Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

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

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

FPCore

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

C

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

Julia

function code(v)
	return Float64(Float64(sqrt(Float64(2.0 * fma(Float64(v * v), -3.0, 1.0))) * Float64(1.0 - Float64(v * v))) / 4.0)
end

Wolfram

code[v_] := N[(N[(N[Sqrt[N[(2.0 * N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 4.0), $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. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

   3. frac-2neg 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/l/ 100.0%

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

   6. fma-udef 100.0%

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

   7. metadata-eval 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. *-commutative 100.0%

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

   10. +-commutative 100.0%

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

   11. sub-neg 100.0%

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

   12. fma-udef 100.0%

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

   13. distribute-neg-in 100.0%

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

   14. metadata-eval 100.0%

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

   15. +-commutative 100.0%

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

   16. sub-neg 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(v)
	return Float64(sqrt(Float64(2.0 * fma(Float64(v * v), -3.0, 1.0))) / Float64(4.0 / Float64(1.0 - Float64(v * v))))
end

Wolfram

code[v_] := N[(N[Sqrt[N[(2.0 * N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 / N[(1.0 - N[(v * v), $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. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

   3. frac-2neg 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/l/ 100.0%

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

   6. fma-udef 100.0%

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

   7. metadata-eval 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. *-commutative 100.0%

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

   10. +-commutative 100.0%

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

   11. sub-neg 100.0%

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

   12. fma-udef 100.0%

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

   13. distribute-neg-in 100.0%

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

   14. metadata-eval 100.0%

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

   15. +-commutative 100.0%

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

   16. sub-neg 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot \left(\left(\left(v \cdot v\right) \cdot -0.625 + 0.25\right) + 0.09375 \cdot {v}^{4}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (sqrt 2.0) (+ (+ (* (* v v) -0.625) 0.25) (* 0.09375 (pow v 4.0)))))

C

double code(double v) {
	return sqrt(2.0) * ((((v * v) * -0.625) + 0.25) + (0.09375 * pow(v, 4.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) * ((((v * v) * (-0.625d0)) + 0.25d0) + (0.09375d0 * (v ** 4.0d0)))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) * ((((v * v) * -0.625) + 0.25) + (0.09375 * Math.pow(v, 4.0)));
}

Python

def code(v):
	return math.sqrt(2.0) * ((((v * v) * -0.625) + 0.25) + (0.09375 * math.pow(v, 4.0)))

Julia

function code(v)
	return Float64(sqrt(2.0) * Float64(Float64(Float64(Float64(v * v) * -0.625) + 0.25) + Float64(0.09375 * (v ^ 4.0))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) * ((((v * v) * -0.625) + 0.25) + (0.09375 * (v ^ 4.0)));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(v * v), $MachinePrecision] * -0.625), $MachinePrecision] + 0.25), $MachinePrecision] + N[(0.09375 * N[Power[v, 4.0], $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. Taylor expanded in v around 0 99.6%

   \[\leadsto \sqrt{2} \cdot \color{blue}{\left(-0.625 \cdot {v}^{2} + \left(0.25 + 0.09375 \cdot {v}^{4}\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r+ 99.6%

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\left(-0.625 \cdot {v}^{2} + 0.25\right) + 0.09375 \cdot {v}^{4}\right)} \]

   2. fma-def 99.6%

      \[\leadsto \sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)} + 0.09375 \cdot {v}^{4}\right) \]

   3. unpow2 99.6%

      \[\leadsto \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.625, \color{blue}{v \cdot v}, 0.25\right) + 0.09375 \cdot {v}^{4}\right) \]

6. Simplified 99.6%

   \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\mathsf{fma}\left(-0.625, v \cdot v, 0.25\right) + 0.09375 \cdot {v}^{4}\right)} \]
7. Step-by-step derivation

   1. fma-udef 99.6%

      \[\leadsto \sqrt{2} \cdot \left(\color{blue}{\left(-0.625 \cdot \left(v \cdot v\right) + 0.25\right)} + 0.09375 \cdot {v}^{4}\right) \]

8. Applied egg-rr 99.6%

   \[\leadsto \sqrt{2} \cdot \left(\color{blue}{\left(-0.625 \cdot \left(v \cdot v\right) + 0.25\right)} + 0.09375 \cdot {v}^{4}\right) \]

9. Final simplification 99.6%

   \[\leadsto \sqrt{2} \cdot \left(\left(\left(v \cdot v\right) \cdot -0.625 + 0.25\right) + 0.09375 \cdot {v}^{4}\right) \]

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (sqrt(2.0d0) / 4.0d0) * (1.0d0 + ((v * v) * (-2.5d0)))
end function

Java

public static double code(double v) {
	return (Math.sqrt(2.0) / 4.0) * (1.0 + ((v * v) * -2.5));
}

Python

def code(v):
	return (math.sqrt(2.0) / 4.0) * (1.0 + ((v * v) * -2.5))

Julia

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

MATLAB

function tmp = code(v)
	tmp = (sqrt(2.0) / 4.0) * (1.0 + ((v * v) * -2.5));
end

Wolfram

code[v_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[(1.0 + N[(N[(v * v), $MachinePrecision] * -2.5), $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 99.5%

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

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (sqrt(2.0d0) / 4.0d0) * (1.0d0 + (v * (v * (-2.5d0))))
end function

