Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 10.2s

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, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot -0.25}{\frac{1}{\mathsf{fma}\left(v, v, -1\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (* (sqrt (fma v (* v -6.0) 2.0)) -0.25) (/ 1.0 (fma v v -1.0))))

C

double code(double v) {
	return (sqrt(fma(v, (v * -6.0), 2.0)) * -0.25) / (1.0 / fma(v, v, -1.0));
}

Julia

function code(v)
	return Float64(Float64(sqrt(fma(v, Float64(v * -6.0), 2.0)) * -0.25) / Float64(1.0 / fma(v, v, -1.0)))
end

Wolfram

code[v_] := N[(N[(N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * -0.25), $MachinePrecision] / N[(1.0 / N[(v * v + -1.0), $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. div-inv N/A

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

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

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

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

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

   4. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

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

   1. *-commutative N/A

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

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

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

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

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. *-rgt-identity N/A

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

   11. sqrt-unprod N/A

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

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

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

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

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

   14. metadata-eval N/A

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

   15. sub-neg N/A

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

   16. *-rgt-identity N/A

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

   17. *-commutative N/A

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

6. Applied egg-rr 100.0%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. /-rgt-identity N/A

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

   10. associate-/r/ N/A

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

   11. frac-2neg N/A

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

   12. /-lowering-/.f64 N/A

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

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot -0.25}{\frac{1}{\mathsf{fma}\left(v, v, -1\right)}}} \]
9. Add Preprocessing

Alternative 2: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot -0.25\right) \cdot \mathsf{fma}\left(v, v, -1\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (* (sqrt (fma v (* v -6.0) 2.0)) -0.25) (fma v v -1.0)))

C

double code(double v) {
	return (sqrt(fma(v, (v * -6.0), 2.0)) * -0.25) * fma(v, v, -1.0);
}

Julia

function code(v)
	return Float64(Float64(sqrt(fma(v, Float64(v * -6.0), 2.0)) * -0.25) * fma(v, v, -1.0))
end

Wolfram

code[v_] := N[(N[(N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * -0.25), $MachinePrecision] * N[(v * v + -1.0), $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. Step-by-step derivation

   1. div-inv N/A

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

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

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

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

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

   4. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

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

   1. *-commutative N/A

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

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

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

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

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. *-rgt-identity N/A

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

   11. sqrt-unprod N/A

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

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

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

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

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

   14. metadata-eval N/A

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

   15. sub-neg N/A

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

   16. *-rgt-identity N/A

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

   17. *-commutative N/A

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

6. Applied egg-rr 100.0%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. associate-*l* N/A

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

   9. /-rgt-identity N/A

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

   10. frac-2neg N/A

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

   11. metadata-eval N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, v, -1\right) \cdot \left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot -0.25\right)} \]

9. Final simplification 100.0%

   \[\leadsto \left(\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot -0.25\right) \cdot \mathsf{fma}\left(v, v, -1\right) \]
10. Add Preprocessing

Alternative 3: 99.5% accurate, 2.3× speedup

Math

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

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(Float64(v * v), -0.625, 0.25))
end

Wolfram

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

4. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(v \cdot \left(v \cdot \left(v \cdot v\right)\right), -9, 1\right)} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(v, 3 \cdot v, 1\right)}}\right)}\right) \cdot \left(1 - v \cdot v\right) \]

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

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

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

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

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

   14. *-commutative N/A

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

   15. accelerator-lowering-fma.f64 N/A

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

7. Simplified 99.8%

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

Alternative 4: 98.9% accurate, 3.9× 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

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

   2. sqrt-lowering-sqrt.f64 99.4

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Reproduce

herbie shell --seed 2024204 
(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))))