Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 11.2s

Alternatives: 3

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. sqrt-unprod 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. metadata-eval 100.0%

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

   6. swap-sqr 100.0%

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

   7. sqrt-unprod 43.4%

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

   8. add-sqr-sqrt 97.2%

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

   9. metadata-eval 97.2%

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

   10. distribute-lft-neg-in 97.2%

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

   11. add-log-exp 97.2%

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

   12. neg-log 97.2%

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

   13. add-sqr-sqrt 97.2%

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

   14. sqrt-unprod 97.2%

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

   15. swap-sqr 97.2%

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

   16. metadata-eval 97.2%

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \log \left(\frac{1}{e^{\sqrt{\color{blue}{9} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}}}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   17. metadata-eval 97.2%

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

   18. swap-sqr 97.2%

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

   19. sqrt-unprod 43.4%

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

   20. add-sqr-sqrt 100.0%

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

   21. exp-prod 100.0%

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

   22. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

   2. pow1/2 100.0%

      \[\leadsto \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{1 - \log \left(\frac{1}{{\left(e^{-3}\right)}^{\left({v}^{2}\right)}}\right)}}{4} \cdot \left(1 - v \cdot v\right) \]

   3. pow1/2 100.0%

      \[\leadsto \frac{{2}^{0.5} \cdot \color{blue}{{\left(1 - \log \left(\frac{1}{{\left(e^{-3}\right)}^{\left({v}^{2}\right)}}\right)\right)}^{0.5}}}{4} \cdot \left(1 - v \cdot v\right) \]

   4. pow-prod-down 100.0%

      \[\leadsto \frac{\color{blue}{{\left(2 \cdot \left(1 - \log \left(\frac{1}{{\left(e^{-3}\right)}^{\left({v}^{2}\right)}}\right)\right)\right)}^{0.5}}}{4} \cdot \left(1 - v \cdot v\right) \]

   5. pow-flip 100.0%

      \[\leadsto \frac{{\left(2 \cdot \left(1 - \log \color{blue}{\left({\left(e^{-3}\right)}^{\left(-{v}^{2}\right)}\right)}\right)\right)}^{0.5}}{4} \cdot \left(1 - v \cdot v\right) \]

   6. log-pow 100.0%

      \[\leadsto \frac{{\left(2 \cdot \left(1 - \color{blue}{\left(-{v}^{2}\right) \cdot \log \left(e^{-3}\right)}\right)\right)}^{0.5}}{4} \cdot \left(1 - v \cdot v\right) \]

   7. rem-log-exp 100.0%

      \[\leadsto \frac{{\left(2 \cdot \left(1 - \left(-{v}^{2}\right) \cdot \color{blue}{-3}\right)\right)}^{0.5}}{4} \cdot \left(1 - v \cdot v\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(1 - \left(-{v}^{2}\right) \cdot -3\right)\right)}^{0.5}}{4}} \cdot \left(1 - v \cdot v\right) \]
7. Step-by-step derivation

   1. unpow1/2 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-lft-in 100.0%

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

   4. metadata-eval 100.0%

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

8. Simplified 100.0%

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

   1. pow2 100.0%

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

10. Applied egg-rr 100.0%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. sqrt-unprod 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. metadata-eval 100.0%

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

   6. swap-sqr 100.0%

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

   7. sqrt-unprod 43.4%

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

   8. add-sqr-sqrt 97.2%

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

   9. metadata-eval 97.2%

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

   10. distribute-lft-neg-in 97.2%

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

   11. add-log-exp 97.2%

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

   12. neg-log 97.2%

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

   13. add-sqr-sqrt 97.2%

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

   14. sqrt-unprod 97.2%

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

   15. swap-sqr 97.2%

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

   16. metadata-eval 97.2%

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \log \left(\frac{1}{e^{\sqrt{\color{blue}{9} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}}}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   17. metadata-eval 97.2%

