Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 6

Speedup: 1.0×

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(-3, {v}^{2}, 1\right) \cdot 0.125} \cdot \left(1 - v \cdot v\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (sqrt (* (fma -3.0 (pow v 2.0) 1.0) 0.125)) (- 1.0 (* v v))))

C

double code(double v) {
	return sqrt((fma(-3.0, pow(v, 2.0), 1.0) * 0.125)) * (1.0 - (v * v));
}

Julia

function code(v)
	return Float64(sqrt(Float64(fma(-3.0, (v ^ 2.0), 1.0) * 0.125)) * Float64(1.0 - Float64(v * v)))
end

Wolfram

code[v_] := N[(N[Sqrt[N[(N[(-3.0 * N[Power[v, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] * 0.125), $MachinePrecision]], $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 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 99.9%

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

   7. cancel-sign-sub-inv 99.9%

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

   8. metadata-eval 99.9%

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

   9. +-commutative 99.9%

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

   10. fma-def 99.9%

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

   11. pow2 99.9%

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

   12. frac-times 99.9%

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

   13. rem-square-sqrt 100.0%

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

   14. metadata-eval 100.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(-3, {v}^{2}, 1\right) \cdot \frac{2}{\color{blue}{16}}} \cdot \left(1 - v \cdot v\right) \]

   15. metadata-eval 100.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(-3, {v}^{2}, 1\right) \cdot \color{blue}{0.125}} \cdot \left(1 - v \cdot v\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-3, {v}^{2}, 1\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]

5. Final simplification 100.0%

   \[\leadsto \sqrt{\mathsf{fma}\left(-3, {v}^{2}, 1\right) \cdot 0.125} \cdot \left(1 - v \cdot v\right) \]
6. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Final simplification 100.0%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in v around 0 99.7%

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

   1. *-commutative 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 5: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - v \cdot v\right) \cdot {0.125}^{0.5} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (- 1.0 (* v v)) (pow 0.125 0.5)))

C

double code(double v) {
	return (1.0 - (v * v)) * pow(0.125, 0.5);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * (0.125d0 ** 0.5d0)
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * Math.pow(0.125, 0.5);
}

Python

def code(v):
	return (1.0 - (v * v)) * math.pow(0.125, 0.5)

Julia

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) * (0.125 ^ 0.5))
end

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * (0.125 ^ 0.5);
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Power[0.125, 0.5], $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 98.5%

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

   2. pow2 98.5%

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

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{\mathsf{fma}\left(-3, {v}^{2}, 1\right) \cdot 0.125}}\right)}^{2}} \cdot \left(1 - v \cdot v\right) \]

5. Taylor expanded in v around 0 97.7%

   \[\leadsto {\color{blue}{\left(\sqrt{\sqrt{0.125}}\right)}}^{2} \cdot \left(1 - v \cdot v\right) \]
6. Step-by-step derivation

   1. unpow2 97.7%

      \[\leadsto \color{blue}{\left(\sqrt{\sqrt{0.125}} \cdot \sqrt{\sqrt{0.125}}\right)} \cdot \left(1 - v \cdot v\right) \]

   2. sqrt-unprod 99.2%

      \[\leadsto \color{blue}{\sqrt{\sqrt{0.125} \cdot \sqrt{0.125}}} \cdot \left(1 - v \cdot v\right) \]

   3. pow1/2 99.2%

      \[\leadsto \color{blue}{{\left(\sqrt{0.125} \cdot \sqrt{0.125}\right)}^{0.5}} \cdot \left(1 - v \cdot v\right) \]

   4. sqr-pow 97.7%

      \[\leadsto \color{blue}{\left({\left(\sqrt{0.125} \cdot \sqrt{0.125}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\sqrt{0.125} \cdot \sqrt{0.125}\right)}^{\left(\frac{0.5}{2}\right)}\right)} \cdot \left(1 - v \cdot v\right) \]

