Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. associate-*r/ 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. fma-udef 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-*r* 100.0%

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

   7. *-commutative 100.0%

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

   8. metadata-eval 100.0%

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

   9. cancel-sign-sub-inv 100.0%

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

   10. sqrt-unprod 100.0%

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

   11. cancel-sign-sub-inv 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

   14. associate-*r* 100.0%

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

   15. +-commutative 100.0%

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

   16. fma-udef 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-/r/ 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 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]

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. Final simplification 100.0%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2}}{\frac{1}{0.25 - \frac{{v}^{2}}{1.6}}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (sqrt 2.0) (/ 1.0 (- 0.25 (/ (pow v 2.0) 1.6)))))

C

double code(double v) {
	return sqrt(2.0) / (1.0 / (0.25 - (pow(v, 2.0) / 1.6)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) / (1.0d0 / (0.25d0 - ((v ** 2.0d0) / 1.6d0)))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) / (1.0 / (0.25 - (Math.pow(v, 2.0) / 1.6)));
}

Python

def code(v):
	return math.sqrt(2.0) / (1.0 / (0.25 - (math.pow(v, 2.0) / 1.6)))

Julia

function code(v)
	return Float64(sqrt(2.0) / Float64(1.0 / Float64(0.25 - Float64((v ^ 2.0) / 1.6))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) / (1.0 / (0.25 - ((v ^ 2.0) / 1.6)));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] / N[(1.0 / N[(0.25 - N[(N[Power[v, 2.0], $MachinePrecision] / 1.6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.4%

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

   1. flip3-+ 99.4%

      \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{{0.25}^{3} + {\left(-0.625 \cdot {v}^{2}\right)}^{3}}{0.25 \cdot 0.25 + \left(\left(-0.625 \cdot {v}^{2}\right) \cdot \left(-0.625 \cdot {v}^{2}\right) - 0.25 \cdot \left(-0.625 \cdot {v}^{2}\right)\right)}} \]

   2. associate-*r/ 99.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \left({0.25}^{3} + {\left(-0.625 \cdot {v}^{2}\right)}^{3}\right)}{0.25 \cdot 0.25 + \left(\left(-0.625 \cdot {v}^{2}\right) \cdot \left(-0.625 \cdot {v}^{2}\right) - 0.25 \cdot \left(-0.625 \cdot {v}^{2}\right)\right)}} \]

   3. associate-/l* 99.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{0.25 \cdot 0.25 + \left(\left(-0.625 \cdot {v}^{2}\right) \cdot \left(-0.625 \cdot {v}^{2}\right) - 0.25 \cdot \left(-0.625 \cdot {v}^{2}\right)\right)}{{0.25}^{3} + {\left(-0.625 \cdot {v}^{2}\right)}^{3}}}} \]

   4. *-un-lft-identity 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{\color{blue}{1 \cdot \left(0.25 \cdot 0.25 + \left(\left(-0.625 \cdot {v}^{2}\right) \cdot \left(-0.625 \cdot {v}^{2}\right) - 0.25 \cdot \left(-0.625 \cdot {v}^{2}\right)\right)\right)}}{{0.25}^{3} + {\left(-0.625 \cdot {v}^{2}\right)}^{3}}} \]

   5. associate-/l* 99.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{1}{\frac{{0.25}^{3} + {\left(-0.625 \cdot {v}^{2}\right)}^{3}}{0.25 \cdot 0.25 + \left(\left(-0.625 \cdot {v}^{2}\right) \cdot \left(-0.625 \cdot {v}^{2}\right) - 0.25 \cdot \left(-0.625 \cdot {v}^{2}\right)\right)}}}} \]

   6. flip3-+ 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{0.25 + -0.625 \cdot {v}^{2}}}} \]

   7. +-commutative 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{-0.625 \cdot {v}^{2} + 0.25}}} \]

   8. fma-def 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{\mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}}} \]

6. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{1}{\mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}}} \]
7. Step-by-step derivation

   1. /-rgt-identity 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}{1}}}} \]

   2. frac-2neg 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{\frac{-\mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}{-1}}}} \]

