Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 9.7s

Alternatives: 5

Speedup: 1.0×

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{\mathsf{fma}\left({v}^{6}, -27, 1\right)}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(3 \cdot {v}^{2}, v \cdot \sqrt{3}\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (*
  (*
   (/ (sqrt 2.0) 4.0)
   (/
    (sqrt (fma (pow v 6.0) -27.0 1.0))
    (hypot 1.0 (hypot (* 3.0 (pow v 2.0)) (* v (sqrt 3.0))))))
  (- 1.0 (* v v))))

C

double code(double v) {
	return ((sqrt(2.0) / 4.0) * (sqrt(fma(pow(v, 6.0), -27.0, 1.0)) / hypot(1.0, hypot((3.0 * pow(v, 2.0)), (v * sqrt(3.0)))))) * (1.0 - (v * v));
}

Julia

function code(v)
	return Float64(Float64(Float64(sqrt(2.0) / 4.0) * Float64(sqrt(fma((v ^ 6.0), -27.0, 1.0)) / hypot(1.0, hypot(Float64(3.0 * (v ^ 2.0)), Float64(v * sqrt(3.0)))))) * Float64(1.0 - Float64(v * v)))
end

Wolfram

code[v_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[(N[Sqrt[N[(N[Power[v, 6.0], $MachinePrecision] * -27.0 + 1.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[Sqrt[N[(3.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(v * N[Sqrt[3.0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left({v}^{6}, -27, 1\right)}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(3 \cdot {v}^{2}, v \cdot \sqrt{3}\right)\right)}}\right) \cdot \left(1 - v \cdot v\right) \]

5. Final simplification 100.0%

   \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{\mathsf{fma}\left({v}^{6}, -27, 1\right)}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(3 \cdot {v}^{2}, v \cdot \sqrt{3}\right)\right)}\right) \cdot \left(1 - v \cdot v\right) \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (* (- 1.0 (* v v)) (sqrt (fma (* v -0.375) v 0.125))))

C

double code(double v) {
	return (1.0 - (v * v)) * sqrt(fma((v * -0.375), v, 0.125));
}

Julia

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) * sqrt(fma(Float64(v * -0.375), v, 0.125)))
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(v * -0.375), $MachinePrecision] * v + 0.125), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-undefine 100.0%

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

   3. log1p-undefine 100.0%

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

   4. add-exp-log 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-define 100.0%

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

   7. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

   1. pow1 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(\sqrt{\left(1 - \left(3 \cdot {v}^{2} + 0\right)\right) \cdot 0.125}\right)}^{1}} \cdot \left(1 - v \cdot v\right) \]
7. Step-by-step derivation

   1. unpow1 100.0%

      \[\leadsto \color{blue}{\sqrt{\left(1 - \left(3 \cdot {v}^{2} + 0\right)\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]

   2. *-commutative 100.0%

      \[\leadsto \sqrt{\color{blue}{0.125 \cdot \left(1 - \left(3 \cdot {v}^{2} + 0\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

   3. sub-neg 100.0%

      \[\leadsto \sqrt{0.125 \cdot \color{blue}{\left(1 + \left(-\left(3 \cdot {v}^{2} + 0\right)\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

   4. +-rgt-identity 100.0%

      \[\leadsto \sqrt{0.125 \cdot \left(1 + \left(-\color{blue}{3 \cdot {v}^{2}}\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   5. distribute-lft-in 100.0%

      \[\leadsto \sqrt{\color{blue}{0.125 \cdot 1 + 0.125 \cdot \left(-3 \cdot {v}^{2}\right)}} \cdot \left(1 - v \cdot v\right) \]

   6. metadata-eval 100.0%

      \[\leadsto \sqrt{\color{blue}{0.125} + 0.125 \cdot \left(-3 \cdot {v}^{2}\right)} \cdot \left(1 - v \cdot v\right) \]

   7. *-commutative 100.0%

      \[\leadsto \sqrt{0.125 + 0.125 \cdot \left(-\color{blue}{{v}^{2} \cdot 3}\right)} \cdot \left(1 - v \cdot v\right) \]

   8. distribute-rgt-neg-in 100.0%

      \[\leadsto \sqrt{0.125 + 0.125 \cdot \color{blue}{\left({v}^{2} \cdot \left(-3\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

   9. metadata-eval 100.0%

      \[\leadsto \sqrt{0.125 + 0.125 \cdot \left({v}^{2} \cdot \color{blue}{-3}\right)} \cdot \left(1 - v \cdot v\right) \]

8. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{0.125 + 0.125 \cdot \left({v}^{2} \cdot -3\right)}} \cdot \left(1 - v \cdot v\right) \]

9. Taylor expanded in v around 0 100.0%

   \[\leadsto \sqrt{0.125 + \color{blue}{-0.375 \cdot {v}^{2}}} \cdot \left(1 - v \cdot v\right) \]
10. Step-by-step derivation

    1. *-commutative 100.0%

       \[\leadsto \sqrt{0.125 + \color{blue}{{v}^{2} \cdot -0.375}} \cdot \left(1 - v \cdot v\right) \]

11. Simplified 100.0%

    \[\leadsto \sqrt{0.125 + \color{blue}{{v}^{2} \cdot -0.375}} \cdot \left(1 - v \cdot v\right) \]
12. Step-by-step derivation

    1. +-commutative 100.0%

       \[\leadsto \sqrt{\color{blue}{{v}^{2} \cdot -0.375 + 0.125}} \cdot \left(1 - v \cdot v\right) \]

    2. pow2 100.0%

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

    3. *-commutative 100.0%

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

    4. associate-*r* 100.0%

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

    5. fma-define 100.0%

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

13. Applied egg-rr 100.0%

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

14. Final simplification 100.0%

    \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot -0.375, v, 0.125\right)} \]
15. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l* 100.0%

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

   2. sqr-neg 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. metadata-eval 100.0%

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

   5. sqr-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in v around 0 99.3%

   \[\leadsto \frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + -2.5 \cdot {v}^{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{{v}^{2} \cdot -2.5}\right) \]

7. Simplified 99.3%

   \[\leadsto \frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + {v}^{2} \cdot -2.5\right)} \]

8. Taylor expanded in v around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. distribute-rgt-out 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

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

Alternative 4: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.125], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-undefine 100.0%

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

   3. log1p-undefine 100.0%

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

   4. add-exp-log 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-define 100.0%

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

   7. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

   1. add-log-exp 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. sqrt-unprod 100.0%

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

   4. *-commutative 100.0%

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

   5. *-commutative 100.0%

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

   6. swap-sqr 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\log \left(e^{\sqrt{\left(1 - \left(3 \cdot {v}^{2} + 0\right)\right) \cdot 0.125}}\right)} \cdot \left(1 - v \cdot v\right) \]

7. Taylor expanded in v around 0 98.6%

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

8. Final simplification 98.6%

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

Alternative 5: 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 98.6%

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

4. Taylor expanded in v around 0 98.6%

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

5. Final simplification 98.6%

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

Reproduce

herbie shell --seed 2024039 
(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))))