Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[Sqrt[N[(N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision] * 0.125), $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. Step-by-step derivation

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. fma-def 100.0%

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

   12. metadata-eval 100.0%

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

   13. frac-times 100.0%

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

   14. add-sqr-sqrt 100.0%

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

   15. metadata-eval 100.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot \frac{2}{\color{blue}{16}}} \cdot \left(1 - v \cdot v\right) \]

   16. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(-1.0 / N[(N[(4.0 / N[(v * N[(v * 2.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 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. 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 99.1%

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

   1. unpow2 99.1%

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

6. Simplified 99.1%

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

   1. associate-*l/ 99.1%

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

   2. frac-2neg 99.1%

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

   3. +-commutative 99.1%

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

   4. associate-*r* 99.1%

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

   5. fma-def 99.1%

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

   6. metadata-eval 99.1%

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

8. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{-\sqrt{2} \cdot \mathsf{fma}\left(-2.5 \cdot v, v, 1\right)}{-4}} \]
9. Step-by-step derivation

   1. distribute-rgt-neg-in 99.1%

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

   2. associate-/l* 99.1%

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

   3. fma-udef 99.1%

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

   4. +-commutative 99.1%

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

   5. distribute-neg-in 99.1%

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

   6. metadata-eval 99.1%

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

   7. associate-*l* 99.1%

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

   8. distribute-lft-neg-in 99.1%

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

   9. metadata-eval 99.1%

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

   10. *-commutative 99.1%

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

   11. associate-*l* 99.1%

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

10. Simplified 99.1%

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

    1. frac-2neg 99.1%

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

    2. div-inv 99.1%

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

    3. distribute-neg-frac 99.1%

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

    4. metadata-eval 99.1%

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

    5. +-commutative 99.1%

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

    6. fma-def 99.1%

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

12. Applied egg-rr 99.1%

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

14. Applied egg-rr 99.1%

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

15. Final simplification 99.1%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / N[(4.0 / N[(v * N[(v * 2.5), $MachinePrecision] + -1.0), $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. Taylor expanded in v around 0 99.1%

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

   1. unpow2 99.1%

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

6. Simplified 99.1%

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

   1. associate-*l/ 99.1%

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

   2. frac-2neg 99.1%

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

   3. +-commutative 99.1%

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

   4. associate-*r* 99.1%

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

   5. fma-def 99.1%

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

   6. metadata-eval 99.1%

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

8. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{-\sqrt{2} \cdot \mathsf{fma}\left(-2.5 \cdot v, v, 1\right)}{-4}} \]
9. Step-by-step derivation

   1. distribute-rgt-neg-in 99.1%

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

   2. associate-/l* 99.1%

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

   3. fma-udef 99.1%

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

   4. +-commutative 99.1%

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

   5. distribute-neg-in 99.1%

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

   6. metadata-eval 99.1%

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

   7. associate-*l* 99.1%

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

   8. distribute-lft-neg-in 99.1%

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

   9. metadata-eval 99.1%

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

   10. *-commutative 99.1%

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

   11. associate-*l* 99.1%

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

10. Simplified 99.1%

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

    1. frac-2neg 99.1%

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

    2. div-inv 99.1%

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

    3. distribute-neg-frac 99.1%

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

    4. metadata-eval 99.1%

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

    5. +-commutative 99.1%

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

    6. fma-def 99.1%

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

12. Applied egg-rr 99.1%

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

13. Final simplification 99.1%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(v \cdot -2.5, v, 1\right)}{\frac{1}{\sqrt{0.125}}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (fma (* v -2.5) v 1.0) (/ 1.0 (sqrt 0.125))))

C

double code(double v) {
	return fma((v * -2.5), v, 1.0) / (1.0 / sqrt(0.125));
}

Julia

function code(v)
	return Float64(fma(Float64(v * -2.5), v, 1.0) / Float64(1.0 / sqrt(0.125)))
end

Wolfram

code[v_] := N[(N[(N[(v * -2.5), $MachinePrecision] * v + 1.0), $MachinePrecision] / N[(1.0 / N[Sqrt[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. 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 99.1%

