Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 4

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] / N[(4.0 / N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - v \cdot v\right), \color{blue}{\left(\frac{4}{\sqrt{2} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)}\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(v \cdot v\right)\right), \left(\frac{\color{blue}{4}}{\sqrt{2} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \left(\frac{4}{\sqrt{2} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \color{blue}{\left(\sqrt{2} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}\right)\right) \]

   9. sqrt-unprod N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(1 + \left(\mathsf{neg}\left(3 \cdot \left(v \cdot v\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(\left(1 \cdot 2 + \left(\mathsf{neg}\left(3 \cdot \left(v \cdot v\right)\right)\right) \cdot 2\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(\left(2 + \left(\mathsf{neg}\left(3 \cdot \left(v \cdot v\right)\right)\right) \cdot 2\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\mathsf{neg}\left(3 \cdot \left(v \cdot v\right)\right)\right) \cdot 2\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(3 \cdot \left(v \cdot v\right)\right)\right), 2\right)\right)\right)\right)\right) \]

4. Applied egg-rr 100.0%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\left(v \cdot v\right) \cdot \left(-3 \cdot 2\right)\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(v \cdot v\right), \left(-3 \cdot 2\right)\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(-3 \cdot 2\right)\right)\right)\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(4, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right)\right) \]

6. Applied egg-rr 100.0%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2}}{\frac{1}{0.25 + \left(v \cdot v\right) \cdot -0.625}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (sqrt 2.0) (/ 1.0 (+ 0.25 (* (* v v) -0.625)))))

C

double code(double v) {
	return sqrt(2.0) / (1.0 / (0.25 + ((v * v) * -0.625)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) / (1.0d0 / (0.25d0 + ((v * v) * (-0.625d0))))
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) / (1.0 / (0.25 + ((v * v) * -0.625)));
}

Python

def code(v):
	return math.sqrt(2.0) / (1.0 / (0.25 + ((v * v) * -0.625)))

Julia

function code(v)
	return Float64(sqrt(2.0) / Float64(1.0 / Float64(0.25 + Float64(Float64(v * v) * -0.625))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) / (1.0 / (0.25 + ((v * v) * -0.625)));
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] / N[(1.0 / N[(0.25 + N[(N[(v * v), $MachinePrecision] * -0.625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({v}^{2} \cdot \sqrt{2}\right) + \frac{1}{4} \cdot \sqrt{2}\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \sqrt{2} + \frac{-3}{8} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \sqrt{2} + \left(\frac{-3}{8} \cdot {v}^{2}\right) \cdot \sqrt{2}\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   3. distribute-rgt-out N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2} \cdot \left(\frac{1}{4} + \frac{-3}{8} \cdot {v}^{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\frac{1}{4} + \frac{-3}{8} \cdot {v}^{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{1}{4} + \frac{-3}{8} \cdot {v}^{2}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{-3}{8} \cdot {v}^{2}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left({v}^{2} \cdot \frac{-3}{8}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({v}^{2}\right), \frac{-3}{8}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(v \cdot v\right), \frac{-3}{8}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   10. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \frac{-3}{8}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

5. Simplified 99.7%

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

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{-5}{8} \cdot \left({v}^{2} \cdot \sqrt{2}\right) + \frac{1}{4} \cdot \sqrt{2}} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-5}{8} \cdot {v}^{2}\right) \cdot \sqrt{2} + \color{blue}{\frac{1}{4}} \cdot \sqrt{2} \]

   2. distribute-rgt-out N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right)}\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{\frac{-5}{8} \cdot {v}^{2}} + \frac{1}{4}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{1}{4} + \color{blue}{\frac{-5}{8} \cdot {v}^{2}}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-5}{8} \cdot {v}^{2}\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left({v}^{2} \cdot \color{blue}{\frac{-5}{8}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({v}^{2}\right), \color{blue}{\frac{-5}{8}}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(v \cdot v\right), \frac{-5}{8}\right)\right)\right) \]

