Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

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

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

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. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{\sqrt{2}}{4}}\right)\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \left(0 + \color{blue}{\frac{\sqrt{2}}{4}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   6. add-sqr-sqrt 98.5%

      \[\leadsto \left(0 + \color{blue}{\sqrt{\frac{\sqrt{2}}{4}} \cdot \sqrt{\frac{\sqrt{2}}{4}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   7. sqrt-unprod 100.0%

      \[\leadsto \left(0 + \color{blue}{\sqrt{\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   8. frac-times 100.0%

      \[\leadsto \left(0 + \sqrt{\color{blue}{\frac{\sqrt{2} \cdot \sqrt{2}}{4 \cdot 4}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   9. rem-square-sqrt 100.0%

      \[\leadsto \left(0 + \sqrt{\frac{\color{blue}{2}}{4 \cdot 4}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   10. metadata-eval 100.0%

       \[\leadsto \left(0 + \sqrt{\frac{2}{\color{blue}{16}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \left(0 + \sqrt{\color{blue}{0.125}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(0 + \sqrt{0.125}\right)} \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]
7. Step-by-step derivation

   1. +-lft-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, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + \left(\mathsf{fma}\left(v, v, 1\right) \cdot -2.5 + 2.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* 0.25 (* (sqrt 2.0) (+ 1.0 (+ (* (fma v v 1.0) -2.5) 2.5)))))

C

double code(double v) {
	return 0.25 * (sqrt(2.0) * (1.0 + ((fma(v, v, 1.0) * -2.5) + 2.5)));
}

Julia

function code(v)
	return Float64(0.25 * Float64(sqrt(2.0) * Float64(1.0 + Float64(Float64(fma(v, v, 1.0) * -2.5) + 2.5))))
end

Wolfram

code[v_] := N[(0.25 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 + N[(N[(N[(v * v + 1.0), $MachinePrecision] * -2.5), $MachinePrecision] + 2.5), $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. Taylor expanded in v around 0 99.6%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} + 0.25 \cdot \left({v}^{2} \cdot \left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)} \]
6. 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. *-rgt-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. distribute-rgt-out 99.6%

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

   5. associate-*l* 99.6%

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

   6. distribute-lft-out 99.6%

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

   7. metadata-eval 99.6%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + \color{blue}{-2.5} \cdot {v}^{2}\right)\right) \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot {v}^{2}\right)\right)} \]
8. Step-by-step derivation

   1. pow2 99.6%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \color{blue}{\left(v \cdot v\right)}\right)\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \color{blue}{\left(v \cdot v\right)}\right)\right) \]
10. Step-by-step derivation

    1. expm1-log1p-u 99.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot v\right)\right)}\right)\right) \]

    2. log1p-define 99.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(1 + v \cdot v\right)}\right)\right)\right) \]

    3. expm1-define 99.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \color{blue}{\left(e^{\log \left(1 + v \cdot v\right)} - 1\right)}\right)\right) \]

    4. add-exp-log 99.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \left(\color{blue}{\left(1 + v \cdot v\right)} - 1\right)\right)\right) \]

    5. pow2 99.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \left(\left(1 + \color{blue}{{v}^{2}}\right) - 1\right)\right)\right) \]

    6. sub-neg 99.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \color{blue}{\left(\left(1 + {v}^{2}\right) + \left(-1\right)\right)}\right)\right) \]

    7. pow2 99.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \left(\left(1 + \color{blue}{v \cdot v}\right) + \left(-1\right)\right)\right)\right) \]

    8. distribute-rgt-in 99.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + \color{blue}{\left(\left(1 + v \cdot v\right) \cdot -2.5 + \left(-1\right) \cdot -2.5\right)}\right)\right) \]

    9. +-commutative 99.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + \left(\color{blue}{\left(v \cdot v + 1\right)} \cdot -2.5 + \left(-1\right) \cdot -2.5\right)\right)\right) \]

    10. fma-define 99.6%

        \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + \left(\color{blue}{\mathsf{fma}\left(v, v, 1\right)} \cdot -2.5 + \left(-1\right) \cdot -2.5\right)\right)\right) \]

    11. metadata-eval 99.6%

        \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + \left(\mathsf{fma}\left(v, v, 1\right) \cdot -2.5 + \color{blue}{-1} \cdot -2.5\right)\right)\right) \]

    12. metadata-eval 99.6%

        \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + \left(\mathsf{fma}\left(v, v, 1\right) \cdot -2.5 + \color{blue}{2.5}\right)\right)\right) \]

11. Applied egg-rr 99.6%

    \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + \color{blue}{\left(\mathsf{fma}\left(v, v, 1\right) \cdot -2.5 + 2.5\right)}\right)\right) \]
12. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(\sqrt{2} + \left(v \cdot v\right) \cdot \left(\sqrt{2} \cdot -2.5\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* 0.25 (+ (sqrt 2.0) (* (* v v) (* (sqrt 2.0) -2.5)))))

C

double code(double v) {
	return 0.25 * (sqrt(2.0) + ((v * v) * (sqrt(2.0) * -2.5)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = 0.25d0 * (sqrt(2.0d0) + ((v * v) * (sqrt(2.0d0) * (-2.5d0))))
end function

Java

public static double code(double v) {
	return 0.25 * (Math.sqrt(2.0) + ((v * v) * (Math.sqrt(2.0) * -2.5)));
}

Python

def code(v):
	return 0.25 * (math.sqrt(2.0) + ((v * v) * (math.sqrt(2.0) * -2.5)))

