Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(1.0 / N[(4.0 / N[Sqrt[N[(2.0 + N[(v * N[(v * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
2. 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. associate-*l/ 100.0%

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

   2. clear-num 100.0%

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

   3. sqrt-unprod 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. distribute-lft-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

   10. *-commutative 100.0%

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

   11. fma-udef 100.0%

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

5. Applied egg-rr 100.0%

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

   1. fma-udef 100.0%

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

7. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-rgt-in 100.0%

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

   5. metadata-eval 100.0%

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

   6. associate-*r* 100.0%

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

   7. associate-*l* 100.0%

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

   8. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

    3. associate-*l* 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. distribute-lft-neg-in 100.0%

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

   12. metadata-eval 100.0%

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

   13. *-commutative 100.0%

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

   14. fma-udef 100.0%

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

   15. frac-times 100.0%

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

5. Applied egg-rr 100.0%

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

   1. fma-udef 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.9× speedup

Math

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

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(Float64(v * 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[(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. 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. associate-*l/ 100.0%

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

   2. clear-num 100.0%

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

   3. sqrt-unprod 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. distribute-lft-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

   10. *-commutative 100.0%

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

   11. fma-udef 100.0%

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

5. Applied egg-rr 100.0%

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

   1. fma-udef 100.0%

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

7. Applied egg-rr 100.0%

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

   1. add-sqr-sqrt 98.4%

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

   2. sqrt-unprod 100.0%

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

   3. associate-/r/ 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/r/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. swap-sqr 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\sqrt{2 \cdot \left(v \cdot \left(v \cdot -3\right) + 1\right)} \cdot \sqrt{2 \cdot \left(v \cdot \left(v \cdot -3\right) + 1\right)}\right)} \cdot \left(1 - v \cdot v\right) \]

   9. add-sqr-sqrt 100.0%

      \[\leadsto \sqrt{0.0625 \cdot \color{blue}{\left(2 \cdot \left(v \cdot \left(v \cdot -3\right) + 1\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

   10. +-commutative 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \left(2 \cdot \color{blue}{\left(1 + v \cdot \left(v \cdot -3\right)\right)}\right)} \cdot \left(1 - v \cdot v\right) \]

   11. distribute-rgt-in 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \color{blue}{\left(1 \cdot 2 + \left(v \cdot \left(v \cdot -3\right)\right) \cdot 2\right)}} \cdot \left(1 - v \cdot v\right) \]

   12. metadata-eval 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \left(\color{blue}{2} + \left(v \cdot \left(v \cdot -3\right)\right) \cdot 2\right)} \cdot \left(1 - v \cdot v\right) \]

   13. associate-*r* 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \left(2 + \color{blue}{\left(\left(v \cdot v\right) \cdot -3\right)} \cdot 2\right)} \cdot \left(1 - v \cdot v\right) \]

   14. associate-*l* 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \left(2 + \color{blue}{\left(v \cdot v\right) \cdot \left(-3 \cdot 2\right)}\right)} \cdot \left(1 - v \cdot v\right) \]

   15. metadata-eval 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \left(2 + \left(v \cdot v\right) \cdot \color{blue}{-6}\right)} \cdot \left(1 - v \cdot v\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt{0.0625 \cdot \left(2 + \left(v \cdot v\right) \cdot -6\right)}} \cdot \left(1 - v \cdot v\right) \]
10. Step-by-step derivation

    1. distribute-lft-in 100.0%

       \[\leadsto \sqrt{\color{blue}{0.0625 \cdot 2 + 0.0625 \cdot \left(\left(v \cdot v\right) \cdot -6\right)}} \cdot \left(1 - v \cdot v\right) \]

    2. metadata-eval 100.0%

       \[\leadsto \sqrt{\color{blue}{0.125} + 0.0625 \cdot \left(\left(v \cdot v\right) \cdot -6\right)} \cdot \left(1 - v \cdot v\right) \]

    3. *-commutative 100.0%

       \[\leadsto \sqrt{0.125 + 0.0625 \cdot \color{blue}{\left(-6 \cdot \left(v \cdot v\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

    4. associate-*r* 100.0%

       \[\leadsto \sqrt{0.125 + \color{blue}{\left(0.0625 \cdot -6\right) \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right) \]