   10. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \frac{-5}{8}\right)\right)\right) \]

8. Simplified 99.7%

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

   1. flip3-+ N/A

      \[\leadsto \sqrt{2} \cdot \frac{{\frac{1}{4}}^{3} + {\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)}^{3}}{\color{blue}{\frac{1}{4} \cdot \frac{1}{4} + \left(\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) - \frac{1}{4} \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)\right)}} \]

   2. clear-num N/A

      \[\leadsto \sqrt{2} \cdot \frac{1}{\color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{4} + \left(\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) - \frac{1}{4} \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)\right)}{{\frac{1}{4}}^{3} + {\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)}^{3}}}} \]

   3. un-div-inv N/A

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{4} + \left(\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) - \frac{1}{4} \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)\right)}{{\frac{1}{4}}^{3} + {\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)}^{3}}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\frac{\frac{1}{4} \cdot \frac{1}{4} + \left(\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) - \frac{1}{4} \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)\right)}{{\frac{1}{4}}^{3} + {\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)}^{3}}\right)}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{4} + \left(\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) - \frac{1}{4} \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)\right)}}{{\frac{1}{4}}^{3} + {\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)}^{3}}\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{1}{\color{blue}{\frac{{\frac{1}{4}}^{3} + {\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)}^{3}}{\frac{1}{4} \cdot \frac{1}{4} + \left(\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right) - \frac{1}{4} \cdot \left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)\right)}}}\right)\right) \]

   7. flip3-+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{1}{\frac{1}{4} + \color{blue}{\left(v \cdot v\right) \cdot \frac{-5}{8}}}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{4} + \left(v \cdot v\right) \cdot \frac{-5}{8}\right)}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\left(v \cdot v\right) \cdot \frac{-5}{8}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(v \cdot v\right), \color{blue}{\frac{-5}{8}}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \frac{-5}{8}\right)\right)\right)\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{1}{0.25 + \left(v \cdot v\right) \cdot -0.625}}} \]
11. Add Preprocessing

Alternative 3: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({v}^{2} \cdot \sqrt{2}\right) + \frac{1}{4} \cdot \sqrt{2}\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \sqrt{2} + \frac{-3}{8} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \sqrt{2} + \left(\frac{-3}{8} \cdot {v}^{2}\right) \cdot \sqrt{2}\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   3. distribute-rgt-out N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2} \cdot \left(\frac{1}{4} + \frac{-3}{8} \cdot {v}^{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\frac{1}{4} + \frac{-3}{8} \cdot {v}^{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{1}{4} + \frac{-3}{8} \cdot {v}^{2}\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{-3}{8} \cdot {v}^{2}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left({v}^{2} \cdot \frac{-3}{8}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({v}^{2}\right), \frac{-3}{8}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(v \cdot v\right), \frac{-3}{8}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

   10. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \frac{-3}{8}\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right) \]

5. Simplified 99.7%

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

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{-5}{8} \cdot \left({v}^{2} \cdot \sqrt{2}\right) + \frac{1}{4} \cdot \sqrt{2}} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-5}{8} \cdot {v}^{2}\right) \cdot \sqrt{2} + \color{blue}{\frac{1}{4}} \cdot \sqrt{2} \]

   2. distribute-rgt-out N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right)}\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\color{blue}{\frac{-5}{8} \cdot {v}^{2}} + \frac{1}{4}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{1}{4} + \color{blue}{\frac{-5}{8} \cdot {v}^{2}}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-5}{8} \cdot {v}^{2}\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \left({v}^{2} \cdot \color{blue}{\frac{-5}{8}}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({v}^{2}\right), \color{blue}{\frac{-5}{8}}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(v \cdot v\right), \frac{-5}{8}\right)\right)\right) \]

   10. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \frac{-5}{8}\right)\right)\right) \]

8. Simplified 99.7%

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

Alternative 4: 99.0% 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. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 99.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 99.2%

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

6. Final simplification 99.2%

   \[\leadsto \sqrt{2} \cdot 0.25 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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))))