   5. pow1/2 97.7%

      \[\leadsto \left({\left(\color{blue}{{0.125}^{0.5}} \cdot \sqrt{0.125}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\sqrt{0.125} \cdot \sqrt{0.125}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   6. pow1/2 97.7%

      \[\leadsto \left({\left({0.125}^{0.5} \cdot \color{blue}{{0.125}^{0.5}}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\sqrt{0.125} \cdot \sqrt{0.125}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   7. pow-prod-up 97.7%

      \[\leadsto \left({\color{blue}{\left({0.125}^{\left(0.5 + 0.5\right)}\right)}}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\sqrt{0.125} \cdot \sqrt{0.125}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   8. metadata-eval 97.7%

      \[\leadsto \left({\left({0.125}^{\color{blue}{1}}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\sqrt{0.125} \cdot \sqrt{0.125}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   9. metadata-eval 97.7%

      \[\leadsto \left({\color{blue}{0.125}}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\sqrt{0.125} \cdot \sqrt{0.125}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   10. metadata-eval 97.7%

       \[\leadsto \left({0.125}^{\color{blue}{0.25}} \cdot {\left(\sqrt{0.125} \cdot \sqrt{0.125}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   11. pow1/2 97.7%

       \[\leadsto \left({0.125}^{0.25} \cdot {\left(\color{blue}{{0.125}^{0.5}} \cdot \sqrt{0.125}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   12. pow1/2 97.7%

       \[\leadsto \left({0.125}^{0.25} \cdot {\left({0.125}^{0.5} \cdot \color{blue}{{0.125}^{0.5}}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   13. pow-prod-up 97.7%

       \[\leadsto \left({0.125}^{0.25} \cdot {\color{blue}{\left({0.125}^{\left(0.5 + 0.5\right)}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   14. metadata-eval 97.7%

       \[\leadsto \left({0.125}^{0.25} \cdot {\left({0.125}^{\color{blue}{1}}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   15. metadata-eval 97.7%

       \[\leadsto \left({0.125}^{0.25} \cdot {\color{blue}{0.125}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   16. metadata-eval 97.7%

       \[\leadsto \left({0.125}^{0.25} \cdot {0.125}^{\color{blue}{0.25}}\right) \cdot \left(1 - v \cdot v\right) \]

7. Applied egg-rr 97.7%

   \[\leadsto \color{blue}{\left({0.125}^{0.25} \cdot {0.125}^{0.25}\right)} \cdot \left(1 - v \cdot v\right) \]
8. Step-by-step derivation

   1. pow-prod-up 99.2%

      \[\leadsto \color{blue}{{0.125}^{\left(0.25 + 0.25\right)}} \cdot \left(1 - v \cdot v\right) \]

   2. metadata-eval 99.2%

      \[\leadsto {0.125}^{\color{blue}{0.5}} \cdot \left(1 - v \cdot v\right) \]

9. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{0.125}^{0.5}} \cdot \left(1 - v \cdot v\right) \]

10. Final simplification 99.2%

    \[\leadsto \left(1 - v \cdot v\right) \cdot {0.125}^{0.5} \]
11. Add Preprocessing

Alternative 6: 99.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.125} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (sqrt 0.125))

C

double code(double v) {
	return sqrt(0.125);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(0.125d0)
end function

Java

public static double code(double v) {
	return Math.sqrt(0.125);
}

Python

def code(v):
	return math.sqrt(0.125)

Julia

function code(v)
	return sqrt(0.125)
end

MATLAB

function tmp = code(v)
	tmp = sqrt(0.125);
end

Wolfram

code[v_] := N[Sqrt[0.125], $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 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 99.9%

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

   7. cancel-sign-sub-inv 99.9%

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

   8. metadata-eval 99.9%

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

   9. +-commutative 99.9%

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

   10. fma-def 99.9%

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

   11. pow2 99.9%

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

   12. frac-times 99.9%

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

   13. rem-square-sqrt 100.0%

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

   14. metadata-eval 100.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(-3, {v}^{2}, 1\right) \cdot \frac{2}{\color{blue}{16}}} \cdot \left(1 - v \cdot v\right) \]

   15. metadata-eval 100.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(-3, {v}^{2}, 1\right) \cdot \color{blue}{0.125}} \cdot \left(1 - v \cdot v\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-3, {v}^{2}, 1\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]

5. Taylor expanded in v around 0 99.1%

   \[\leadsto \color{blue}{\sqrt{0.125}} \]

6. Final simplification 99.1%

   \[\leadsto \sqrt{0.125} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(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))))