Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

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, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}{\frac{4}{1 - v \cdot v}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (sqrt (+ 2.0 (* v (* v -6.0)))) (/ 4.0 (- 1.0 (* v v)))))

C

double code(double v) {
	return sqrt((2.0 + (v * (v * -6.0)))) / (4.0 / (1.0 - (v * v)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt((2.0d0 + (v * (v * (-6.0d0))))) / (4.0d0 / (1.0d0 - (v * v)))
end function

Java

public static double code(double v) {
	return Math.sqrt((2.0 + (v * (v * -6.0)))) / (4.0 / (1.0 - (v * v)));
}

Python

def code(v):
	return math.sqrt((2.0 + (v * (v * -6.0)))) / (4.0 / (1.0 - (v * v)))

Julia

function code(v)
	return Float64(sqrt(Float64(2.0 + Float64(v * Float64(v * -6.0)))) / Float64(4.0 / Float64(1.0 - Float64(v * v))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt((2.0 + (v * (v * -6.0)))) / (4.0 / (1.0 - (v * v)));
end

Wolfram

code[v_] := N[(N[Sqrt[N[(2.0 + N[(v * N[(v * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 / 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-*r* 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sqrt-unprod 100.0%

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

   5. sub-neg 100.0%

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

   6. +-commutative 100.0%

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

   7. *-commutative 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. distribute-lft-neg-in 100.0%

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

   10. metadata-eval 100.0%

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

   11. *-commutative 100.0%

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

   12. fma-udef 100.0%

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

5. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/l* 100.0%

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

7. Simplified 100.0%

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

   1. fma-udef 100.0%

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

9. Applied egg-rr 100.0%

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

    1. expm1-log1p-u 100.0%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(v \cdot \left(v \cdot -3\right) + 1\right)}\right)\right)}}{\frac{4}{1 - v \cdot v}} \]

    2. expm1-udef 98.5%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(v \cdot \left(v \cdot -3\right) + 1\right)}\right)} - 1}}{\frac{4}{1 - v \cdot v}} \]

    3. distribute-rgt-in 98.5%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(v \cdot \left(v \cdot -3\right)\right) \cdot 2 + 1 \cdot 2}}\right)} - 1}{\frac{4}{1 - v \cdot v}} \]

    4. metadata-eval 98.5%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\sqrt{\left(v \cdot \left(v \cdot -3\right)\right) \cdot 2 + \color{blue}{2}}\right)} - 1}{\frac{4}{1 - v \cdot v}} \]

    5. +-commutative 98.5%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\sqrt{\color{blue}{2 + \left(v \cdot \left(v \cdot -3\right)\right) \cdot 2}}\right)} - 1}{\frac{4}{1 - v \cdot v}} \]

    6. associate-*r* 98.4%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\sqrt{2 + \color{blue}{\left(\left(v \cdot v\right) \cdot -3\right)} \cdot 2}\right)} - 1}{\frac{4}{1 - v \cdot v}} \]

    7. associate-*l* 98.4%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\sqrt{2 + \color{blue}{\left(v \cdot v\right) \cdot \left(-3 \cdot 2\right)}}\right)} - 1}{\frac{4}{1 - v \cdot v}} \]

    8. metadata-eval 98.4%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}\right)} - 1}{\frac{4}{1 - v \cdot v}} \]

11. Applied egg-rr 98.4%

    \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 + \left(v \cdot v\right) \cdot -6}\right)} - 1}}{\frac{4}{1 - v \cdot v}} \]
12. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 + \left(v \cdot v\right) \cdot -6}\right)\right)}}{\frac{4}{1 - v \cdot v}} \]

    2. expm1-log1p 100.0%

       \[\leadsto \frac{\color{blue}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\frac{4}{1 - v \cdot v}} \]

    3. associate-*l* 100.0%

       \[\leadsto \frac{\sqrt{2 + \color{blue}{v \cdot \left(v \cdot -6\right)}}}{\frac{4}{1 - v \cdot v}} \]

