Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(0.25 + \frac{v \cdot v}{-4}\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (sqrt 2.0) (* (sqrt (- 1.0 (* 3.0 (* v v)))) (+ 0.25 (/ (* v v) -4.0)))))

C

double code(double v) {
	return sqrt(2.0) * (sqrt((1.0 - (3.0 * (v * v)))) * (0.25 + ((v * v) / -4.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) * (sqrt((1.0d0 - (3.0d0 * (v * v)))) * (0.25d0 + ((v * v) / (-4.0d0))))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) * (Math.sqrt((1.0 - (3.0 * (v * v)))) * (0.25 + ((v * v) / -4.0)));
}

Python

def code(v):
	return math.sqrt(2.0) * (math.sqrt((1.0 - (3.0 * (v * v)))) * (0.25 + ((v * v) / -4.0)))

Julia

function code(v)
	return Float64(sqrt(2.0) * Float64(sqrt(Float64(1.0 - Float64(3.0 * Float64(v * v)))) * Float64(0.25 + Float64(Float64(v * v) / -4.0))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) * (sqrt((1.0 - (3.0 * (v * v)))) * (0.25 + ((v * v) / -4.0)));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.25 + N[(N[(v * v), $MachinePrecision] / -4.0), $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. Add Preprocessing

Alternative 2: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(0.25 + \frac{v \cdot v}{-4}\right) \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (+ 0.25 (/ (* v v) -4.0)) (sqrt (+ 2.0 (* (* v v) -6.0)))))

C

double code(double v) {
	return (0.25 + ((v * v) / -4.0)) * sqrt((2.0 + ((v * v) * -6.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (0.25d0 + ((v * v) / (-4.0d0))) * sqrt((2.0d0 + ((v * v) * (-6.0d0))))
end function

Java

public static double code(double v) {
	return (0.25 + ((v * v) / -4.0)) * Math.sqrt((2.0 + ((v * v) * -6.0)));
}

Python

def code(v):
	return (0.25 + ((v * v) / -4.0)) * math.sqrt((2.0 + ((v * v) * -6.0)))

Julia

function code(v)
	return Float64(Float64(0.25 + Float64(Float64(v * v) / -4.0)) * sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0))))
end

MATLAB

function tmp = code(v)
	tmp = (0.25 + ((v * v) / -4.0)) * sqrt((2.0 + ((v * v) * -6.0)));
end

Wolfram

code[v_] := N[(N[(0.25 + N[(N[(v * v), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $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. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. sqrt-unprod N/A

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

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

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

   9. sub-neg N/A

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

   10. distribute-lft-in N/A

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

   11. metadata-eval N/A

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

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

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

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

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

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

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

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

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

   18. *-commutative N/A

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

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

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

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

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

   21. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*l* N/A

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

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

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

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

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

   8. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \left(0.25 + \frac{v \cdot v}{-4}\right) \cdot \sqrt{\color{blue}{\left(v \cdot v\right) \cdot -6 + 2}} \]

9. Final simplification 100.0%

   \[\leadsto \left(0.25 + \frac{v \cdot v}{-4}\right) \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6} \]
10. Add Preprocessing

Alternative 3: 99.7% accurate, 1.9× speedup

Math

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

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(Float64(v * 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[(N[(v * v), $MachinePrecision] * N[(-0.625 + N[(N[(v * v), $MachinePrecision] * 0.09375), $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. *-lowering-*.f64 N/A

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 99.5%

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

7. Simplified 99.5%

   \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0.25 + \left(v \cdot v\right) \cdot \left(-0.625 + \left(v \cdot v\right) \cdot 0.09375\right)\right)} \]
8. 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 99.5%

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

7. Simplified 99.5%

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

Alternative 5: 98.9% 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 98.8%

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

Reproduce

herbie shell --seed 2024138 
(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))))