Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

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 rewrites 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 rewrites 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: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v \cdot v, 0.09375, -0.625\right), 0.25\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (sqrt 2.0) (fma (* v v) (fma (* v v) 0.09375 -0.625) 0.25)))

C

double code(double v) {
	return sqrt(2.0) * fma((v * v), fma((v * v), 0.09375, -0.625), 0.25);
}

Julia

function code(v)
	return Float64(sqrt(2.0) * fma(Float64(v * v), fma(Float64(v * v), 0.09375, -0.625), 0.25))
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(v * v), $MachinePrecision] * N[(N[(v * v), $MachinePrecision] * 0.09375 + -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. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \sqrt{2} + {v}^{2} \cdot \left(\frac{1}{4} \cdot \left({v}^{2} \cdot \left(\frac{-9}{8} \cdot \sqrt{2} + \frac{3}{2} \cdot \sqrt{2}\right)\right) + \frac{1}{4} \cdot \left(\frac{-3}{2} \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{v}^{2} \cdot \left(\frac{1}{4} \cdot \left({v}^{2} \cdot \left(\frac{-9}{8} \cdot \sqrt{2} + \frac{3}{2} \cdot \sqrt{2}\right)\right) + \frac{1}{4} \cdot \left(\frac{-3}{2} \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right) + \frac{1}{4} \cdot \sqrt{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left({v}^{2} \cdot \left(\frac{-9}{8} \cdot \sqrt{2} + \frac{3}{2} \cdot \sqrt{2}\right)\right) + \frac{1}{4} \cdot \left(\frac{-3}{2} \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right) \cdot {v}^{2}} + \frac{1}{4} \cdot \sqrt{2} \]

   3. distribute-lft-out N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left({v}^{2} \cdot \left(\frac{-9}{8} \cdot \sqrt{2} + \frac{3}{2} \cdot \sqrt{2}\right) + \left(\frac{-3}{2} \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)\right)} \cdot {v}^{2} + \frac{1}{4} \cdot \sqrt{2} \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left({v}^{2} \cdot \left(\frac{-9}{8} \cdot \sqrt{2} + \frac{3}{2} \cdot \sqrt{2}\right) + \left(\frac{-3}{2} \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right) \cdot {v}^{2}\right)} + \frac{1}{4} \cdot \sqrt{2} \]

   5. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left({v}^{2} \cdot \left(\frac{-9}{8} \cdot \sqrt{2} + \frac{3}{2} \cdot \sqrt{2}\right) + \left(\frac{-3}{2} \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right) \cdot {v}^{2} + \sqrt{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left({v}^{2} \cdot \left(\frac{-9}{8} \cdot \sqrt{2} + \frac{3}{2} \cdot \sqrt{2}\right) + \left(\frac{-3}{2} \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right) \cdot {v}^{2} + \sqrt{2}\right)} \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{0.25 \cdot \mathsf{fma}\left(v \cdot v, \sqrt{2} \cdot \mathsf{fma}\left(v \cdot v, 0.375, -2.5\right), \sqrt{2}\right)} \]

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \sqrt{2} + {v}^{2} \cdot \left(\frac{-5}{8} \cdot \sqrt{2} + \frac{3}{32} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{v}^{2} \cdot \left(\frac{-5}{8} \cdot \sqrt{2} + \frac{3}{32} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right) + \frac{1}{4} \cdot \sqrt{2}} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\frac{-5}{8} \cdot \sqrt{2}\right) \cdot {v}^{2} + \left(\frac{3}{32} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right) \cdot {v}^{2}\right)} + \frac{1}{4} \cdot \sqrt{2} \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\frac{-5}{8} \cdot \left(\sqrt{2} \cdot {v}^{2}\right)} + \left(\frac{3}{32} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right) \cdot {v}^{2}\right) + \frac{1}{4} \cdot \sqrt{2} \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{-5}{8} \cdot \color{blue}{\left({v}^{2} \cdot \sqrt{2}\right)} + \left(\frac{3}{32} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right) \cdot {v}^{2}\right) + \frac{1}{4} \cdot \sqrt{2} \]

   5. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-5}{8} \cdot {v}^{2}\right) \cdot \sqrt{2}} + \left(\frac{3}{32} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right) \cdot {v}^{2}\right) + \frac{1}{4} \cdot \sqrt{2} \]

   6. *-commutative N/A

      \[\leadsto \left(\left(\frac{-5}{8} \cdot {v}^{2}\right) \cdot \sqrt{2} + \color{blue}{{v}^{2} \cdot \left(\frac{3}{32} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right)}\right) + \frac{1}{4} \cdot \sqrt{2} \]

   7. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-5}{8} \cdot {v}^{2}\right) \cdot \sqrt{2} + {v}^{2} \cdot \color{blue}{\left(\left(\frac{3}{32} \cdot {v}^{2}\right) \cdot \sqrt{2}\right)}\right) + \frac{1}{4} \cdot \sqrt{2} \]

   8. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-5}{8} \cdot {v}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left({v}^{2} \cdot \left(\frac{3}{32} \cdot {v}^{2}\right)\right) \cdot \sqrt{2}}\right) + \frac{1}{4} \cdot \sqrt{2} \]

   9. distribute-rgt-out N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\frac{-5}{8} \cdot {v}^{2} + {v}^{2} \cdot \left(\frac{3}{32} \cdot {v}^{2}\right)\right)} + \frac{1}{4} \cdot \sqrt{2} \]

   10. *-commutative N/A

       \[\leadsto \sqrt{2} \cdot \left(\frac{-5}{8} \cdot {v}^{2} + {v}^{2} \cdot \left(\frac{3}{32} \cdot {v}^{2}\right)\right) + \color{blue}{\sqrt{2} \cdot \frac{1}{4}} \]

   11. distribute-lft-out N/A

       \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\left(\frac{-5}{8} \cdot {v}^{2} + {v}^{2} \cdot \left(\frac{3}{32} \cdot {v}^{2}\right)\right) + \frac{1}{4}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\left(\frac{-5}{8} \cdot {v}^{2} + {v}^{2} \cdot \left(\frac{3}{32} \cdot {v}^{2}\right)\right) + \frac{1}{4}\right)} \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{2}} \cdot \left(\left(\frac{-5}{8} \cdot {v}^{2} + {v}^{2} \cdot \left(\frac{3}{32} \cdot {v}^{2}\right)\right) + \frac{1}{4}\right) \]

8. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v \cdot v, 0.09375, -0.625\right), 0.25\right)} \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(v * N[(v * -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. 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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. distribute-rgt-out N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lower-fma.f64 N/A

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

   18. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \mathsf{fma}\left(v, v \cdot -0.625, 0.25\right)} \]
8. 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 99.4

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

5. Applied rewrites 99.4%

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

Reproduce

herbie shell --seed 2024216 
(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))))