   3. metadata-eval 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{-\mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}{\color{blue}{-1}}}} \]

   4. neg-sub0 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{\color{blue}{0 - \mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}}{-1}}} \]

   5. fma-udef 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{0 - \color{blue}{\left(-0.625 \cdot {v}^{2} + 0.25\right)}}{-1}}} \]

   6. +-commutative 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{0 - \color{blue}{\left(0.25 + -0.625 \cdot {v}^{2}\right)}}{-1}}} \]

   7. associate--r+ 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{\color{blue}{\left(0 - 0.25\right) - -0.625 \cdot {v}^{2}}}{-1}}} \]

   8. metadata-eval 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{\color{blue}{-0.25} - -0.625 \cdot {v}^{2}}{-1}}} \]

8. Applied egg-rr 99.4%

   \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{\frac{-0.25 - -0.625 \cdot {v}^{2}}{-1}}}} \]
9. Step-by-step derivation

   1. div-sub 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{\frac{-0.25}{-1} - \frac{-0.625 \cdot {v}^{2}}{-1}}}} \]

   2. metadata-eval 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{0.25} - \frac{-0.625 \cdot {v}^{2}}{-1}}} \]

   3. *-commutative 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - \frac{\color{blue}{{v}^{2} \cdot -0.625}}{-1}}} \]

   4. associate-/l* 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - \color{blue}{\frac{{v}^{2}}{\frac{-1}{-0.625}}}}} \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - \frac{{v}^{2}}{\color{blue}{1.6}}}} \]

10. Simplified 99.4%

    \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{0.25 - \frac{{v}^{2}}{1.6}}}} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - \frac{{v}^{2}}{1.6}}} \]

Alternative 4: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2}}{\frac{1}{0.25 - v \cdot \left(v \cdot 0.625\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (sqrt 2.0) (/ 1.0 (- 0.25 (* v (* v 0.625))))))

C

double code(double v) {
	return sqrt(2.0) / (1.0 / (0.25 - (v * (v * 0.625))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) / (1.0d0 / (0.25d0 - (v * (v * 0.625d0))))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) / (1.0 / (0.25 - (v * (v * 0.625))));
}

Python

def code(v):
	return math.sqrt(2.0) / (1.0 / (0.25 - (v * (v * 0.625))))

Julia

function code(v)
	return Float64(sqrt(2.0) / Float64(1.0 / Float64(0.25 - Float64(v * Float64(v * 0.625)))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) / (1.0 / (0.25 - (v * (v * 0.625))));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] / N[(1.0 / N[(0.25 - N[(v * N[(v * 0.625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.4%

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

   1. flip3-+ 99.4%

      \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{{0.25}^{3} + {\left(-0.625 \cdot {v}^{2}\right)}^{3}}{0.25 \cdot 0.25 + \left(\left(-0.625 \cdot {v}^{2}\right) \cdot \left(-0.625 \cdot {v}^{2}\right) - 0.25 \cdot \left(-0.625 \cdot {v}^{2}\right)\right)}} \]

   2. associate-*r/ 99.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \left({0.25}^{3} + {\left(-0.625 \cdot {v}^{2}\right)}^{3}\right)}{0.25 \cdot 0.25 + \left(\left(-0.625 \cdot {v}^{2}\right) \cdot \left(-0.625 \cdot {v}^{2}\right) - 0.25 \cdot \left(-0.625 \cdot {v}^{2}\right)\right)}} \]

   3. associate-/l* 99.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{0.25 \cdot 0.25 + \left(\left(-0.625 \cdot {v}^{2}\right) \cdot \left(-0.625 \cdot {v}^{2}\right) - 0.25 \cdot \left(-0.625 \cdot {v}^{2}\right)\right)}{{0.25}^{3} + {\left(-0.625 \cdot {v}^{2}\right)}^{3}}}} \]

   4. *-un-lft-identity 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{\color{blue}{1 \cdot \left(0.25 \cdot 0.25 + \left(\left(-0.625 \cdot {v}^{2}\right) \cdot \left(-0.625 \cdot {v}^{2}\right) - 0.25 \cdot \left(-0.625 \cdot {v}^{2}\right)\right)\right)}}{{0.25}^{3} + {\left(-0.625 \cdot {v}^{2}\right)}^{3}}} \]