    5. metadata-eval 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 4: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(v * v), $MachinePrecision] * -0.625), $MachinePrecision] + 0.25), $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 98.9%

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

   1. +-commutative 98.9%

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

   2. associate-*r* 98.9%

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

   3. distribute-rgt-out 98.9%

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

   4. *-commutative 98.9%

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

   5. unpow2 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in v around 0 99.0%

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

   1. unpow2 99.0%

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

   2. associate-*r* 99.0%

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

   3. distribute-rgt-out 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 5: 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. associate-*l/ 100.0%

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

   2. clear-num 100.0%

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

   3. sqrt-unprod 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. distribute-lft-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

   10. *-commutative 100.0%

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

   11. fma-udef 100.0%

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

5. Applied egg-rr 100.0%

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

   1. fma-udef 100.0%

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

7. Applied egg-rr 100.0%

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

   1. add-sqr-sqrt 98.4%

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

   2. sqrt-unprod 100.0%

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

   3. associate-/r/ 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/r/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. swap-sqr 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\sqrt{2 \cdot \left(v \cdot \left(v \cdot -3\right) + 1\right)} \cdot \sqrt{2 \cdot \left(v \cdot \left(v \cdot -3\right) + 1\right)}\right)} \cdot \left(1 - v \cdot v\right) \]

   9. add-sqr-sqrt 100.0%

      \[\leadsto \sqrt{0.0625 \cdot \color{blue}{\left(2 \cdot \left(v \cdot \left(v \cdot -3\right) + 1\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

   10. +-commutative 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \left(2 \cdot \color{blue}{\left(1 + v \cdot \left(v \cdot -3\right)\right)}\right)} \cdot \left(1 - v \cdot v\right) \]

   11. distribute-rgt-in 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \color{blue}{\left(1 \cdot 2 + \left(v \cdot \left(v \cdot -3\right)\right) \cdot 2\right)}} \cdot \left(1 - v \cdot v\right) \]

   12. metadata-eval 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \left(\color{blue}{2} + \left(v \cdot \left(v \cdot -3\right)\right) \cdot 2\right)} \cdot \left(1 - v \cdot v\right) \]

   13. associate-*r* 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \left(2 + \color{blue}{\left(\left(v \cdot v\right) \cdot -3\right)} \cdot 2\right)} \cdot \left(1 - v \cdot v\right) \]

   14. associate-*l* 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \left(2 + \color{blue}{\left(v \cdot v\right) \cdot \left(-3 \cdot 2\right)}\right)} \cdot \left(1 - v \cdot v\right) \]

   15. metadata-eval 100.0%

       \[\leadsto \sqrt{0.0625 \cdot \left(2 + \left(v \cdot v\right) \cdot \color{blue}{-6}\right)} \cdot \left(1 - v \cdot v\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt{0.0625 \cdot \left(2 + \left(v \cdot v\right) \cdot -6\right)}} \cdot \left(1 - v \cdot v\right) \]
10. Step-by-step derivation

    1. distribute-lft-in 100.0%

       \[\leadsto \sqrt{\color{blue}{0.0625 \cdot 2 + 0.0625 \cdot \left(\left(v \cdot v\right) \cdot -6\right)}} \cdot \left(1 - v \cdot v\right) \]

    2. metadata-eval 100.0%

       \[\leadsto \sqrt{\color{blue}{0.125} + 0.0625 \cdot \left(\left(v \cdot v\right) \cdot -6\right)} \cdot \left(1 - v \cdot v\right) \]

    3. *-commutative 100.0%

       \[\leadsto \sqrt{0.125 + 0.0625 \cdot \color{blue}{\left(-6 \cdot \left(v \cdot v\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

    4. associate-*r* 100.0%

       \[\leadsto \sqrt{0.125 + \color{blue}{\left(0.0625 \cdot -6\right) \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right) \]

    5. metadata-eval 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in v around 0 98.5%

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

13. Final simplification 98.5%

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

Alternative 6: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot 0.25 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (sqrt 2.0) 0.25))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * 0.25), $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. *-commutative 100.0%

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

   2. sqr-neg 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-*l* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 98.5%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2}} \]

5. Final simplification 98.5%

   \[\leadsto \sqrt{2} \cdot 0.25 \]

Reproduce

herbie shell --seed 2023287 
(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))))