Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 10.2s

Alternatives: 5

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

\[\begin{array}{l} \\ \sqrt{0.125 + {v}^{2} \cdot -0.375} \cdot \left(1 - v \cdot v\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (sqrt (+ 0.125 (* (pow v 2.0) -0.375))) (- 1.0 (* v v))))

C

double code(double v) {
	return sqrt((0.125 + (pow(v, 2.0) * -0.375))) * (1.0 - (v * v));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt((0.125d0 + ((v ** 2.0d0) * (-0.375d0)))) * (1.0d0 - (v * v))
end function

Java

public static double code(double v) {
	return Math.sqrt((0.125 + (Math.pow(v, 2.0) * -0.375))) * (1.0 - (v * v));
}

Python

def code(v):
	return math.sqrt((0.125 + (math.pow(v, 2.0) * -0.375))) * (1.0 - (v * v))

Julia

function code(v)
	return Float64(sqrt(Float64(0.125 + Float64((v ^ 2.0) * -0.375))) * Float64(1.0 - Float64(v * v)))
end

MATLAB

function tmp = code(v)
	tmp = sqrt((0.125 + ((v ^ 2.0) * -0.375))) * (1.0 - (v * v));
end

Wolfram

code[v_] := N[(N[Sqrt[N[(0.125 + N[(N[Power[v, 2.0], $MachinePrecision] * -0.375), $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. add-sqr-sqrt 100.0%

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

   2. sqrt-unprod 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. metadata-eval 100.0%

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

   6. swap-sqr 100.0%

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

   7. sqrt-unprod 48.4%

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

   8. expm1-log1p-u 48.4%

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-3 \cdot \left(v \cdot v\right)} \cdot \sqrt{-3 \cdot \left(v \cdot v\right)}\right)\right)}}\right) \cdot \left(1 - v \cdot v\right) \]

   9. add-sqr-sqrt 98.7%

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{-3 \cdot \left(v \cdot v\right)}\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   10. log1p-define 98.7%

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   11. metadata-eval 98.7%

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{\left(-3\right)} \cdot \left(v \cdot v\right)\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   12. cancel-sign-sub-inv 98.7%

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \mathsf{expm1}\left(\log \color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   13. expm1-undefine 98.7%

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

   14. add-exp-log 98.7%

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

   15. add-cube-cbrt 98.7%

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

   16. fma-neg 98.7%

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

4. Applied egg-rr 100.0%

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

   1. pow1 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(\sqrt{\left(1 - \left(3 \cdot {v}^{2} + 0\right)\right) \cdot 0.125}\right)}^{1}} \cdot \left(1 - v \cdot v\right) \]
7. Step-by-step derivation

   1. unpow1 100.0%

      \[\leadsto \color{blue}{\sqrt{\left(1 - \left(3 \cdot {v}^{2} + 0\right)\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]

   2. *-commutative 100.0%

      \[\leadsto \sqrt{\color{blue}{0.125 \cdot \left(1 - \left(3 \cdot {v}^{2} + 0\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

   3. sub-neg 100.0%

      \[\leadsto \sqrt{0.125 \cdot \color{blue}{\left(1 + \left(-\left(3 \cdot {v}^{2} + 0\right)\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

   4. distribute-lft-in 100.0%

      \[\leadsto \sqrt{\color{blue}{0.125 \cdot 1 + 0.125 \cdot \left(-\left(3 \cdot {v}^{2} + 0\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

   5. metadata-eval 100.0%

      \[\leadsto \sqrt{\color{blue}{0.125} + 0.125 \cdot \left(-\left(3 \cdot {v}^{2} + 0\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   6. +-rgt-identity 100.0%

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

   7. *-commutative 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

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. metadata-eval 100.0%

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

   2. distribute-lft-in 100.0%

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

   3. *-commutative 100.0%

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

   4. metadata-eval 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. +-rgt-identity 100.0%

      \[\leadsto \sqrt{0.125 \cdot \left(1 - \color{blue}{\left(3 \cdot {v}^{2} + 0\right)}\right)} \cdot \left(1 - v \cdot v\right) \]

   7. *-commutative 100.0%

      \[\leadsto \sqrt{\color{blue}{\left(1 - \left(3 \cdot {v}^{2} + 0\right)\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]

   8. *-un-lft-identity 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\left(1 - \left(3 \cdot {v}^{2} + 0\right)\right) \cdot 0.125}\right)} \cdot \left(1 - v \cdot v\right) \]

   9. *-commutative 100.0%

      \[\leadsto \left(1 \cdot \sqrt{\color{blue}{0.125 \cdot \left(1 - \left(3 \cdot {v}^{2} + 0\right)\right)}}\right) \cdot \left(1 - v \cdot v\right) \]

