Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 4

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, 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]

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%

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

Alternative 2: 99.5% 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%

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

3. Taylor expanded in v around 0 99.7%

   \[\leadsto \color{blue}{\left(-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \sqrt{2}\right)} \cdot \left(1 - v \cdot v\right) \]

4. Taylor expanded in v around 0 99.7%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} + {v}^{2} \cdot \left(-0.375 \cdot \sqrt{2} + -0.25 \cdot \sqrt{2}\right)} \]
5. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot 0.25} + {v}^{2} \cdot \left(-0.375 \cdot \sqrt{2} + -0.25 \cdot \sqrt{2}\right) \]

   2. distribute-rgt-out 99.7%

      \[\leadsto \sqrt{2} \cdot 0.25 + {v}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-0.375 + -0.25\right)\right)} \]

   3. metadata-eval 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

   8. distribute-lft-out 99.7%

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

   9. *-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 98.9% 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. Taylor expanded in v around 0 99.7%

   \[\leadsto \color{blue}{\left(-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \sqrt{2}\right)} \cdot \left(1 - v \cdot v\right) \]
4. Step-by-step derivation

   1. flip-+ 98.2%

      \[\leadsto \color{blue}{\frac{\left(-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right) - \left(0.25 \cdot \sqrt{2}\right) \cdot \left(0.25 \cdot \sqrt{2}\right)}{-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right) - 0.25 \cdot \sqrt{2}}} \cdot \left(1 - v \cdot v\right) \]

5. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{\left(2 \cdot {v}^{4}\right) \cdot 0.140625 + -0.125}{\sqrt{2} \cdot \left({v}^{2} \cdot -0.375 - 0.25\right)}} \cdot \left(1 - v \cdot v\right) \]

6. Taylor expanded in v around 0 97.7%

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

   1. div-inv 97.7%

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

   2. add-sqr-sqrt 97.7%

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

   3. sqrt-unprod 97.7%

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

   4. div-inv 97.7%

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

   5. div-inv 97.7%

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

   6. frac-times 97.7%

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

   7. metadata-eval 97.7%

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

   8. rem-square-sqrt 99.3%

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

   9. metadata-eval 99.3%

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

   10. pow1/2 99.3%

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

8. Applied egg-rr 99.3%

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

9. Final simplification 99.3%

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

Alternative 4: 98.9% 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. Taylor expanded in v around 0 99.7%

   \[\leadsto \color{blue}{\left(-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \sqrt{2}\right)} \cdot \left(1 - v \cdot v\right) \]
4. Step-by-step derivation

   1. add-sqr-sqrt 53.5%

      \[\leadsto \left(\color{blue}{\sqrt{-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)} \cdot \sqrt{-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)}} + 0.25 \cdot \sqrt{2}\right) \cdot \left(1 - v \cdot v\right) \]

   2. *-commutative 53.5%

      \[\leadsto \left(\sqrt{-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)} \cdot \sqrt{-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)} + \color{blue}{\sqrt{2} \cdot 0.25}\right) \cdot \left(1 - v \cdot v\right) \]

   3. metadata-eval 53.5%

      \[\leadsto \left(\sqrt{-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)} \cdot \sqrt{-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)} + \sqrt{2} \cdot \color{blue}{\frac{1}{4}}\right) \cdot \left(1 - v \cdot v\right) \]

   4. div-inv 53.5%

      \[\leadsto \left(\sqrt{-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)} \cdot \sqrt{-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)} + \color{blue}{\frac{\sqrt{2}}{4}}\right) \cdot \left(1 - v \cdot v\right) \]

   5. fma-define 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)}, \sqrt{-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)}, \frac{\sqrt{2}}{4}\right)} \cdot \left(1 - v \cdot v\right) \]

5. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(v \cdot {2}^{0.25}\right) \cdot \sqrt{-0.375}, \left(v \cdot {2}^{0.25}\right) \cdot \sqrt{-0.375}, \sqrt{0.125}\right)} \cdot \left(1 - v \cdot v\right) \]
6. Step-by-step derivation

