Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 15.9s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v)
	tmp = (sqrt(0.125) * sqrt((1.0 - (3.0 * (v * v))))) * ((2.0 - (v * v)) + -1.0);
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[(N[(2.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] + -1.0), $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. expm1-log1p-u 100.0%

      \[\leadsto \left(\sqrt{0.125} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - v \cdot v\right)\right)} \]

   2. pow2 100.0%

      \[\leadsto \left(\sqrt{0.125} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(1 - \color{blue}{{v}^{2}}\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \left(\sqrt{0.125} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - {v}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. expm1-undefine 100.0%

      \[\leadsto \left(\sqrt{0.125} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - {v}^{2}\right)} - 1\right)} \]

   2. sub-neg 100.0%

      \[\leadsto \left(\sqrt{0.125} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 - {v}^{2}\right)} + \left(-1\right)\right)} \]

   3. log1p-undefine 100.0%

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

   4. rem-exp-log 100.0%

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

   5. associate-+r- 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

10. Simplified 100.0%

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

    1. pow2 100.0%

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

12. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. pow2 98.4%

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

   3. pow1/2 98.4%

      \[\leadsto \left(\frac{{\left(\sqrt{\color{blue}{{2}^{0.5}}}\right)}^{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   4. sqrt-pow1 98.4%

      \[\leadsto \left(\frac{{\color{blue}{\left({2}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

   5. metadata-eval 98.4%

      \[\leadsto \left(\frac{{\left({2}^{\color{blue}{0.25}}\right)}^{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

4. Applied egg-rr 98.4%

   \[\leadsto \left(\frac{\color{blue}{{\left({2}^{0.25}\right)}^{2}}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
5. Step-by-step derivation

   1. pow1 98.4%

      \[\leadsto \color{blue}{{\left(\frac{{\left({2}^{0.25}\right)}^{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}^{1}} \cdot \left(1 - v \cdot v\right) \]

6. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

8. Simplified 100.0%

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

   1. pow2 100.0%

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

10. Applied egg-rr 100.0%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-*r* 99.9%

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

   3. distribute-rgt-out 99.9%

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

   4. *-commutative 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in v around 0 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-*r* 99.9%

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

   3. distribute-rgt-out 99.9%

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

   4. *-commutative 99.9%

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

8. Simplified 99.9%

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

   1. pow2 100.0%

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

10. Applied egg-rr 99.9%

    \[\leadsto \sqrt{2} \cdot \left(0.25 + \color{blue}{\left(v \cdot v\right)} \cdot -0.625\right) \]
11. 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 99.5%

   \[\leadsto \color{blue}{\sqrt{0.125}} \cdot \left(1 - v \cdot v\right) \]
8. 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. 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 99.5%

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

8. Taylor expanded in v around 0 99.5%

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

Reproduce

herbie shell --seed 2024143 
(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))))