Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 9.0s

Alternatives: 4

Speedup: 1.7×

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, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.25, v \cdot v, 0.25\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (fma -0.25 (* v v) 0.25) (sqrt (fma v (* v -6.0) 2.0))))

C

double code(double v) {
	return fma(-0.25, (v * v), 0.25) * sqrt(fma(v, (v * -6.0), 2.0));
}

Julia

function code(v)
	return Float64(fma(-0.25, Float64(v * v), 0.25) * sqrt(fma(v, Float64(v * -6.0), 2.0)))
end

Wolfram

code[v_] := N[(N[(-0.25 * N[(v * v), $MachinePrecision] + 0.25), $MachinePrecision] * N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $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. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. associate-*l/ N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

4. Applied egg-rr 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, v \cdot v, 0.25\right) \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}} \]
7. Add Preprocessing

Alternative 2: 99.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-out N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-out N/A

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

   11. distribute-rgt1-in N/A

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

   12. lower-*.f64 N/A

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

5. Simplified 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \left(\color{blue}{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)} \cdot \sqrt{2}\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \left(\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right) \cdot \color{blue}{\sqrt{2}}\right) \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)\right) \cdot \sqrt{2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)\right) \cdot \sqrt{2}} \]

   6. lift-fma.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(v \cdot v, \frac{1}{4} \cdot \frac{-5}{2}, \frac{1}{4}\right)} \cdot \sqrt{2} \]

   13. metadata-eval 99.0

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

7. Applied egg-rr 99.0%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 99.0

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

9. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v \cdot -0.625, v, 0.25\right)} \cdot \sqrt{2} \]
10. Add Preprocessing

Alternative 3: 98.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[((-v) * v + 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. Add Preprocessing

3. Taylor expanded in v around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-out N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-out N/A

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

   11. distribute-rgt1-in N/A

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

   12. lower-*.f64 N/A

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

5. Simplified 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \left(\color{blue}{\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)} \cdot \sqrt{2}\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \left(\mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right) \cdot \color{blue}{\sqrt{2}}\right) \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)\right) \cdot \sqrt{2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \mathsf{fma}\left(v \cdot v, \frac{-5}{2}, 1\right)\right) \cdot \sqrt{2}} \]

   6. lift-fma.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(v \cdot v, \frac{1}{4} \cdot \frac{-5}{2}, \frac{1}{4}\right)} \cdot \sqrt{2} \]

   13. metadata-eval 99.0

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

7. Applied egg-rr 99.0%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 99.0

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

9. Applied egg-rr 99.0%

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

10. Taylor expanded in v around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 98.6

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

12. Simplified 98.6%

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

13. Final simplification 98.6%

    \[\leadsto \sqrt{2} \cdot \mathsf{fma}\left(-v, v, 0.25\right) \]
14. Add Preprocessing

Alternative 4: 98.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \sqrt{2}} \]

   2. lower-sqrt.f64 98.6

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

5. Simplified 98.6%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))