Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (sqrt(0.125d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

public static double code(double v) {
	return (Math.sqrt(0.125) * Math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

Python

def code(v):
	return (math.sqrt(0.125) * math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v))

Julia

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

MATLAB

function tmp = code(v)
	tmp = (sqrt(0.125) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
end

Wolfram

code[v_] := N[(N[(N[Sqrt[0.125], $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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. add-sqr-sqrt 98.5%

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

   4. sqrt-unprod 100.0%

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

   5. frac-times 100.0%

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

   6. rem-square-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

   8. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. *-rgt-identity 100.0%

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

6. Simplified 100.0%

   \[\leadsto \left(\color{blue}{\sqrt{0.125}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - v \cdot v\right) \cdot \sqrt{0.125 \cdot \left(1 + {v}^{2} \cdot -3\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (- 1.0 (* v v)) (sqrt (* 0.125 (+ 1.0 (* (pow v 2.0) -3.0))))))

C

double code(double v) {
	return (1.0 - (v * v)) * sqrt((0.125 * (1.0 + (pow(v, 2.0) * -3.0))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * sqrt((0.125d0 * (1.0d0 + ((v ** 2.0d0) * (-3.0d0)))))
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * Math.sqrt((0.125 * (1.0 + (Math.pow(v, 2.0) * -3.0))));
}

Python

def code(v):
	return (1.0 - (v * v)) * math.sqrt((0.125 * (1.0 + (math.pow(v, 2.0) * -3.0))))

Julia

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) * sqrt(Float64(0.125 * Float64(1.0 + Float64((v ^ 2.0) * -3.0)))))
end

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * sqrt((0.125 * (1.0 + ((v ^ 2.0) * -3.0))));
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.125 * N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
2. 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 99.9%

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

   7. sub-neg 99.9%

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

   8. +-commutative 99.9%

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

   9. *-commutative 99.9%

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

   10. distribute-rgt-neg-in 99.9%

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

   11. fma-define 99.9%

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

   12. pow2 99.9%

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

   13. metadata-eval 99.9%

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

   14. frac-times 99.9%

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

   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. Step-by-step derivation

   1. fma-undefine 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt{0.125 \cdot \left(1 + {v}^{2} \cdot -3\right)} \]
8. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.125} + \left(v \cdot v\right) \cdot \left(-0.625 \cdot \sqrt{2}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (+ (sqrt 0.125) (* (* v v) (* -0.625 (sqrt 2.0)))))

C

double code(double v) {
	return sqrt(0.125) + ((v * v) * (-0.625 * sqrt(2.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(0.125d0) + ((v * v) * ((-0.625d0) * sqrt(2.0d0)))
end function

Java

public static double code(double v) {
	return Math.sqrt(0.125) + ((v * v) * (-0.625 * Math.sqrt(2.0)));
}

Python

def code(v):
	return math.sqrt(0.125) + ((v * v) * (-0.625 * math.sqrt(2.0)))

Julia

function code(v)
	return Float64(sqrt(0.125) + Float64(Float64(v * v) * Float64(-0.625 * sqrt(2.0))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(0.125) + ((v * v) * (-0.625 * sqrt(2.0)));
end

Wolfram

code[v_] := N[(N[Sqrt[0.125], $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * N[(-0.625 * N[Sqrt[2.0], $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. 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. neg-mul-1 99.6%

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

   3. distribute-lft-in 99.6%

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

   4. associate-+r+ 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-+r+ 99.6%

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

   8. distribute-lft-in 99.6%

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

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. associate-*r* 99.6%

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

   2. distribute-rgt-out 99.6%

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

   3. +-commutative 99.6%

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

   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. +-commutative 99.6%

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

   2. distribute-lft-in 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. add-sqr-sqrt 98.1%

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

   6. sqrt-unprod 99.6%

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

   7. swap-sqr 99.6%

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

   8. rem-square-sqrt 99.6%

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

   9. metadata-eval 99.6%

      \[\leadsto {v}^{2} \cdot \left(-0.625 \cdot \sqrt{2}\right) + \sqrt{2 \cdot \color{blue}{0.0625}} \]

   10. metadata-eval 99.6%

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

10. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{{v}^{2} \cdot \left(-0.625 \cdot \sqrt{2}\right) + \sqrt{0.125}} \]
11. Step-by-step derivation

    1. unpow2 99.6%

       \[\leadsto \color{blue}{\left(v \cdot v\right)} \cdot \left(-0.625 \cdot \sqrt{2}\right) + \sqrt{0.125} \]

12. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\left(v \cdot v\right)} \cdot \left(-0.625 \cdot \sqrt{2}\right) + \sqrt{0.125} \]

13. Final simplification 99.6%

    \[\leadsto \sqrt{0.125} + \left(v \cdot v\right) \cdot \left(-0.625 \cdot \sqrt{2}\right) \]
14. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 + N[(N[Power[v, 2.0], $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. neg-mul-1 99.6%

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

   3. distribute-lft-in 99.6%

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

   4. associate-+r+ 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-+r+ 99.6%

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

   8. distribute-lft-in 99.6%

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

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. associate-*r* 99.6%

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

   2. distribute-rgt-out 99.6%

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

   3. +-commutative 99.6%

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

   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. Add Preprocessing

Alternative 5: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[Sqrt[0.125], $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 99.9%

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

   7. sub-neg 99.9%

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

   8. +-commutative 99.9%

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

   9. *-commutative 99.9%

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

   10. distribute-rgt-neg-in 99.9%

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

   11. fma-define 99.9%

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

   12. pow2 99.9%

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

   13. metadata-eval 99.9%

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

   14. frac-times 99.9%

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

   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 98.4%

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

Alternative 6: 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 99.9%

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

   7. sub-neg 99.9%

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

   8. +-commutative 99.9%

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

   9. *-commutative 99.9%

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

   10. distribute-rgt-neg-in 99.9%

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

   11. fma-define 99.9%

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

   12. pow2 99.9%

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

   13. metadata-eval 99.9%

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

   14. frac-times 99.9%

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

   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 98.4%

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

6. Taylor expanded in v around 0 98.4%

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

Reproduce

herbie shell --seed 2024111 
(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))))