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

   18. swap-sqr 97.2%

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

   19. sqrt-unprod 43.4%

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

   20. add-sqr-sqrt 100.0%

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

   21. exp-prod 100.0%

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

   22. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in v around 0 98.7%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} + 0.25 \cdot \left({v}^{2} \cdot \left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + 0.25 \cdot \color{blue}{\left(\left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right) \cdot {v}^{2}\right)} \]

   2. associate-*r* 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(0.25 \cdot \left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right) \cdot {v}^{2}} \]

   3. *-commutative 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(\left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right) \cdot 0.25\right)} \cdot {v}^{2} \]

   4. distribute-rgt-out 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(\color{blue}{\left(\sqrt{2} \cdot \left(-1.5 + -1\right)\right)} \cdot 0.25\right) \cdot {v}^{2} \]

   5. associate-*l* 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(\sqrt{2} \cdot \left(\left(-1.5 + -1\right) \cdot 0.25\right)\right)} \cdot {v}^{2} \]

   6. metadata-eval 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(\sqrt{2} \cdot \left(\color{blue}{-2.5} \cdot 0.25\right)\right) \cdot {v}^{2} \]

   7. metadata-eval 98.7%

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

   8. *-commutative 98.7%

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

   9. associate-*r* 98.7%

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

   10. *-commutative 98.7%

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

   11. associate-*r* 98.7%

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

   12. distribute-rgt-out 98.7%

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

7. Simplified 98.7%

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

   1. pow2 100.0%

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

9. Applied egg-rr 98.7%

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

10. Final simplification 98.7%

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

Alternative 3: 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. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 100.0%

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

   2. sqrt-unprod 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. metadata-eval 100.0%

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

   6. swap-sqr 100.0%

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

   7. sqrt-unprod 43.4%

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

   8. add-sqr-sqrt 97.2%

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

   9. metadata-eval 97.2%

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

   10. distribute-lft-neg-in 97.2%

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

   11. add-log-exp 97.2%

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

   12. neg-log 97.2%

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

   13. add-sqr-sqrt 97.2%

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

   14. sqrt-unprod 97.2%

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

   15. swap-sqr 97.2%

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

   16. metadata-eval 97.2%

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \log \left(\frac{1}{e^{\sqrt{\color{blue}{9} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}}}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   17. metadata-eval 97.2%

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

   18. swap-sqr 97.2%

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

   19. sqrt-unprod 43.4%

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

   20. add-sqr-sqrt 100.0%

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

   21. exp-prod 100.0%

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

   22. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in v around 0 98.7%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} + 0.25 \cdot \left({v}^{2} \cdot \left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + 0.25 \cdot \color{blue}{\left(\left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right) \cdot {v}^{2}\right)} \]

   2. associate-*r* 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(0.25 \cdot \left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right) \cdot {v}^{2}} \]

   3. *-commutative 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(\left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right) \cdot 0.25\right)} \cdot {v}^{2} \]

   4. distribute-rgt-out 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(\color{blue}{\left(\sqrt{2} \cdot \left(-1.5 + -1\right)\right)} \cdot 0.25\right) \cdot {v}^{2} \]

   5. associate-*l* 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(\sqrt{2} \cdot \left(\left(-1.5 + -1\right) \cdot 0.25\right)\right)} \cdot {v}^{2} \]

   6. metadata-eval 98.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(\sqrt{2} \cdot \left(\color{blue}{-2.5} \cdot 0.25\right)\right) \cdot {v}^{2} \]

   7. metadata-eval 98.7%

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

   8. *-commutative 98.7%

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

   9. associate-*r* 98.7%

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

   10. *-commutative 98.7%

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

   11. associate-*r* 98.7%

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

   12. distribute-rgt-out 98.7%

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

7. Simplified 98.7%

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

8. Taylor expanded in v around 0 97.2%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2}} \]

9. Final simplification 97.2%

   \[\leadsto \sqrt{2} \cdot 0.25 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(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))))