13. Simplified 100.0%

    \[\leadsto \frac{\color{blue}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}}{\frac{4}{1 - v \cdot v}} \]

14. Final simplification 100.0%

    \[\leadsto \frac{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}{\frac{4}{1 - v \cdot v}} \]

Alternative 2: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 - v \cdot v\right) \cdot \sqrt{0.125 + v \cdot \left(v \cdot -0.375\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (- 1.0 (* v v)) (sqrt (+ 0.125 (* v (* v -0.375))))))

C

double code(double v) {
	return (1.0 - (v * v)) * sqrt((0.125 + (v * (v * -0.375))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * sqrt((0.125d0 + (v * (v * (-0.375d0)))))
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * Math.sqrt((0.125 + (v * (v * -0.375))));
}

Python

def code(v):
	return (1.0 - (v * v)) * math.sqrt((0.125 + (v * (v * -0.375))))

Julia

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) * sqrt(Float64(0.125 + Float64(v * Float64(v * -0.375)))))
end

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * sqrt((0.125 + (v * (v * -0.375))));
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.125 + N[(v * N[(v * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-*r* 100.0%

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

3. Simplified 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-lft-in 100.0%

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

5. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-rgt1-in 100.0%

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

   3. +-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. *-commutative 100.0%

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

7. Simplified 100.0%

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

   1. fma-udef 100.0%

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

9. Applied egg-rr 100.0%

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

    1. expm1-log1p-u 100.0%

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

    2. expm1-udef 98.4%

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

    3. +-commutative 98.4%

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

    4. distribute-rgt-in 98.4%

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

    5. metadata-eval 98.4%

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

    6. associate-*r* 98.4%

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

    7. associate-*l* 98.4%

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

    8. metadata-eval 98.4%

       \[\leadsto \left(1 - v \cdot v\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{0.125 + \left(v \cdot v\right) \cdot \color{blue}{-0.375}}\right)} - 1\right) \]

11. Applied egg-rr 98.4%

    \[\leadsto \left(1 - v \cdot v\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.125 + \left(v \cdot v\right) \cdot -0.375}\right)} - 1\right)} \]
12. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \left(1 - v \cdot v\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.125 + \left(v \cdot v\right) \cdot -0.375}\right)\right)} \]

    2. expm1-log1p 100.0%

       \[\leadsto \left(1 - v \cdot v\right) \cdot \color{blue}{\sqrt{0.125 + \left(v \cdot v\right) \cdot -0.375}} \]

    3. associate-*l* 100.0%

       \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt{0.125 + \color{blue}{v \cdot \left(v \cdot -0.375\right)}} \]

13. Simplified 100.0%

    \[\leadsto \left(1 - v \cdot v\right) \cdot \color{blue}{\sqrt{0.125 + v \cdot \left(v \cdot -0.375\right)}} \]

14. Final simplification 100.0%

    \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt{0.125 + v \cdot \left(v \cdot -0.375\right)} \]

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. Step-by-step derivation

   1. associate-*r* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*r* 99.4%

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

   3. distribute-rgt-out 99.4%

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

   4. *-commutative 99.4%

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

   5. unpow2 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in v around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*r* 99.4%

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

   3. distribute-rgt-out 99.4%

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

   4. unpow2 99.4%

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 98.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.125], $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-*r* 100.0%

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

3. Simplified 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-lft-in 100.0%

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

5. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-rgt1-in 100.0%

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

   3. +-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. *-commutative 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in v around 0 99.0%

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

9. Final simplification 99.0%

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

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. Step-by-step derivation

   1. associate-*r* 100.0%

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

3. Simplified 100.0%

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

   1. sub-neg 100.0%

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

   2. distribute-lft-in 100.0%

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

5. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-rgt1-in 100.0%

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

   3. +-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. cancel-sign-sub-inv 100.0%

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

   6. *-commutative 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in v around 0 99.0%

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

9. Taylor expanded in v around 0 99.0%

   \[\leadsto \color{blue}{\sqrt{0.125}} \]

10. Final simplification 99.0%

    \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023275 
(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))))