Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

public static double code(double v) {
	return ((Math.sqrt(2.0) / 4.0) * Math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

Python

def code(v):
	return ((math.sqrt(2.0) / 4.0) * math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v))

Julia

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

MATLAB

function tmp = code(v)
	tmp = ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
end

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

public static double code(double v) {
	return ((Math.sqrt(2.0) / 4.0) * Math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

Python

def code(v):
	return ((math.sqrt(2.0) / 4.0) * math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v))

Julia

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

MATLAB

function tmp = code(v)
	tmp = ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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(sqrt(0.125) * Float64(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[Sqrt[0.125], $MachinePrecision] * N[(N[Sqrt[N[(1.0 + N[(-3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $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. 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. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. add-sqr-sqrt 98.5%

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

   4. sqrt-unprod 100.0%

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

   5. frac-times 100.0%

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

   6. rem-square-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

   8. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-rgt-identity 100.0%

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

8. Simplified 100.0%

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

Alternative 2: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 - v \cdot v\right) \cdot \left(\sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.375\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (- 1.0 (* v v)) (* (sqrt 2.0) (+ 0.25 (* (* v v) -0.375)))))

C

double code(double v) {
	return (1.0 - (v * v)) * (sqrt(2.0) * (0.25 + ((v * v) * -0.375)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * (sqrt(2.0d0) * (0.25d0 + ((v * v) * (-0.375d0))))
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * (Math.sqrt(2.0) * (0.25 + ((v * v) * -0.375)));
}

Python

def code(v):
	return (1.0 - (v * v)) * (math.sqrt(2.0) * (0.25 + ((v * v) * -0.375)))

Julia

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) * Float64(sqrt(2.0) * Float64(0.25 + Float64(Float64(v * v) * -0.375))))
end

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * (sqrt(2.0) * (0.25 + ((v * v) * -0.375)));
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 + N[(N[(v * v), $MachinePrecision] * -0.375), $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 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-*r* 99.8%

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

   3. distribute-rgt-out 99.8%

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

   4. *-commutative 99.8%

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

5. Simplified 99.8%

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

   1. pow2 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

   \[\leadsto \left(1 - v \cdot v\right) \cdot \left(\sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.375\right)\right) \]
9. Add Preprocessing

Alternative 3: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.375\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (sqrt 2.0) (+ 0.25 (* (* v v) -0.375))))

C

double code(double v) {
	return sqrt(2.0) * (0.25 + ((v * v) * -0.375));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) * (0.25d0 + ((v * v) * (-0.375d0)))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) * (0.25 + ((v * v) * -0.375));
}

Python

def code(v):
	return math.sqrt(2.0) * (0.25 + ((v * v) * -0.375))

Julia

function code(v)
	return Float64(sqrt(2.0) * Float64(0.25 + Float64(Float64(v * v) * -0.375)))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) * (0.25 + ((v * v) * -0.375));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 + N[(N[(v * v), $MachinePrecision] * -0.375), $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 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-*r* 99.8%

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

   3. distribute-rgt-out 99.8%

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

   4. *-commutative 99.8%

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

5. Simplified 99.8%

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

   1. pow2 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in v around 0 99.6%

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

9. Final simplification 99.6%

   \[\leadsto \sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.375\right) \]
10. Add Preprocessing

Alternative 4: 99.1% 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. Taylor expanded in v around 0 99.6%

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

   1. add-sqr-sqrt 98.1%

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

   2. pow2 98.1%

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

   3. pow1/2 98.1%

      \[\leadsto \left(0.25 \cdot {\left(\sqrt{\color{blue}{{2}^{0.5}}}\right)}^{2}\right) \cdot \left(1 - v \cdot v\right) \]

   4. sqrt-pow1 98.1%

      \[\leadsto \left(0.25 \cdot {\color{blue}{\left({2}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}\right) \cdot \left(1 - v \cdot v\right) \]

   5. metadata-eval 98.1%

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

5. Applied egg-rr 98.1%

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

   1. expm1-log1p-u 98.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.25 \cdot {\left({2}^{0.25}\right)}^{2}\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   2. expm1-undefine 98.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(0.25 \cdot {\left({2}^{0.25}\right)}^{2}\right)} - 1\right)} \cdot \left(1 - v \cdot v\right) \]

   3. pow-pow 98.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(0.25 \cdot \color{blue}{{2}^{\left(0.25 \cdot 2\right)}}\right)} - 1\right) \cdot \left(1 - v \cdot v\right) \]

   4. metadata-eval 98.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(0.25 \cdot {2}^{\color{blue}{0.5}}\right)} - 1\right) \cdot \left(1 - v \cdot v\right) \]

   5. pow1/2 98.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(0.25 \cdot \color{blue}{\sqrt{2}}\right)} - 1\right) \cdot \left(1 - v \cdot v\right) \]

   6. add-sqr-sqrt 98.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{0.25 \cdot \sqrt{2}} \cdot \sqrt{0.25 \cdot \sqrt{2}}}\right)} - 1\right) \cdot \left(1 - v \cdot v\right) \]

   7. sqrt-unprod 98.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(0.25 \cdot \sqrt{2}\right) \cdot \left(0.25 \cdot \sqrt{2}\right)}}\right)} - 1\right) \cdot \left(1 - v \cdot v\right) \]

   8. swap-sqr 98.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(0.25 \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}}\right)} - 1\right) \cdot \left(1 - v \cdot v\right) \]

   9. metadata-eval 98.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{0.0625} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}\right)} - 1\right) \cdot \left(1 - v \cdot v\right) \]

   10. rem-square-sqrt 98.1%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{0.0625 \cdot \color{blue}{2}}\right)} - 1\right) \cdot \left(1 - v \cdot v\right) \]

   11. metadata-eval 98.1%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{0.125}}\right)} - 1\right) \cdot \left(1 - v \cdot v\right) \]

7. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.125}\right)} - 1\right)} \cdot \left(1 - v \cdot v\right) \]
8. Step-by-step derivation

   1. log1p-undefine 98.1%

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

   2. rem-exp-log 98.1%

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

   3. +-commutative 98.1%

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

   4. associate--l+ 99.6%

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

   5. metadata-eval 99.6%

      \[\leadsto \left(\sqrt{0.125} + \color{blue}{0}\right) \cdot \left(1 - v \cdot v\right) \]

   6. +-rgt-identity 99.6%

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

9. Simplified 99.6%

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

Alternative 5: 99.0% 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-*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. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. add-sqr-sqrt 98.5%

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

   4. sqrt-unprod 100.0%

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

   5. frac-times 100.0%

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

   6. rem-square-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

   8. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-rgt-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in v around 0 99.6%

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

Reproduce

herbie shell --seed 2024160 
(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))))