Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

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

\[\begin{array}{l} \\ \frac{1 - v \cdot v}{\frac{4}{\sqrt{2 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (- 1.0 (* v v)) (/ 4.0 (sqrt (* 2.0 (fma (* v v) -3.0 1.0))))))

C

double code(double v) {
	return (1.0 - (v * v)) / (4.0 / sqrt((2.0 * fma((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 * fma(Float64(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[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-*l/ 100.0%

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

   2. associate-/r/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. neg-sub0 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]

   13. associate-+l- 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]

   14. sub0-neg 100.0%

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

   15. neg-mul-1 100.0%

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

   16. associate-/r* 100.0%

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

   17. metadata-eval 100.0%

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

   18. fma-neg 100.0%

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

   19. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. associate-*r/ 100.0%

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

   2. frac-2neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. associate-/r/ 100.0%

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

   5. sqrt-unprod 100.0%

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

   6. fma-udef 100.0%

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

   7. metadata-eval 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. *-commutative 100.0%

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

   10. +-commutative 100.0%

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

   11. sub-neg 100.0%

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

   12. sqrt-unprod 100.0%

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

   13. associate-*l/ 100.0%

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

   14. fma-udef 100.0%

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

   15. distribute-neg-in 100.0%

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

   16. metadata-eval 100.0%

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

   17. +-commutative 100.0%

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

   18. sub-neg 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-/r/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. neg-sub0 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]

   13. associate-+l- 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]

   14. sub0-neg 100.0%

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

   15. neg-mul-1 100.0%

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

   16. associate-/r* 100.0%

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

   17. metadata-eval 100.0%

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

   18. fma-neg 100.0%

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

   19. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. associate-*r/ 100.0%

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

   2. frac-2neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. associate-/r/ 100.0%

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

   5. sqrt-unprod 100.0%

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

   6. fma-udef 100.0%

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

   7. metadata-eval 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. *-commutative 100.0%

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

   10. +-commutative 100.0%

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

   11. sub-neg 100.0%

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

   12. sqrt-unprod 100.0%

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

   13. associate-*l/ 100.0%

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

   14. fma-udef 100.0%

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

   15. distribute-neg-in 100.0%

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

   16. metadata-eval 100.0%

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

   17. +-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-rgt1-in 100.0%

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

   3. +-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. *-commutative 100.0%

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

   7. fma-udef 100.0%

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

   8. associate-*r* 100.0%

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

   9. fma-def 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-/r/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. neg-sub0 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]

   13. associate-+l- 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]

   14. sub0-neg 100.0%

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

   15. neg-mul-1 100.0%

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

   16. associate-/r* 100.0%

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

   17. metadata-eval 100.0%

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

   18. fma-neg 100.0%

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

   19. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.6%

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

   1. fma-def 99.6%

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

   2. unpow2 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 99.0% 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. associate-*l/ 100.0%

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

   2. associate-/r/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. neg-sub0 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]

   13. associate-+l- 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]

   14. sub0-neg 100.0%

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

   15. neg-mul-1 100.0%

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

   16. associate-/r* 100.0%

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

   17. metadata-eval 100.0%

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

   18. fma-neg 100.0%

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

   19. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. associate-*r/ 100.0%

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

   2. frac-2neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. associate-/r/ 100.0%

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

   5. sqrt-unprod 100.0%

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

   6. fma-udef 100.0%

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

   7. metadata-eval 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. *-commutative 100.0%

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

   10. +-commutative 100.0%

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

   11. sub-neg 100.0%

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

   12. sqrt-unprod 100.0%

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

   13. associate-*l/ 100.0%

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

   14. fma-udef 100.0%

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

   15. distribute-neg-in 100.0%

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

   16. metadata-eval 100.0%

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

   17. +-commutative 100.0%

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

   18. sub-neg 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in v around 0 99.1%

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

9. Final simplification 99.1%

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

Alternative 5: 99.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.125} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (sqrt 0.125))

C

double code(double v) {
	return sqrt(0.125);
}

Fortran

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

Java

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

Python

def code(v):
	return math.sqrt(0.125)

Julia

function code(v)
	return sqrt(0.125)
end

MATLAB

function tmp = code(v)
	tmp = sqrt(0.125);
end

Wolfram

code[v_] := N[Sqrt[0.125], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-*l/ 100.0%

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

   2. associate-/r/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. neg-sub0 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]

   13. associate-+l- 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]

   14. sub0-neg 100.0%

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

   15. neg-mul-1 100.0%

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

   16. associate-/r* 100.0%

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

   17. metadata-eval 100.0%

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

   18. fma-neg 100.0%

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

   19. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. associate-*r/ 100.0%

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

   2. frac-2neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. associate-/r/ 100.0%

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

   5. sqrt-unprod 100.0%

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

   6. fma-udef 100.0%

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

   7. metadata-eval 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. *-commutative 100.0%

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

   10. +-commutative 100.0%

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

   11. sub-neg 100.0%

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

   12. sqrt-unprod 100.0%

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

   13. associate-*l/ 100.0%

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

   14. fma-udef 100.0%

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

   15. distribute-neg-in 100.0%

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

   16. metadata-eval 100.0%

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

   17. +-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-rgt1-in 100.0%

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

   3. +-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. *-commutative 100.0%

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

   7. fma-udef 100.0%

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

   8. associate-*r* 100.0%

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

   9. fma-def 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in v around 0 99.1%

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

9. Final simplification 99.1%

   \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023173 
(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))))