Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 5

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, 1.0× 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 98.4%

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

   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, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.125 + N[(N[(0.125 * N[(v * v), $MachinePrecision]), $MachinePrecision] * -3.0), $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. 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 98.4%

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

   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. Step-by-step derivation

   1. pow1 100.0%

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

   2. sqrt-unprod 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. *-commutative 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(\sqrt{0.125 + 0.125 \cdot \left({v}^{2} \cdot -3\right)}\right)}^{1}} \cdot \left(1 - v \cdot v\right) \]
9. Step-by-step derivation

   1. unpow1 100.0%

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

   2. associate-*r* 100.0%

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

10. Simplified 100.0%

    \[\leadsto \color{blue}{\sqrt{0.125 + \left(0.125 \cdot {v}^{2}\right) \cdot -3}} \cdot \left(1 - v \cdot v\right) \]
11. Step-by-step derivation

    1. pow2 100.0%

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

12. Applied egg-rr 100.0%

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

13. Final simplification 100.0%

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

Alternative 3: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.125} \cdot \left(1 + \left(v \cdot v\right) \cdot -2.5\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (sqrt 0.125) (+ 1.0 (* (* v v) -2.5))))

C

double code(double v) {
	return sqrt(0.125) * (1.0 + ((v * v) * -2.5));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(0.125d0) * (1.0d0 + ((v * v) * (-2.5d0)))
end function

Java

public static double code(double v) {
	return Math.sqrt(0.125) * (1.0 + ((v * v) * -2.5));
}

Python

def code(v):
	return math.sqrt(0.125) * (1.0 + ((v * v) * -2.5))

Julia

function code(v)
	return Float64(sqrt(0.125) * Float64(1.0 + Float64(Float64(v * v) * -2.5)))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(0.125) * (1.0 + ((v * v) * -2.5));
end

Wolfram

code[v_] := N[(N[Sqrt[0.125], $MachinePrecision] * N[(1.0 + N[(N[(v * v), $MachinePrecision] * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-*l* 100.0%

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

   2. sqr-neg 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. metadata-eval 100.0%

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

   5. sqr-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in v around 0 99.7%

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

   1. *-commutative 99.7%

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

7. Simplified 99.7%

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

   1. pow2 100.0%

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

9. Applied egg-rr 99.7%

   \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{\left(v \cdot v\right)} \cdot -2.5\right) \]
10. 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 98.4%

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

    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%

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

11. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\left(\sqrt{0.125} \cdot 1\right)} \cdot \left(1 + \left(v \cdot v\right) \cdot -2.5\right) \]
12. Step-by-step derivation

    1. *-rgt-identity 100.0%

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

13. Simplified 99.7%

    \[\leadsto \color{blue}{\sqrt{0.125}} \cdot \left(1 + \left(v \cdot v\right) \cdot -2.5\right) \]
14. Add Preprocessing

Alternative 4: 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. *-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 98.4%

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

   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. Taylor expanded in v around 0 99.3%

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

Alternative 5: 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. *-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 98.4%

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

   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. Step-by-step derivation

   1. pow1 100.0%

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

   2. sqrt-unprod 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. *-commutative 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(\sqrt{0.125 + 0.125 \cdot \left({v}^{2} \cdot -3\right)}\right)}^{1}} \cdot \left(1 - v \cdot v\right) \]
9. Step-by-step derivation

   1. unpow1 100.0%

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

   2. associate-*r* 100.0%

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

10. Simplified 100.0%

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

11. Taylor expanded in v around 0 99.3%

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

Reproduce

herbie shell --seed 2024139 
(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))))