Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

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

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

FPCore

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

C

double code(double v) {
	return (sqrt(0.125) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (sqrt(0.125d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

public static double code(double v) {
	return (Math.sqrt(0.125) * Math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

Python

def code(v):
	return (math.sqrt(0.125) * math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v))

Julia

function code(v)
	return Float64(Float64(sqrt(0.125) * sqrt(Float64(1.0 - Float64(3.0 * Float64(v * v))))) * Float64(1.0 - Float64(v * v)))
end

MATLAB

function tmp = code(v)
	tmp = (sqrt(0.125) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
end

Wolfram

code[v_] := N[(N[(N[Sqrt[0.125], $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]

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. *-un-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. add-sqr-sqrt 98.4%

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

   4. sqrt-unprod 100.0%

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

   5. frac-times 100.0%

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

   6. rem-square-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto \left(\left(\sqrt{\color{blue}{0.125}} \cdot 1\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \left(\color{blue}{\left(\sqrt{0.125} \cdot 1\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
5. Step-by-step derivation

   1. *-rgt-identity 100.0%

      \[\leadsto \left(\color{blue}{\sqrt{0.125}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

6. Simplified 100.0%

   \[\leadsto \left(\color{blue}{\sqrt{0.125}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 - v \cdot v\right) \cdot \sqrt{0.125 + 0.125 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (- 1.0 (* v v)) (sqrt (+ 0.125 (* 0.125 (* (* v v) -3.0))))))

C

double code(double v) {
	return (1.0 - (v * v)) * sqrt((0.125 + (0.125 * ((v * v) * -3.0))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * sqrt((0.125d0 + (0.125d0 * ((v * v) * (-3.0d0)))))
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * Math.sqrt((0.125 + (0.125 * ((v * v) * -3.0))));
}

Python

def code(v):
	return (1.0 - (v * v)) * math.sqrt((0.125 + (0.125 * ((v * v) * -3.0))))

Julia

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) * sqrt(Float64(0.125 + Float64(0.125 * Float64(Float64(v * v) * -3.0)))))
end

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * sqrt((0.125 + (0.125 * ((v * v) * -3.0))));
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.125 + N[(0.125 * 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. Step-by-step derivation

   1. add-sqr-sqrt 98.4%

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

   2. pow2 98.4%

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

   3. pow1/2 98.4%

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

   4. sqrt-pow1 98.4%

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

   5. metadata-eval 98.4%

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

4. Applied egg-rr 98.4%

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

   1. pow-pow 100.0%

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

   2. metadata-eval 100.0%

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

   3. pow1/2 100.0%

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

   4. div-inv 100.0%

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

   5. metadata-eval 100.0%

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

   6. *-commutative 100.0%

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

   7. add-sqr-sqrt 98.4%

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

   8. sqrt-unprod 100.0%

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

   9. swap-sqr 100.0%

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

   10. metadata-eval 100.0%

       \[\leadsto \left(\sqrt{\color{blue}{0.0625} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   11. rem-square-sqrt 100.0%

       \[\leadsto \left(\sqrt{0.0625 \cdot \color{blue}{2}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   12. metadata-eval 100.0%

       \[\leadsto \left(\sqrt{\color{blue}{0.125}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   13. pow1 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(\sqrt{0.125 + 0.125 \cdot \left({v}^{2} \cdot -3\right)}\right)}^{1}} \cdot \left(1 - v \cdot v\right) \]
7. Step-by-step derivation

   1. unpow1 100.0%

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

8. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{0.125 + 0.125 \cdot \left({v}^{2} \cdot -3\right)}} \cdot \left(1 - v \cdot v\right) \]
9. Step-by-step derivation

   1. pow2 100.0%

      \[\leadsto \sqrt{0.125 + 0.125 \cdot \left(\color{blue}{\left(v \cdot v\right)} \cdot -3\right)} \cdot \left(1 - v \cdot v\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \sqrt{0.125 + 0.125 \cdot \left(\color{blue}{\left(v \cdot v\right)} \cdot -3\right)} \cdot \left(1 - v \cdot v\right) \]

11. Final simplification 100.0%

    \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt{0.125 + 0.125 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \]
12. Add Preprocessing

Alternative 3: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0 99.7%

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

   1. distribute-lft-out 99.7%

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

   2. distribute-rgt-out 99.7%

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

   3. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in v around 0 99.7%

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

   1. associate-*r* 99.7%

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

   2. distribute-rgt-out 99.7%

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

   3. *-commutative 99.7%

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

8. Simplified 99.7%

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

   1. pow2 100.0%

      \[\leadsto \sqrt{0.125 + 0.125 \cdot \left(\color{blue}{\left(v \cdot v\right)} \cdot -3\right)} \cdot \left(1 - v \cdot v\right) \]

10. Applied egg-rr 99.7%

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

Alternative 4: 99.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (v) :precision binary64 (* (sqrt 0.125) (- 1.0 (* v v))))

C

double code(double v) {
	return sqrt(0.125) * (1.0 - (v * v));
}

Fortran

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

Java

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

Python

def code(v):
	return math.sqrt(0.125) * (1.0 - (v * v))

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[Sqrt[0.125], $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. *-un-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. add-sqr-sqrt 98.4%

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

   4. sqrt-unprod 100.0%

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

   5. frac-times 100.0%

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

   6. rem-square-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto \left(\left(\sqrt{\color{blue}{0.125}} \cdot 1\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \left(\color{blue}{\left(\sqrt{0.125} \cdot 1\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
5. Step-by-step derivation

   1. *-rgt-identity 100.0%

      \[\leadsto \left(\color{blue}{\sqrt{0.125}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

6. Simplified 100.0%

   \[\leadsto \left(\color{blue}{\sqrt{0.125}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

7. Taylor expanded in v around 0 99.1%

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

Alternative 5: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {0.125}^{0.5} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (pow 0.125 0.5))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(v)
	return 0.125 ^ 0.5
end

MATLAB

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

Wolfram

code[v_] := N[Power[0.125, 0.5], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0 99.1%

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

4. Taylor expanded in v around 0 99.1%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 97.6%

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

   2. sqrt-unprod 99.1%

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

   3. swap-sqr 99.1%

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

   4. metadata-eval 99.1%

      \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)} \]

   5. rem-square-sqrt 99.1%

      \[\leadsto \sqrt{0.0625 \cdot \color{blue}{2}} \]

   6. metadata-eval 99.1%

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

   7. pow1/2 99.1%

      \[\leadsto \color{blue}{{0.125}^{0.5}} \]

6. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{{0.125}^{0.5}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))))