Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 9.4s

Alternatives: 4

Speedup: 1.9×

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, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot \left(0.25 - v \cdot \frac{v}{4}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (sqrt (+ 2.0 (* -6.0 (* v v)))) (- 0.25 (* v (/ v 4.0)))))

C

double code(double v) {
	return sqrt((2.0 + (-6.0 * (v * v)))) * (0.25 - (v * (v / 4.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt((2.0d0 + ((-6.0d0) * (v * v)))) * (0.25d0 - (v * (v / 4.0d0)))
end function

Java

public static double code(double v) {
	return Math.sqrt((2.0 + (-6.0 * (v * v)))) * (0.25 - (v * (v / 4.0)));
}

Python

def code(v):
	return math.sqrt((2.0 + (-6.0 * (v * v)))) * (0.25 - (v * (v / 4.0)))

Julia

function code(v)
	return Float64(sqrt(Float64(2.0 + Float64(-6.0 * Float64(v * v)))) * Float64(0.25 - Float64(v * Float64(v / 4.0))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt((2.0 + (-6.0 * (v * v)))) * (0.25 - (v * (v / 4.0)));
end

Wolfram

code[v_] := N[(N[Sqrt[N[(2.0 + N[(-6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.25 - N[(v * N[(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. *-commutative 100.0%

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

   2. sqr-neg 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-*l* 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/l* 100.0%

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

7. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-log-exp 100.0%

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

   6. div-inv 100.0%

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

   7. fma-udef 100.0%

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

   8. distribute-lft-in 100.0%

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

   9. metadata-eval 100.0%

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

   10. fma-def 100.0%

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

   11. associate-*r* 100.0%

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

   12. clear-num 100.0%

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

   13. div-sub 100.0%

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

   14. metadata-eval 100.0%

       \[\leadsto 0 + \sqrt{\mathsf{fma}\left(2, \left(v \cdot v\right) \cdot -3, 2\right)} \cdot \left(\color{blue}{0.25} - \frac{v \cdot v}{4}\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \sqrt{\mathsf{fma}\left(2, \left(v \cdot v\right) \cdot -3, 2\right)} \cdot \left(0.25 - \frac{v \cdot v}{4}\right)} \]
10. Step-by-step derivation

    1. +-lft-identity 100.0%

       \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(2, \left(v \cdot v\right) \cdot -3, 2\right)} \cdot \left(0.25 - \frac{v \cdot v}{4}\right)} \]

    2. fma-udef 100.0%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(v \cdot v\right) \cdot -3\right) + 2}} \cdot \left(0.25 - \frac{v \cdot v}{4}\right) \]

    3. +-commutative 100.0%

       \[\leadsto \sqrt{\color{blue}{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}} \cdot \left(0.25 - \frac{v \cdot v}{4}\right) \]

    4. *-commutative 100.0%

       \[\leadsto \sqrt{2 + 2 \cdot \color{blue}{\left(-3 \cdot \left(v \cdot v\right)\right)}} \cdot \left(0.25 - \frac{v \cdot v}{4}\right) \]

    5. associate-*r* 100.0%

       \[\leadsto \sqrt{2 + \color{blue}{\left(2 \cdot -3\right) \cdot \left(v \cdot v\right)}} \cdot \left(0.25 - \frac{v \cdot v}{4}\right) \]

    6. metadata-eval 100.0%

       \[\leadsto \sqrt{2 + \color{blue}{-6} \cdot \left(v \cdot v\right)} \cdot \left(0.25 - \frac{v \cdot v}{4}\right) \]

    7. associate-/l* 100.0%

       \[\leadsto \sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot \left(0.25 - \color{blue}{\frac{v}{\frac{4}{v}}}\right) \]

    8. associate-/r/ 100.0%

       \[\leadsto \sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot \left(0.25 - \color{blue}{\frac{v}{4} \cdot v}\right) \]

11. Simplified 100.0%

    \[\leadsto \color{blue}{\sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot \left(0.25 - \frac{v}{4} \cdot v\right)} \]