   10. +-rgt-identity 100.0%

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

   11. cancel-sign-sub-inv 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

   14. distribute-rgt-in 100.0%

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

   15. metadata-eval 100.0%

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

   16. associate-*l* 100.0%

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

   17. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{\left(1 \cdot \sqrt{0.125 + {v}^{2} \cdot -0.375}\right)} \cdot \left(1 - v \cdot v\right) \]
11. Step-by-step derivation

    1. *-lft-identity 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

    \[\leadsto \sqrt{0.125 + {v}^{2} \cdot -0.375} \cdot \left(1 - v \cdot v\right) \]
14. 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. Taylor expanded in v around 0 99.3%

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

   1. *-commutative 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in v around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. distribute-rgt-out 99.3%

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

   4. *-commutative 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.125 + {v}^{2} \cdot -0.625} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (sqrt (+ 0.125 (* (pow v 2.0) -0.625))))

C

double code(double v) {
	return sqrt((0.125 + (pow(v, 2.0) * -0.625)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt((0.125d0 + ((v ** 2.0d0) * (-0.625d0))))
end function

Java

public static double code(double v) {
	return Math.sqrt((0.125 + (Math.pow(v, 2.0) * -0.625)));
}

Python

def code(v):
	return math.sqrt((0.125 + (math.pow(v, 2.0) * -0.625)))

Julia

function code(v)
	return sqrt(Float64(0.125 + Float64((v ^ 2.0) * -0.625)))
end

MATLAB

function tmp = code(v)
	tmp = sqrt((0.125 + ((v ^ 2.0) * -0.625)));
end

Wolfram

code[v_] := N[Sqrt[N[(0.125 + 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. Taylor expanded in v around 0 99.3%

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

   1. *-commutative 99.3%

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

7. Simplified 99.3%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 99.3%

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

   3. swap-sqr 99.3%

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

   4. frac-times 99.3%

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

   5. rem-square-sqrt 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{16}} \cdot \left(\left(1 + {v}^{2} \cdot -2.5\right) \cdot \left(1 + {v}^{2} \cdot -2.5\right)\right)} \]

   7. metadata-eval 99.3%

      \[\leadsto \sqrt{\color{blue}{0.125} \cdot \left(\left(1 + {v}^{2} \cdot -2.5\right) \cdot \left(1 + {v}^{2} \cdot -2.5\right)\right)} \]

   8. pow2 99.3%

      \[\leadsto \sqrt{0.125 \cdot \color{blue}{{\left(1 + {v}^{2} \cdot -2.5\right)}^{2}}} \]

   9. +-commutative 99.3%

      \[\leadsto \sqrt{0.125 \cdot {\color{blue}{\left({v}^{2} \cdot -2.5 + 1\right)}}^{2}} \]

   10. fma-define 99.3%

       \[\leadsto \sqrt{0.125 \cdot {\color{blue}{\left(\mathsf{fma}\left({v}^{2}, -2.5, 1\right)\right)}}^{2}} \]

9. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\sqrt{0.125 \cdot {\left(\mathsf{fma}\left({v}^{2}, -2.5, 1\right)\right)}^{2}}} \]

10. Taylor expanded in v around 0 99.2%

    \[\leadsto \sqrt{0.125 \cdot \color{blue}{\left(1 + -5 \cdot {v}^{2}\right)}} \]
11. Step-by-step derivation

    1. *-commutative 99.2%

       \[\leadsto \sqrt{0.125 \cdot \left(1 + \color{blue}{{v}^{2} \cdot -5}\right)} \]

12. Simplified 99.2%

    \[\leadsto \sqrt{0.125 \cdot \color{blue}{\left(1 + {v}^{2} \cdot -5\right)}} \]
13. Step-by-step derivation

    1. *-un-lft-identity 99.2%

       \[\leadsto \color{blue}{1 \cdot \sqrt{0.125 \cdot \left(1 + {v}^{2} \cdot -5\right)}} \]

    2. distribute-rgt-in 99.2%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{1 \cdot 0.125 + \left({v}^{2} \cdot -5\right) \cdot 0.125}} \]

    3. metadata-eval 99.2%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{0.125} + \left({v}^{2} \cdot -5\right) \cdot 0.125} \]

    4. associate-*l* 99.2%

       \[\leadsto 1 \cdot \sqrt{0.125 + \color{blue}{{v}^{2} \cdot \left(-5 \cdot 0.125\right)}} \]

    5. metadata-eval 99.2%

       \[\leadsto 1 \cdot \sqrt{0.125 + {v}^{2} \cdot \color{blue}{-0.625}} \]

14. Applied egg-rr 99.2%

    \[\leadsto \color{blue}{1 \cdot \sqrt{0.125 + {v}^{2} \cdot -0.625}} \]
15. Step-by-step derivation

    1. *-lft-identity 99.2%

       \[\leadsto \color{blue}{\sqrt{0.125 + {v}^{2} \cdot -0.625}} \]

