Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

Alternatives: 8

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{{\left({\left(2 \cdot \mathsf{fma}\left(-3, {v}^{2}, 1\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{-4} \cdot \mathsf{fma}\left(v, v, -1\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (*
  (/
   (pow (pow (* 2.0 (fma -3.0 (pow v 2.0) 1.0)) 1.5) 0.3333333333333333)
   -4.0)
  (fma v v -1.0)))

C

double code(double v) {
	return (pow(pow((2.0 * fma(-3.0, pow(v, 2.0), 1.0)), 1.5), 0.3333333333333333) / -4.0) * fma(v, v, -1.0);
}

Julia

function code(v)
	return Float64(Float64(((Float64(2.0 * fma(-3.0, (v ^ 2.0), 1.0)) ^ 1.5) ^ 0.3333333333333333) / -4.0) * fma(v, v, -1.0))
end

Wolfram

code[v_] := N[(N[(N[Power[N[Power[N[(2.0 * N[(-3.0 * N[Power[v, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision] / -4.0), $MachinePrecision] * N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. associate-*r/ 100.0%

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

   2. fma-udef 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. sqrt-unprod 100.0%

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

   7. +-commutative 100.0%

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

   8. associate-*l* 100.0%

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

   9. fma-def 100.0%

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

   10. pow2 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-/r/ 100.0%

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

8. Simplified 100.0%

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

   1. fma-udef 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. *-commutative 100.0%

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

   4. distribute-rgt-in 100.0%

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

   5. fma-udef 100.0%

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

   6. add-cbrt-cube 98.4%

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

   7. pow1/3 100.0%

      \[\leadsto \frac{\color{blue}{{\left(\left(\sqrt{2 \cdot \mathsf{fma}\left({v}^{2}, -3, 1\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left({v}^{2}, -3, 1\right)}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left({v}^{2}, -3, 1\right)}\right)}^{0.3333333333333333}}}{-4} \cdot \mathsf{fma}\left(v, v, -1\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left({\left(2 \cdot \mathsf{fma}\left(-3, {v}^{2}, 1\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{-4} \cdot \mathsf{fma}\left(v, v, -1\right) \]

11. Final simplification 100.0%

    \[\leadsto \frac{{\left({\left(2 \cdot \mathsf{fma}\left(-3, {v}^{2}, 1\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{-4} \cdot \mathsf{fma}\left(v, v, -1\right) \]
12. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. associate-*r/ 100.0%

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

   2. fma-udef 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. sqrt-unprod 100.0%

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

   7. +-commutative 100.0%

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

   8. associate-*l* 100.0%

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

   9. fma-def 100.0%

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

   10. pow2 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-/r/ 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 3: 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]

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. Final simplification 100.0%

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

Alternative 4: 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%

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

3. Simplified 100.0%

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

5. Taylor expanded in v around 0 99.2%

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

6. Final simplification 99.2%

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

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(v)
	return Float64(fma(v, v, -1.0) * Float64(sqrt(2.0) / -4.0))
end

Wolfram

code[v_] := N[(N[(v * v + -1.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. associate-*r/ 100.0%

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

   2. fma-udef 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. sqrt-unprod 100.0%

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

   7. +-commutative 100.0%

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

   8. associate-*l* 100.0%

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

   9. fma-def 100.0%

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

   10. pow2 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-/r/ 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in v around 0 98.4%

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

10. Final simplification 98.4%

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

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2}}{\frac{-4}{\mathsf{fma}\left(v, v, -1\right)}} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (/ (sqrt 2.0) (/ -4.0 (fma v v -1.0))))

C

double code(double v) {
	return sqrt(2.0) / (-4.0 / fma(v, v, -1.0));
}

Julia

function code(v)
	return Float64(sqrt(2.0) / Float64(-4.0 / fma(v, v, -1.0)))
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] / N[(-4.0 / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. associate-*r/ 100.0%

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

   2. fma-udef 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. +-commutative 100.0%

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

   6. sqrt-unprod 100.0%

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

   7. +-commutative 100.0%

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

   8. associate-*l* 100.0%

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

   9. fma-def 100.0%

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

   10. pow2 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-/r/ 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in v around 0 98.4%

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

    1. associate-*l/ 98.4%

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

    2. associate-/l* 98.4%

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

11. Applied egg-rr 98.4%

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

12. Final simplification 98.4%

    \[\leadsto \frac{\sqrt{2}}{\frac{-4}{\mathsf{fma}\left(v, v, -1\right)}} \]
13. Add Preprocessing

Alternative 7: 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%

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

3. Simplified 100.0%

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

5. Taylor expanded in v around 0 98.4%

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

6. Final simplification 98.4%

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

Alternative 8: 24.4% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (v) :precision binary64 0.3333333333333333)

C

double code(double v) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = 0.3333333333333333d0
end function

Java

public static double code(double v) {
	return 0.3333333333333333;
}

Python

def code(v):
	return 0.3333333333333333

Julia

function code(v)
	return 0.3333333333333333
end

MATLAB

function tmp = code(v)
	tmp = 0.3333333333333333;
end

Wolfram

code[v_] := 0.3333333333333333

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in v around 0 99.2%

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

   1. distribute-lft-in 99.2%

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

   2. flip-+ 97.7%

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

   3. *-commutative 97.7%

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

   4. *-commutative 97.7%

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

   5. *-commutative 97.7%

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

   6. associate-*l* 97.7%

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

   7. *-commutative 97.7%

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

   8. associate-*l* 97.7%

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

   9. *-commutative 97.7%

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

   10. *-commutative 97.7%

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

   11. associate-*l* 97.7%

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

7. Applied egg-rr 97.7%

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

9. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{0.125 + {v}^{4} \cdot -0.78125}{\sqrt{2} \cdot \left(0.25 - {v}^{2} \cdot -0.625\right)}} \]

10. Taylor expanded in v around 0 96.9%

    \[\leadsto \color{blue}{\frac{0.5}{\sqrt{2}}} \]

12. Applied egg-rr 24.4%

    \[\leadsto \color{blue}{0.3333333333333333} \]

13. Final simplification 24.4%

    \[\leadsto 0.3333333333333333 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))