Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-*l* 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-*l* 100.0%

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

   2. fma-udef 100.0%

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

   3. *-commutative 100.0%

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

   4. fma-def 100.0%

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

   5. *-commutative 100.0%

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

   6. fma-udef 100.0%

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

   7. +-commutative 100.0%

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

   8. distribute-rgt-neg-out 100.0%

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

   9. unsub-neg 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 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. Final simplification 100.0%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot -2.5\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (/ (sqrt 2.0) 4.0) (+ 1.0 (* (* v v) -2.5))))

C

double code(double v) {
	return (sqrt(2.0) / 4.0) * (1.0 + ((v * v) * -2.5));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (sqrt(2.0d0) / 4.0d0) * (1.0d0 + ((v * v) * (-2.5d0)))
end function

Java

public static double code(double v) {
	return (Math.sqrt(2.0) / 4.0) * (1.0 + ((v * v) * -2.5));
}

Python

def code(v):
	return (math.sqrt(2.0) / 4.0) * (1.0 + ((v * v) * -2.5))

Julia

function code(v)
	return Float64(Float64(sqrt(2.0) / 4.0) * Float64(1.0 + Float64(Float64(v * v) * -2.5)))
end

MATLAB

function tmp = code(v)
	tmp = (sqrt(2.0) / 4.0) * (1.0 + ((v * v) * -2.5));
end

Wolfram

code[v_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[(1.0 + N[(N[(v * v), $MachinePrecision] * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-*l* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.7%

   \[\leadsto \frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + -2.5 \cdot {v}^{2}\right)} \]
5. Step-by-step derivation

   1. unpow2 99.7%

      \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(1 + -2.5 \cdot \color{blue}{\left(v \cdot v\right)}\right) \]

6. Simplified 99.7%

   \[\leadsto \frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + -2.5 \cdot \left(v \cdot v\right)\right)} \]

7. Final simplification 99.7%

   \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot -2.5\right) \]

Alternative 4: 99.0% 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. Step-by-step derivation

   1. associate-*l* 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-*l* 100.0%

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

   2. fma-udef 100.0%

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

   3. *-commutative 100.0%

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

   4. fma-def 100.0%

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

   5. *-commutative 100.0%

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

   6. fma-udef 100.0%

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

   7. +-commutative 100.0%

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

   8. distribute-rgt-neg-out 100.0%

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

   9. unsub-neg 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in v around 0 99.4%

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

9. Final simplification 99.4%

   \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023182 
(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))))