   1. fma-undefine 0.0%

      \[\leadsto \color{blue}{\left(\left(\left(v \cdot {2}^{0.25}\right) \cdot \sqrt{-0.375}\right) \cdot \left(\left(v \cdot {2}^{0.25}\right) \cdot \sqrt{-0.375}\right) + \sqrt{0.125}\right)} \cdot \left(1 - v \cdot v\right) \]

   2. +-commutative 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{0.125} + \left(\left(v \cdot {2}^{0.25}\right) \cdot \sqrt{-0.375}\right) \cdot \left(\left(v \cdot {2}^{0.25}\right) \cdot \sqrt{-0.375}\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   3. associate-*l* 0.0%

      \[\leadsto \left(\sqrt{0.125} + \color{blue}{\left(v \cdot \left({2}^{0.25} \cdot \sqrt{-0.375}\right)\right)} \cdot \left(\left(v \cdot {2}^{0.25}\right) \cdot \sqrt{-0.375}\right)\right) \cdot \left(1 - v \cdot v\right) \]

   4. associate-*l* 0.0%

      \[\leadsto \left(\sqrt{0.125} + \left(v \cdot \left({2}^{0.25} \cdot \sqrt{-0.375}\right)\right) \cdot \color{blue}{\left(v \cdot \left({2}^{0.25} \cdot \sqrt{-0.375}\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   5. swap-sqr 0.0%

      \[\leadsto \left(\sqrt{0.125} + \color{blue}{\left(v \cdot v\right) \cdot \left(\left({2}^{0.25} \cdot \sqrt{-0.375}\right) \cdot \left({2}^{0.25} \cdot \sqrt{-0.375}\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   6. unpow2 0.0%

      \[\leadsto \left(\sqrt{0.125} + \color{blue}{{v}^{2}} \cdot \left(\left({2}^{0.25} \cdot \sqrt{-0.375}\right) \cdot \left({2}^{0.25} \cdot \sqrt{-0.375}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   7. *-commutative 0.0%

      \[\leadsto \left(\sqrt{0.125} + {v}^{2} \cdot \left(\color{blue}{\left(\sqrt{-0.375} \cdot {2}^{0.25}\right)} \cdot \left({2}^{0.25} \cdot \sqrt{-0.375}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   8. *-commutative 0.0%

      \[\leadsto \left(\sqrt{0.125} + {v}^{2} \cdot \left(\left(\sqrt{-0.375} \cdot {2}^{0.25}\right) \cdot \color{blue}{\left(\sqrt{-0.375} \cdot {2}^{0.25}\right)}\right)\right) \cdot \left(1 - v \cdot v\right) \]

   9. swap-sqr 0.0%

      \[\leadsto \left(\sqrt{0.125} + {v}^{2} \cdot \color{blue}{\left(\left(\sqrt{-0.375} \cdot \sqrt{-0.375}\right) \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   10. rem-square-sqrt 99.7%

       \[\leadsto \left(\sqrt{0.125} + {v}^{2} \cdot \left(\color{blue}{-0.375} \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right)\right) \cdot \left(1 - v \cdot v\right) \]

   11. pow-sqr 99.7%

       \[\leadsto \left(\sqrt{0.125} + {v}^{2} \cdot \left(-0.375 \cdot \color{blue}{{2}^{\left(2 \cdot 0.25\right)}}\right)\right) \cdot \left(1 - v \cdot v\right) \]

   12. metadata-eval 99.7%

       \[\leadsto \left(\sqrt{0.125} + {v}^{2} \cdot \left(-0.375 \cdot {2}^{\color{blue}{0.5}}\right)\right) \cdot \left(1 - v \cdot v\right) \]

7. Simplified 99.7%

   \[\leadsto \color{blue}{\left(\sqrt{0.125} + {v}^{2} \cdot \left(-0.375 \cdot {2}^{0.5}\right)\right)} \cdot \left(1 - v \cdot v\right) \]

8. Taylor expanded in v around 0 99.2%

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

9. Final simplification 99.2%

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

Reproduce

herbie shell --seed 2024078 
(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))))