Julia

function code(v)
	return Float64(0.25 * Float64(sqrt(2.0) + Float64(Float64(v * v) * Float64(sqrt(2.0) * -2.5))))
end

MATLAB

function tmp = code(v)
	tmp = 0.25 * (sqrt(2.0) + ((v * v) * (sqrt(2.0) * -2.5)));
end

Wolfram

code[v_] := N[(0.25 * N[(N[Sqrt[2.0], $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -2.5), $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. Taylor expanded in v around 0 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. distribute-rgt-out 99.6%

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

   3. metadata-eval 99.6%

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

7. Simplified 99.6%

   \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{2} + {v}^{2} \cdot \left(\sqrt{2} \cdot -2.5\right)\right)} \]
8. Step-by-step derivation

   1. pow2 99.6%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \color{blue}{\left(v \cdot v\right)}\right)\right) \]

9. Applied egg-rr 99.6%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + \left(v \cdot v\right) \cdot -2.5\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* 0.25 (* (sqrt 2.0) (+ 1.0 (* (* v v) -2.5)))))

C

double code(double v) {
	return 0.25 * (sqrt(2.0) * (1.0 + ((v * v) * -2.5)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = 0.25d0 * (sqrt(2.0d0) * (1.0d0 + ((v * v) * (-2.5d0))))
end function

Java

public static double code(double v) {
	return 0.25 * (Math.sqrt(2.0) * (1.0 + ((v * v) * -2.5)));
}

Python

def code(v):
	return 0.25 * (math.sqrt(2.0) * (1.0 + ((v * v) * -2.5)))

Julia

function code(v)
	return Float64(0.25 * Float64(sqrt(2.0) * Float64(1.0 + Float64(Float64(v * v) * -2.5))))
end

MATLAB

function tmp = code(v)
	tmp = 0.25 * (sqrt(2.0) * (1.0 + ((v * v) * -2.5)));
end

Wolfram

code[v_] := N[(0.25 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 + N[(N[(v * v), $MachinePrecision] * -2.5), $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. Taylor expanded in v around 0 99.6%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} + 0.25 \cdot \left({v}^{2} \cdot \left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)} \]
6. 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. *-rgt-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. distribute-rgt-out 99.6%

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

   5. associate-*l* 99.6%

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

   6. distribute-lft-out 99.6%

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

   7. metadata-eval 99.6%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + \color{blue}{-2.5} \cdot {v}^{2}\right)\right) \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot {v}^{2}\right)\right)} \]
8. Step-by-step derivation

   1. pow2 99.6%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \color{blue}{\left(v \cdot v\right)}\right)\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + -2.5 \cdot \color{blue}{\left(v \cdot v\right)}\right)\right) \]

10. Final simplification 99.6%

    \[\leadsto 0.25 \cdot \left(\sqrt{2} \cdot \left(1 + \left(v \cdot v\right) \cdot -2.5\right)\right) \]
11. 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. 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 99.1%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{\sqrt{2}}{4}}\right)\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \left(0 + \color{blue}{\frac{\sqrt{2}}{4}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   6. add-sqr-sqrt 98.5%

      \[\leadsto \left(0 + \color{blue}{\sqrt{\frac{\sqrt{2}}{4}} \cdot \sqrt{\frac{\sqrt{2}}{4}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   7. sqrt-unprod 100.0%

      \[\leadsto \left(0 + \color{blue}{\sqrt{\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   8. frac-times 100.0%

      \[\leadsto \left(0 + \sqrt{\color{blue}{\frac{\sqrt{2} \cdot \sqrt{2}}{4 \cdot 4}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   9. rem-square-sqrt 100.0%

      \[\leadsto \left(0 + \sqrt{\frac{\color{blue}{2}}{4 \cdot 4}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   10. metadata-eval 100.0%

       \[\leadsto \left(0 + \sqrt{\frac{2}{\color{blue}{16}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \left(0 + \sqrt{\color{blue}{0.125}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

7. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\left(0 + \sqrt{0.125}\right)} \cdot \left(1 \cdot \left(1 - v \cdot v\right)\right) \]
8. Step-by-step derivation

   1. +-lft-identity 100.0%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Alternative 6: 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. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{\sqrt{2}}{4}}\right)\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   5. add-log-exp 100.0%

      \[\leadsto \left(0 + \color{blue}{\frac{\sqrt{2}}{4}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   6. add-sqr-sqrt 98.5%

      \[\leadsto \left(0 + \color{blue}{\sqrt{\frac{\sqrt{2}}{4}} \cdot \sqrt{\frac{\sqrt{2}}{4}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   7. sqrt-unprod 100.0%

      \[\leadsto \left(0 + \color{blue}{\sqrt{\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   8. frac-times 100.0%

      \[\leadsto \left(0 + \sqrt{\color{blue}{\frac{\sqrt{2} \cdot \sqrt{2}}{4 \cdot 4}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   9. rem-square-sqrt 100.0%

      \[\leadsto \left(0 + \sqrt{\frac{\color{blue}{2}}{4 \cdot 4}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   10. metadata-eval 100.0%

       \[\leadsto \left(0 + \sqrt{\frac{2}{\color{blue}{16}}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \left(0 + \sqrt{\color{blue}{0.125}}\right) \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(0 + \sqrt{0.125}\right)} \cdot \left(\sqrt{1 + -3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right) \]
7. Step-by-step derivation

   1. +-lft-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 99.0%

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

Reproduce

herbie shell --seed 2024172 
(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))))