12. Final simplification 100.0%

    \[\leadsto \sqrt{2 + -6 \cdot \left(v \cdot v\right)} \cdot \left(0.25 - v \cdot \frac{v}{4}\right) \]

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-*r* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 98.7%

   \[\leadsto \color{blue}{\left(-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \sqrt{2}\right)} \cdot \left(1 - v \cdot v\right) \]
5. Step-by-step derivation

   1. associate-*r* 98.7%

      \[\leadsto \left(\color{blue}{\left(-0.375 \cdot {v}^{2}\right) \cdot \sqrt{2}} + 0.25 \cdot \sqrt{2}\right) \cdot \left(1 - v \cdot v\right) \]

   2. distribute-rgt-out 98.7%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(-0.375 \cdot {v}^{2} + 0.25\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   3. unpow2 98.7%

      \[\leadsto \left(\sqrt{2} \cdot \left(-0.375 \cdot \color{blue}{\left(v \cdot v\right)} + 0.25\right)\right) \cdot \left(1 - v \cdot v\right) \]

   4. associate-*r* 98.7%

      \[\leadsto \left(\sqrt{2} \cdot \left(\color{blue}{\left(-0.375 \cdot v\right) \cdot v} + 0.25\right)\right) \cdot \left(1 - v \cdot v\right) \]

6. Simplified 98.7%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\left(-0.375 \cdot v\right) \cdot v + 0.25\right)\right)} \cdot \left(1 - v \cdot v\right) \]

7. Taylor expanded in v around 0 98.7%

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

   1. +-commutative 98.7%

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

   2. associate-*r* 98.7%

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

   3. distribute-rgt-out 98.7%

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

   4. *-commutative 98.7%

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

   5. unpow2 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(v \cdot v\right) \cdot -0.625 + 0.125} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[Sqrt[N[(N[(N[(v * v), $MachinePrecision] * -0.625), $MachinePrecision] + 0.125), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. sqr-neg 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-*l* 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/l* 100.0%

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

7. Simplified 100.0%

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

   1. add-sqr-sqrt 98.4%

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

   2. sqrt-unprod 100.0%

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

   3. frac-times 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fma-udef 100.0%

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

   6. distribute-lft-in 100.0%

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

   7. metadata-eval 100.0%

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

   8. fma-def 100.0%

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

   9. associate-*r* 100.0%

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

   10. frac-times 100.0%

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

   11. metadata-eval 100.0%

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

   12. pow2 100.0%

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

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{fma}\left(2, \left(v \cdot v\right) \cdot -3, 2\right)}{\frac{16}{{\left(1 - v \cdot v\right)}^{2}}}}} \]
10. Step-by-step derivation

    1. associate-/r/ 100.0%

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

    2. *-commutative 100.0%

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

    3. fma-udef 100.0%

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

    4. +-commutative 100.0%

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

    5. *-commutative 100.0%

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

    6. associate-*r* 100.0%

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

    7. metadata-eval 100.0%

       \[\leadsto \sqrt{{\left(1 - v \cdot v\right)}^{2} \cdot \frac{2 + \color{blue}{-6} \cdot \left(v \cdot v\right)}{16}} \]

11. Simplified 100.0%

    \[\leadsto \color{blue}{\sqrt{{\left(1 - v \cdot v\right)}^{2} \cdot \frac{2 + -6 \cdot \left(v \cdot v\right)}{16}}} \]

12. Taylor expanded in v around 0 98.6%

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

    1. *-commutative 98.6%

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

    2. unpow2 98.6%

       \[\leadsto \sqrt{0.125 + \color{blue}{\left(v \cdot v\right)} \cdot -0.625} \]

14. Simplified 98.6%

    \[\leadsto \sqrt{\color{blue}{0.125 + \left(v \cdot v\right) \cdot -0.625}} \]

15. Final simplification 98.6%

    \[\leadsto \sqrt{\left(v \cdot v\right) \cdot -0.625 + 0.125} \]

Alternative 4: 99.1% 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-*r* 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. distribute-lft-neg-in 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

   14. fma-udef 100.0%

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

   15. frac-times 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in v around 0 98.0%

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

7. Final simplification 98.0%

   \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023283 
(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))))