{4} \cdot \frac{1}{4} - \left(v \cdot \left(v \cdot \frac{-5}{8}\right)\right) \cdot \left(v \cdot \left(v \cdot \frac{-5}{8}\right)\right)}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{4} - v \cdot \left(v \cdot \frac{-5}{8}\right)}{\frac{1}{4} \cdot \frac{1}{4} - \left(v \cdot \left(v \cdot \frac{-5}{8}\right)\right) \cdot \left(v \cdot \left(v \cdot \frac{-5}{8}\right)\right)}\right)}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{4} - \left(v \cdot \left(v \cdot \frac{-5}{8}\right)\right) \cdot \left(v \cdot \left(v \cdot \frac{-5}{8}\right)\right)}{\frac{1}{4} - v \cdot \left(v \cdot \frac{-5}{8}\right)}}}\right)\right)\right) \]

   5. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \left(\frac{1}{\frac{1}{4} + \color{blue}{v \cdot \left(v \cdot \frac{-5}{8}\right)}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{4} + v \cdot \left(v \cdot \frac{-5}{8}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(v \cdot \left(v \cdot \frac{-5}{8}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \color{blue}{\left(v \cdot \frac{-5}{8}\right)}\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 99.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \color{blue}{\frac{-5}{8}}\right)\right)\right)\right)\right)\right) \]

9. Applied egg-rr 99.2%

   \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{1}{\frac{1}{0.25 + v \cdot \left(v \cdot -0.625\right)}}} \]
10. Step-by-step derivation

    1. un-div-inv N/A

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

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

       \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\frac{1}{\frac{1}{4} + v \cdot \left(v \cdot \frac{-5}{8}\right)}\right)}\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{\color{blue}{1}}{\frac{1}{4} + v \cdot \left(v \cdot \frac{-5}{8}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{4} + v \cdot \left(v \cdot \frac{-5}{8}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(v \cdot \left(v \cdot \frac{-5}{8}\right)\right)}\right)\right)\right) \]

    6. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(v \cdot v\right) \cdot \color{blue}{\frac{-5}{8}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(v \cdot v\right), \color{blue}{\frac{-5}{8}}\right)\right)\right)\right) \]

    8. *-lowering-*.f64 99.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \frac{-5}{8}\right)\right)\right)\right) \]

11. Applied egg-rr 99.2%

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{1}{0.25 + \left(v \cdot v\right) \cdot -0.625}}} \]
12. Add Preprocessing

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

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(v * Float64(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[(v * N[(v * -0.625), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{4} \cdot \left(1 - v \cdot v\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{4}} \cdot \left(1 - v \cdot v\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)}{\color{blue}{4}}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \color{blue}{\frac{1 - v \cdot v}{4}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right), \color{blue}{\left(\frac{1 - v \cdot v}{4}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - 3 \cdot \left(v \cdot v\right)\right)\right), \left(\frac{\color{blue}{1 - v \cdot v}}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(3 \cdot \left(v \cdot v\right)\right)\right)\right), \left(\frac{\color{blue}{1} - v \cdot v}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \left(v \cdot v\right)\right)\right)\right), \left(\frac{1 - v \cdot v}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1 - v \cdot v}{4}\right)\right)\right) \]

   13. div-sub N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} - \color{blue}{\frac{v \cdot v}{4}}\right)\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} + \color{blue}{\left(\mathsf{neg}\left(\frac{v \cdot v}{4}\right)\right)}\right)\right)\right) \]

   15. distribute-frac-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} + \frac{\mathsf{neg}\left(v \cdot v\right)}{\color{blue}{4}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{4}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(v \cdot v\right)}{4}\right)}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{\color{blue}{\mathsf{neg}\left(v \cdot v\right)}}{4}\right)\right)\right)\right) \]

   18. distribute-frac-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\mathsf{neg}\left(\frac{v \cdot v}{4}\right)\right)\right)\right)\right) \]

   19. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{v \cdot v}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left(v \cdot v\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(0.25 + \frac{v \cdot v}{-4}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-out N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\frac{1}{4} + \frac{-5}{8} \cdot {v}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{\frac{1}{4}} + \frac{-5}{8} \cdot {v}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-5}{8} \cdot {v}^{2}\right)}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left({v}^{2} \cdot \color{blue}{\frac{-5}{8}}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(v \cdot \color{blue}{\left(v \cdot \frac{-5}{8}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \color{blue}{\left(v \cdot \frac{-5}{8}\right)}\right)\right)\right) \]

   14. *-lowering-*.f64 99.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \color{blue}{\frac{-5}{8}}\right)\right)\right)\right) \]

7. Simplified 99.2%

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

Alternative 5: 99.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot 0.25 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (sqrt 2.0) 0.25))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * 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. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{4} \cdot \left(1 - v \cdot v\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{4}} \cdot \left(1 - v \cdot v\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)}{\color{blue}{4}}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \color{blue}{\frac{1 - v \cdot v}{4}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right), \color{blue}{\left(\frac{1 - v \cdot v}{4}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - 3 \cdot \left(v \cdot v\right)\right)\right), \left(\frac{\color{blue}{1 - v \cdot v}}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(3 \cdot \left(v \cdot v\right)\right)\right)\right), \left(\frac{\color{blue}{1} - v \cdot v}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \left(v \cdot v\right)\right)\right)\right), \left(\frac{1 - v \cdot v}{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1 - v \cdot v}{4}\right)\right)\right) \]

   13. div-sub N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} - \color{blue}{\frac{v \cdot v}{4}}\right)\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} + \color{blue}{\left(\mathsf{neg}\left(\frac{v \cdot v}{4}\right)\right)}\right)\right)\right) \]

   15. distribute-frac-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \left(\frac{1}{4} + \frac{\mathsf{neg}\left(v \cdot v\right)}{\color{blue}{4}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{4}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(v \cdot v\right)}{4}\right)}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{\color{blue}{\mathsf{neg}\left(v \cdot v\right)}}{4}\right)\right)\right)\right) \]

   18. distribute-frac-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\mathsf{neg}\left(\frac{v \cdot v}{4}\right)\right)\right)\right)\right) \]

   19. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{v \cdot v}{\color{blue}{\mathsf{neg}\left(4\right)}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left(v \cdot v\right), \color{blue}{\left(\mathsf{neg}\left(4\right)\right)}\right)\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(0.25 + \frac{v \cdot v}{-4}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \color{blue}{\frac{1}{4}}\right) \]
6. Step-by-step derivation

7. Simplified 98.5%

   \[\leadsto \sqrt{2} \cdot \color{blue}{0.25} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))))