Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

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

\[\begin{array}{l} \\ \sqrt{0.125 + \left(v \cdot v\right) \cdot -0.375} \cdot \left(1 - v \cdot v\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (sqrt (+ 0.125 (* (* v v) -0.375))) (- 1.0 (* v v))))

C

double code(double v) {
	return sqrt((0.125 + ((v * v) * -0.375))) * (1.0 - (v * v));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt((0.125d0 + ((v * v) * (-0.375d0)))) * (1.0d0 - (v * v))
end function

Java

public static double code(double v) {
	return Math.sqrt((0.125 + ((v * v) * -0.375))) * (1.0 - (v * v));
}

Python

def code(v):
	return math.sqrt((0.125 + ((v * v) * -0.375))) * (1.0 - (v * v))

Julia

function code(v)
	return Float64(sqrt(Float64(0.125 + Float64(Float64(v * v) * -0.375))) * Float64(1.0 - Float64(v * v)))
end

MATLAB

function tmp = code(v)
	tmp = sqrt((0.125 + ((v * v) * -0.375))) * (1.0 - (v * v));
end

Wolfram

code[v_] := N[(N[Sqrt[N[(0.125 + N[(N[(v * v), $MachinePrecision] * -0.375), $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. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. fma-define 100.0%

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

   12. pow2 100.0%

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

   13. metadata-eval 100.0%

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

   14. frac-times 100.0%

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

   15. rem-square-sqrt 100.0%

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

   16. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left({v}^{2}, -3, 1\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]

5. Taylor expanded in v around 0 100.0%

   \[\leadsto \sqrt{\color{blue}{0.125 + -0.375 \cdot {v}^{2}}} \cdot \left(1 - v \cdot v\right) \]
6. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto \sqrt{0.125 + \color{blue}{{v}^{2} \cdot -0.375}} \cdot \left(1 - v \cdot v\right) \]

7. Simplified 100.0%

   \[\leadsto \sqrt{\color{blue}{0.125 + {v}^{2} \cdot -0.375}} \cdot \left(1 - v \cdot v\right) \]
8. Step-by-step derivation

   1. pow2 100.0%

      \[\leadsto \sqrt{0.125 + \color{blue}{\left(v \cdot v\right)} \cdot -0.375} \cdot \left(1 - v \cdot v\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.125 + \color{blue}{\left(v \cdot v\right)} \cdot -0.375} \cdot \left(1 - v \cdot v\right) \]
10. Add Preprocessing

Alternative 2: 99.5% 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. Add Preprocessing

3. Taylor expanded in v around 0 99.6%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} + 0.25 \cdot \left({v}^{2} \cdot \left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)} \]
4. Step-by-step derivation

   1. distribute-lft-out 99.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{2} + {v}^{2} \cdot \left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)} \]

   2. distribute-rgt-out 99.6%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} + {v}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-1.5 + -1\right)\right)}\right) \]

   3. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in v around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. distribute-rgt-out 99.6%

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

   4. *-commutative 99.6%

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

8. Simplified 99.6%

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

   1. pow2 100.0%

      \[\leadsto \sqrt{0.125 + \color{blue}{\left(v \cdot v\right)} \cdot -0.375} \cdot \left(1 - v \cdot v\right) \]

10. Applied egg-rr 99.6%

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

Alternative 3: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * sqrt(0.125d0)
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * Math.sqrt(0.125);
}

Python

def code(v):
	return (1.0 - (v * v)) * math.sqrt(0.125)

Julia

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

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * sqrt(0.125);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. fma-define 100.0%

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

   12. pow2 100.0%

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

   13. metadata-eval 100.0%

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

   14. frac-times 100.0%

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

   15. rem-square-sqrt 100.0%

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

   16. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left({v}^{2}, -3, 1\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]

5. Taylor expanded in v around 0 98.7%

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

6. Final simplification 98.7%

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

Alternative 4: 98.9% 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. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. fma-define 100.0%

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

   12. pow2 100.0%

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

   13. metadata-eval 100.0%

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

   14. frac-times 100.0%

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

   15. rem-square-sqrt 100.0%

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

   16. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left({v}^{2}, -3, 1\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]

5. Taylor expanded in v around 0 100.0%

   \[\leadsto \sqrt{\color{blue}{0.125 + -0.375 \cdot {v}^{2}}} \cdot \left(1 - v \cdot v\right) \]
6. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto \sqrt{0.125 + \color{blue}{{v}^{2} \cdot -0.375}} \cdot \left(1 - v \cdot v\right) \]

7. Simplified 100.0%

   \[\leadsto \sqrt{\color{blue}{0.125 + {v}^{2} \cdot -0.375}} \cdot \left(1 - v \cdot v\right) \]
8. Step-by-step derivation

   1. pow2 100.0%

      \[\leadsto \sqrt{0.125 + \color{blue}{\left(v \cdot v\right)} \cdot -0.375} \cdot \left(1 - v \cdot v\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.125 + \color{blue}{\left(v \cdot v\right)} \cdot -0.375} \cdot \left(1 - v \cdot v\right) \]

10. Taylor expanded in v around 0 98.6%

    \[\leadsto \color{blue}{\sqrt{0.125}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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))))