Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

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. Add Preprocessing

Alternative 2: 99.6% 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. Step-by-step derivation

   1. associate-*l* 100.0%

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

   2. sqr-neg 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. metadata-eval 100.0%

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

   5. sqr-neg 100.0%

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

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. Add Preprocessing
5. Step-by-step derivation

   1. flip-+ 100.0%

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

   2. sqrt-div 100.0%

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

   3. metadata-eval 100.0%

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

   4. *-commutative 100.0%

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

   5. *-commutative 100.0%

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

   6. swap-sqr 100.0%

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

   7. pow2 100.0%

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

   8. pow2 100.0%

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

   9. pow-prod-up 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

       \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(\frac{\sqrt{1 - {v}^{4} \cdot \color{blue}{9}}}{\sqrt{1 - -3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)\right) \]

   12. cancel-sign-sub-inv 100.0%

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

   13. metadata-eval 100.0%

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

   14. pow2 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in v around 0 99.5%

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

    1. *-commutative 99.5%

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

    2. associate-*r* 99.5%

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

    3. *-commutative 99.5%

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

    4. *-commutative 99.5%

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

    5. distribute-rgt-out 99.5%

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

    6. metadata-eval 99.5%

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

    7. associate-*l* 99.5%

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

    8. metadata-eval 99.5%

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

    9. associate-*r* 99.5%

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

    10. *-commutative 99.5%

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

    11. associate-*r* 99.5%

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

    12. distribute-rgt-out 99.5%

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

    13. *-commutative 99.5%

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

11. Simplified 99.5%

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

Alternative 3: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(N[Sqrt[2.0], $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. Step-by-step derivation

   1. associate-*l* 100.0%

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

   2. sqr-neg 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. metadata-eval 100.0%

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

   5. sqr-neg 100.0%

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

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. Add Preprocessing

5. Taylor expanded in v around 0 99.1%

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

6. Final simplification 99.1%

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

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%

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

   2. sqr-neg 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. metadata-eval 100.0%

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

   5. sqr-neg 100.0%

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

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. Add Preprocessing

5. Taylor expanded in v around 0 99.1%

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

   1. add-sqr-sqrt 97.5%

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

   2. pow2 97.5%

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

   3. add-sqr-sqrt 97.5%

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

   4. sqrt-unprod 97.5%

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

   5. frac-times 97.5%

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

   6. rem-square-sqrt 97.5%

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

   7. metadata-eval 97.5%

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

   8. metadata-eval 97.5%

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

   9. *-lft-identity 97.5%

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

   10. pow2 97.5%

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

7. Applied egg-rr 97.5%

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

8. Taylor expanded in v around 0 99.0%

   \[\leadsto \color{blue}{\sqrt{0.125}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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))))