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

   1. unpow2 99.1%

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

6. Simplified 99.1%

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

   1. *-commutative 99.1%

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

   2. clear-num 99.1%

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

   3. un-div-inv 99.1%

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

   4. +-commutative 99.1%

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

   5. associate-*r* 99.1%

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

   6. fma-def 99.1%

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

   7. clear-num 99.1%

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

   8. add-sqr-sqrt 97.6%

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

   9. sqrt-unprod 99.1%

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

   10. frac-times 99.1%

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

   11. add-sqr-sqrt 99.1%

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

   12. metadata-eval 99.1%

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

   13. metadata-eval 99.1%

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

8. Applied egg-rr 99.1%

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

9. Final simplification 99.1%

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

Alternative 5: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot \left(\left(v \cdot v\right) \cdot -0.625 + 0.25\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (sqrt 2.0) (+ (* (* v v) -0.625) 0.25)))

C

double code(double v) {
	return sqrt(2.0) * (((v * v) * -0.625) + 0.25);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) * (((v * v) * (-0.625d0)) + 0.25d0)
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) * (((v * v) * -0.625) + 0.25);
}

Python

def code(v):
	return math.sqrt(2.0) * (((v * v) * -0.625) + 0.25)

Julia

function code(v)
	return Float64(sqrt(2.0) * Float64(Float64(Float64(v * v) * -0.625) + 0.25))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) * (((v * v) * -0.625) + 0.25);
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(v * v), $MachinePrecision] * -0.625), $MachinePrecision] + 0.25), $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 99.1%

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

   1. unpow2 99.1%

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

6. Simplified 99.1%

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

   1. associate-*l/ 99.1%

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

   2. frac-2neg 99.1%

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

   3. +-commutative 99.1%

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

   4. associate-*r* 99.1%

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

   5. fma-def 99.1%

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

   6. metadata-eval 99.1%

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

8. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{-\sqrt{2} \cdot \mathsf{fma}\left(-2.5 \cdot v, v, 1\right)}{-4}} \]
9. Step-by-step derivation

   1. distribute-rgt-neg-in 99.1%

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

   2. associate-/l* 99.1%

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

   3. fma-udef 99.1%

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

   4. +-commutative 99.1%

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

   5. distribute-neg-in 99.1%

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

   6. metadata-eval 99.1%

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

   7. associate-*l* 99.1%

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

   8. distribute-lft-neg-in 99.1%

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

   9. metadata-eval 99.1%

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

   10. *-commutative 99.1%

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

   11. associate-*l* 99.1%

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

10. Simplified 99.1%

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

11. Taylor expanded in v around 0 99.1%

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

    1. +-commutative 99.1%

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

    2. *-commutative 99.1%

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

    3. associate-*r* 99.1%

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

    4. metadata-eval 99.1%

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

    5. associate-*r* 99.1%

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

    6. *-commutative 99.1%

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

    7. unpow2 99.1%

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

    8. associate-*r* 99.1%

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

    9. *-commutative 99.1%

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

    10. *-commutative 99.1%

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

    11. distribute-lft-out 99.1%

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

    12. associate-*r* 99.1%

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

    13. unpow2 99.1%

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

    14. *-commutative 99.1%

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

    15. associate-*r* 99.1%

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

    16. metadata-eval 99.1%

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

    17. unpow2 99.1%

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

13. Simplified 99.1%

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

14. Final simplification 99.1%

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

Alternative 6: 99.2% 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. Step-by-step derivation

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. fma-def 100.0%

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

   12. metadata-eval 100.0%

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

   13. frac-times 100.0%

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

   14. add-sqr-sqrt 100.0%

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

   15. metadata-eval 100.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot \frac{2}{\color{blue}{16}}} \cdot \left(1 - v \cdot v\right) \]

   16. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in v around 0 98.3%

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

5. Final simplification 98.3%

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

Alternative 7: 99.2% 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. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. fma-def 100.0%

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

   12. metadata-eval 100.0%

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

   13. frac-times 100.0%

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

   14. add-sqr-sqrt 100.0%

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

   15. metadata-eval 100.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot \frac{2}{\color{blue}{16}}} \cdot \left(1 - v \cdot v\right) \]

   16. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in v around 0 98.3%

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

5. Taylor expanded in v around 0 98.2%

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

6. Final simplification 98.2%

   \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023192 
(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))))