   5. associate-/l* 99.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{1}{\frac{{0.25}^{3} + {\left(-0.625 \cdot {v}^{2}\right)}^{3}}{0.25 \cdot 0.25 + \left(\left(-0.625 \cdot {v}^{2}\right) \cdot \left(-0.625 \cdot {v}^{2}\right) - 0.25 \cdot \left(-0.625 \cdot {v}^{2}\right)\right)}}}} \]

   6. flip3-+ 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{0.25 + -0.625 \cdot {v}^{2}}}} \]

   7. +-commutative 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{-0.625 \cdot {v}^{2} + 0.25}}} \]

   8. fma-def 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{\mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}}} \]

6. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{1}{\mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}}} \]
7. Step-by-step derivation

   1. /-rgt-identity 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}{1}}}} \]

   2. frac-2neg 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{\frac{-\mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}{-1}}}} \]

   3. metadata-eval 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{-\mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}{\color{blue}{-1}}}} \]

   4. neg-sub0 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{\color{blue}{0 - \mathsf{fma}\left(-0.625, {v}^{2}, 0.25\right)}}{-1}}} \]

   5. fma-udef 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{0 - \color{blue}{\left(-0.625 \cdot {v}^{2} + 0.25\right)}}{-1}}} \]

   6. +-commutative 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{0 - \color{blue}{\left(0.25 + -0.625 \cdot {v}^{2}\right)}}{-1}}} \]

   7. associate--r+ 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{\color{blue}{\left(0 - 0.25\right) - -0.625 \cdot {v}^{2}}}{-1}}} \]

   8. metadata-eval 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\frac{\color{blue}{-0.25} - -0.625 \cdot {v}^{2}}{-1}}} \]

8. Applied egg-rr 99.4%

   \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{\frac{-0.25 - -0.625 \cdot {v}^{2}}{-1}}}} \]
9. Step-by-step derivation

   1. div-sub 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{\frac{-0.25}{-1} - \frac{-0.625 \cdot {v}^{2}}{-1}}}} \]

   2. metadata-eval 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{0.25} - \frac{-0.625 \cdot {v}^{2}}{-1}}} \]

   3. *-commutative 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - \frac{\color{blue}{{v}^{2} \cdot -0.625}}{-1}}} \]

   4. associate-/l* 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - \color{blue}{\frac{{v}^{2}}{\frac{-1}{-0.625}}}}} \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - \frac{{v}^{2}}{\color{blue}{1.6}}}} \]

10. Simplified 99.4%

    \[\leadsto \frac{\sqrt{2}}{\frac{1}{\color{blue}{0.25 - \frac{{v}^{2}}{1.6}}}} \]
11. Step-by-step derivation

    1. div-inv 99.4%

       \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - \color{blue}{{v}^{2} \cdot \frac{1}{1.6}}}} \]

    2. unpow2 99.4%

       \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - \color{blue}{\left(v \cdot v\right)} \cdot \frac{1}{1.6}}} \]

    3. associate-*l* 99.4%

       \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - \color{blue}{v \cdot \left(v \cdot \frac{1}{1.6}\right)}}} \]

    4. metadata-eval 99.4%

       \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - v \cdot \left(v \cdot \color{blue}{0.625}\right)}} \]

12. Applied egg-rr 99.4%

    \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - \color{blue}{v \cdot \left(v \cdot 0.625\right)}}} \]

13. Final simplification 99.4%

    \[\leadsto \frac{\sqrt{2}}{\frac{1}{0.25 - v \cdot \left(v \cdot 0.625\right)}} \]

Alternative 5: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot 0.25 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (sqrt 2.0) 0.25))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * 0.25), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 98.6%

   \[\leadsto \sqrt{2} \cdot \color{blue}{0.25} \]

5. Final simplification 98.6%

   \[\leadsto \sqrt{2} \cdot 0.25 \]

Reproduce

herbie shell --seed 2023336 
(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))))