Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 12.4s

Alternatives: 4

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

\[\begin{array}{l} \\ \frac{1 - v \cdot v}{\frac{4}{\sqrt{2 \cdot \frac{9 \cdot {v}^{4} + -1}{v \cdot \left(v \cdot -3\right) + -1}}}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/
  (- 1.0 (* v v))
  (/
   4.0
   (sqrt (* 2.0 (/ (+ (* 9.0 (pow v 4.0)) -1.0) (+ (* v (* v -3.0)) -1.0)))))))

C

double code(double v) {
	return (1.0 - (v * v)) / (4.0 / sqrt((2.0 * (((9.0 * pow(v, 4.0)) + -1.0) / ((v * (v * -3.0)) + -1.0)))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) / (4.0d0 / sqrt((2.0d0 * (((9.0d0 * (v ** 4.0d0)) + (-1.0d0)) / ((v * (v * (-3.0d0))) + (-1.0d0))))))
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) / (4.0 / Math.sqrt((2.0 * (((9.0 * Math.pow(v, 4.0)) + -1.0) / ((v * (v * -3.0)) + -1.0)))));
}

Python

def code(v):
	return (1.0 - (v * v)) / (4.0 / math.sqrt((2.0 * (((9.0 * math.pow(v, 4.0)) + -1.0) / ((v * (v * -3.0)) + -1.0)))))

Julia

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) / Float64(4.0 / sqrt(Float64(2.0 * Float64(Float64(Float64(9.0 * (v ^ 4.0)) + -1.0) / Float64(Float64(v * Float64(v * -3.0)) + -1.0))))))
end

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) / (4.0 / sqrt((2.0 * (((9.0 * (v ^ 4.0)) + -1.0) / ((v * (v * -3.0)) + -1.0)))));
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] / N[(4.0 / N[Sqrt[N[(2.0 * N[(N[(N[(9.0 * N[Power[v, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(v * N[(v * -3.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. sqr-neg 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-*l* 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-/l* 100.0%

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

7. Simplified 100.0%

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

   1. fma-udef 100.0%

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

   2. flip-+ 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 + N[(v * N[(v * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $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. *-commutative 100.0%

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

   2. sqr-neg 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-*l* 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-*l* 100.0%

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

7. Simplified 100.0%

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

   1. associate-*r* 100.0%

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

   2. sqrt-prod 100.0%

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

   3. add-cube-cbrt 98.4%

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

   4. unpow3 98.4%

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

   5. unpow2 98.4%

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

   6. sqrt-prod 98.4%

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

   7. add-sqr-sqrt 98.4%

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

   8. *-commutative 98.4%

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

   9. unpow3 98.4%

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

   10. add-cube-cbrt 100.0%

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

9. Applied egg-rr 100.0%

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

    1. fma-udef 100.0%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 3: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] / N[(4.0 / N[Sqrt[2.0], $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. *-commutative 100.0%

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

   2. sqr-neg 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-*l* 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in v around 0 98.5%

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

9. Final simplification 98.5%

   \[\leadsto \frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}} \]

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. *-commutative 100.0%

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

   2. sqr-neg 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-*l* 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-*l* 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in v around 0 98.4%

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

9. Final simplification 98.4%

   \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023274 
(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))))