Java

public static double code(double v) {
	return (Math.sqrt(2.0) / 4.0) * (1.0 + (v * (v * -2.5)));
}

Python

def code(v):
	return (math.sqrt(2.0) / 4.0) * (1.0 + (v * (v * -2.5)))

Julia

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

MATLAB

function tmp = code(v)
	tmp = (sqrt(2.0) / 4.0) * (1.0 + (v * (v * -2.5)));
end

Wolfram

code[v_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[(1.0 + N[(v * N[(v * -2.5), $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 99.5%

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

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in v around 0 99.5%

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

   1. unpow2 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*l* 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) / ((-4.0d0) / ((v * (v * 2.5d0)) + (-1.0d0)))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) / (-4.0 / ((v * (v * 2.5)) + -1.0));
}

Python

def code(v):
	return math.sqrt(2.0) / (-4.0 / ((v * (v * 2.5)) + -1.0))

Julia

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

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) / (-4.0 / ((v * (v * 2.5)) + -1.0));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] / N[(-4.0 / N[(N[(v * N[(v * 2.5), $MachinePrecision]), $MachinePrecision] + -1.0), $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 99.5%

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

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

   1. associate-*l/ 99.5%

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

   2. frac-2neg 99.5%

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

   3. +-commutative 99.5%

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

   4. *-commutative 99.5%

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

   5. fma-def 99.5%

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

   6. metadata-eval 99.5%

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

8. Applied egg-rr 99.5%

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

   1. distribute-rgt-neg-in 99.5%

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

   2. associate-/l* 99.5%

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

   3. fma-udef 99.5%

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

   4. distribute-neg-in 99.5%

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

   5. distribute-rgt-neg-in 99.5%

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

   6. metadata-eval 99.5%

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

   7. associate-*l* 99.5%

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

   8. metadata-eval 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 7: 99.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(v)
	return sqrt(0.125)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-*l* 100.0%

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

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 99.5%

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

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

   1. distribute-rgt-in 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. flip-+ 98.0%

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

8. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{0.125 - \left(\left(\left(v \cdot v\right) \cdot -2.5\right) \cdot \sqrt{0.125}\right) \cdot \left(\left(\left(v \cdot v\right) \cdot -2.5\right) \cdot \sqrt{0.125}\right)}{\sqrt{0.125} - \left(\left(v \cdot v\right) \cdot -2.5\right) \cdot \sqrt{0.125}}} \]
9. Step-by-step derivation

10. Simplified 98.0%

    \[\leadsto \color{blue}{\frac{0.125 + 0.125 \cdot \left({v}^{4} \cdot -6.25\right)}{\mathsf{fma}\left(v \cdot v, 2.5, 1\right) \cdot \sqrt{0.125}}} \]

11. Taylor expanded in v around 0 97.4%

    \[\leadsto \color{blue}{\frac{0.125}{\sqrt{0.125}}} \]
12. Step-by-step derivation

    1. expm1-log1p-u 97.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{\sqrt{0.125}}\right)\right)} \]

    2. expm1-udef 97.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.125}{\sqrt{0.125}}\right)} - 1} \]

    3. log1p-udef 97.4%

       \[\leadsto e^{\color{blue}{\log \left(1 + \frac{0.125}{\sqrt{0.125}}\right)}} - 1 \]

    4. add-sqr-sqrt 97.4%

       \[\leadsto e^{\log \left(1 + \color{blue}{\sqrt{\frac{0.125}{\sqrt{0.125}}} \cdot \sqrt{\frac{0.125}{\sqrt{0.125}}}}\right)} - 1 \]

    5. sqrt-unprod 97.4%

       \[\leadsto e^{\log \left(1 + \color{blue}{\sqrt{\frac{0.125}{\sqrt{0.125}} \cdot \frac{0.125}{\sqrt{0.125}}}}\right)} - 1 \]

    6. frac-times 97.4%

       \[\leadsto e^{\log \left(1 + \sqrt{\color{blue}{\frac{0.125 \cdot 0.125}{\sqrt{0.125} \cdot \sqrt{0.125}}}}\right)} - 1 \]

    7. metadata-eval 97.4%

       \[\leadsto e^{\log \left(1 + \sqrt{\frac{\color{blue}{0.015625}}{\sqrt{0.125} \cdot \sqrt{0.125}}}\right)} - 1 \]

    8. add-sqr-sqrt 97.4%

       \[\leadsto e^{\log \left(1 + \sqrt{\frac{0.015625}{\color{blue}{0.125}}}\right)} - 1 \]

    9. metadata-eval 97.4%

       \[\leadsto e^{\log \left(1 + \sqrt{\color{blue}{0.125}}\right)} - 1 \]

    10. add-exp-log 97.4%

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

13. Applied egg-rr 97.4%

    \[\leadsto \color{blue}{\left(1 + \sqrt{0.125}\right) - 1} \]
14. Step-by-step derivation

    1. +-commutative 97.4%

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

    2. associate--l+ 98.9%

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

    3. metadata-eval 98.9%

       \[\leadsto \sqrt{0.125} + \color{blue}{0} \]

    4. +-rgt-identity 98.9%

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

15. Simplified 98.9%

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

16. Final simplification 98.9%

    \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023240 
(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))))