16. Simplified 99.2%

    \[\leadsto \color{blue}{\sqrt{0.125 + {v}^{2} \cdot -0.625}} \]

17. Final simplification 99.2%

    \[\leadsto \sqrt{0.125 + {v}^{2} \cdot -0.625} \]
18. Add Preprocessing

Alternative 4: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. sqrt-unprod 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. metadata-eval 100.0%

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

   6. swap-sqr 100.0%

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

   7. sqrt-unprod 48.4%

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

   8. expm1-log1p-u 48.4%

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-3 \cdot \left(v \cdot v\right)} \cdot \sqrt{-3 \cdot \left(v \cdot v\right)}\right)\right)}}\right) \cdot \left(1 - v \cdot v\right) \]

   9. add-sqr-sqrt 98.7%

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{-3 \cdot \left(v \cdot v\right)}\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   10. log1p-define 98.7%

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   11. metadata-eval 98.7%

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{\left(-3\right)} \cdot \left(v \cdot v\right)\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   12. cancel-sign-sub-inv 98.7%

       \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \mathsf{expm1}\left(\log \color{blue}{\left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   13. expm1-undefine 98.7%

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

   14. add-exp-log 98.7%

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

   15. add-cube-cbrt 98.7%

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

   16. fma-neg 98.7%

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

4. Applied egg-rr 100.0%

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

   1. pow1 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(\sqrt{\left(1 - \left(3 \cdot {v}^{2} + 0\right)\right) \cdot 0.125}\right)}^{1}} \cdot \left(1 - v \cdot v\right) \]
7. Step-by-step derivation

   1. unpow1 100.0%

      \[\leadsto \color{blue}{\sqrt{\left(1 - \left(3 \cdot {v}^{2} + 0\right)\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]

   2. *-commutative 100.0%

      \[\leadsto \sqrt{\color{blue}{0.125 \cdot \left(1 - \left(3 \cdot {v}^{2} + 0\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

   3. sub-neg 100.0%

      \[\leadsto \sqrt{0.125 \cdot \color{blue}{\left(1 + \left(-\left(3 \cdot {v}^{2} + 0\right)\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

   4. distribute-lft-in 100.0%

      \[\leadsto \sqrt{\color{blue}{0.125 \cdot 1 + 0.125 \cdot \left(-\left(3 \cdot {v}^{2} + 0\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

   5. metadata-eval 100.0%

      \[\leadsto \sqrt{\color{blue}{0.125} + 0.125 \cdot \left(-\left(3 \cdot {v}^{2} + 0\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   6. +-rgt-identity 100.0%

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

   7. *-commutative 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

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. Taylor expanded in v around 0 98.7%

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

10. Final simplification 98.7%

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

Alternative 5: 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. 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 99.3%

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

   1. *-commutative 99.3%

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

7. Simplified 99.3%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 99.3%

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

   3. swap-sqr 99.3%

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

   4. frac-times 99.3%

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

   5. rem-square-sqrt 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{16}} \cdot \left(\left(1 + {v}^{2} \cdot -2.5\right) \cdot \left(1 + {v}^{2} \cdot -2.5\right)\right)} \]

   7. metadata-eval 99.3%

      \[\leadsto \sqrt{\color{blue}{0.125} \cdot \left(\left(1 + {v}^{2} \cdot -2.5\right) \cdot \left(1 + {v}^{2} \cdot -2.5\right)\right)} \]

   8. pow2 99.3%

      \[\leadsto \sqrt{0.125 \cdot \color{blue}{{\left(1 + {v}^{2} \cdot -2.5\right)}^{2}}} \]

   9. +-commutative 99.3%

      \[\leadsto \sqrt{0.125 \cdot {\color{blue}{\left({v}^{2} \cdot -2.5 + 1\right)}}^{2}} \]

   10. fma-define 99.3%

       \[\leadsto \sqrt{0.125 \cdot {\color{blue}{\left(\mathsf{fma}\left({v}^{2}, -2.5, 1\right)\right)}}^{2}} \]

9. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\sqrt{0.125 \cdot {\left(\mathsf{fma}\left({v}^{2}, -2.5, 1\right)\right)}^{2}}} \]

10. Taylor expanded in v around 0 99.2%

    \[\leadsto \sqrt{0.125 \cdot \color{blue}{\left(1 + -5 \cdot {v}^{2}\right)}} \]
11. Step-by-step derivation

    1. *-commutative 99.2%

       \[\leadsto \sqrt{0.125 \cdot \left(1 + \color{blue}{{v}^{2} \cdot -5}\right)} \]

12. Simplified 99.2%

    \[\leadsto \sqrt{0.125 \cdot \color{blue}{\left(1 + {v}^{2} \cdot -5\right)}} \]

13. Taylor expanded in v around 0 98.7%

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

14. Final simplification 98.7%

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

Reproduce

herbie shell --seed 2024067 
(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))))