Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 5

Speedup: 1.9×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 + (1.0d0 + ((-1.0d0) - (v * (v * 3.0d0))))))) * (1.0d0 - (v * v))
end function

Java

public static double code(double v) {
	return ((Math.sqrt(2.0) / 4.0) * Math.sqrt((1.0 + (1.0 + (-1.0 - (v * (v * 3.0))))))) * (1.0 - (v * v));
}

Python

def code(v):
	return ((math.sqrt(2.0) / 4.0) * math.sqrt((1.0 + (1.0 + (-1.0 - (v * (v * 3.0))))))) * (1.0 - (v * v))

Julia

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

MATLAB

function tmp = code(v)
	tmp = ((sqrt(2.0) / 4.0) * sqrt((1.0 + (1.0 + (-1.0 - (v * (v * 3.0))))))) * (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[(1.0 + N[(-1.0 - N[(v * N[(v * 3.0), $MachinePrecision]), $MachinePrecision]), $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. 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-udef 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-udef 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%

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

   6. associate-*l* 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * sqrt((0.125d0 + (v * (v * (-0.375d0)))))
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * Math.sqrt((0.125 + (v * (v * -0.375))));
}

Python

def code(v):
	return (1.0 - (v * v)) * math.sqrt((0.125 + (v * (v * -0.375))))

Julia

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

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * sqrt((0.125 + (v * (v * -0.375))));
end

Wolfram

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

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

   6. associate-*l* 100.0%

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

3. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 98.4%

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

5. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(1 - v \cdot \left(v \cdot 3\right)\right) \cdot 0.125}\right)} - 1\right)} \cdot \left(1 - v \cdot v\right) \]
6. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(1 - v \cdot \left(v \cdot 3\right)\right) \cdot 0.125}\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. sub-neg 100.0%

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

   5. distribute-lft-in 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*r* 100.0%

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

   8. unpow2 100.0%

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

   9. distribute-rgt-neg-in 100.0%

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

   10. unpow2 100.0%

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

   11. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in v around 0 100.0%

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

   1. *-commutative 100.0%

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

   2. unpow2 100.0%

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

   3. associate-*l* 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 3: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.125 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (sqrt (* 0.125 (- 1.0 (* v (* v 3.0))))))

C

double code(double v) {
	return sqrt((0.125 * (1.0 - (v * (v * 3.0)))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt((0.125d0 * (1.0d0 - (v * (v * 3.0d0)))))
end function

Java

public static double code(double v) {
	return Math.sqrt((0.125 * (1.0 - (v * (v * 3.0)))));
}

Python

def code(v):
	return math.sqrt((0.125 * (1.0 - (v * (v * 3.0)))))

Julia

function code(v)
	return sqrt(Float64(0.125 * Float64(1.0 - Float64(v * Float64(v * 3.0)))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt((0.125 * (1.0 - (v * (v * 3.0)))));
end

Wolfram

code[v_] := N[Sqrt[N[(0.125 * N[(1.0 - N[(v * N[(v * 3.0), $MachinePrecision]), $MachinePrecision]), $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. add-sqr-sqrt 98.4%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. pow2 100.0%

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

   7. *-commutative 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in v around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. unpow2 99.7%

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

6. Simplified 99.7%

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

   1. fma-udef 99.7%

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

   2. associate-*r* 99.7%

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

   3. metadata-eval 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

   5. *-commutative 99.7%

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

   6. +-commutative 99.7%

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

   7. sub-neg 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-*l* 99.7%

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in v around 0 99.0%

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

10. Final simplification 99.0%

    \[\leadsto \sqrt{0.125 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)} \]

Alternative 4: 98.9% 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. 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-udef 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-udef 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%

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

   6. associate-*l* 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in v around 0 99.0%

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

   1. sub-neg 99.0%

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

   2. distribute-lft-in 99.0%

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

   3. *-commutative 99.0%

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

   4. *-un-lft-identity 99.0%

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

   5. add-sqr-sqrt 97.5%

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

   6. sqrt-unprod 99.0%

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

   7. swap-sqr 99.0%

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

   8. metadata-eval 99.0%

      \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)} + \left(0.25 \cdot \sqrt{2}\right) \cdot \left(-v \cdot v\right) \]

   9. add-sqr-sqrt 99.0%

      \[\leadsto \sqrt{0.0625 \cdot \color{blue}{2}} + \left(0.25 \cdot \sqrt{2}\right) \cdot \left(-v \cdot v\right) \]

   10. metadata-eval 99.0%

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

   11. add-sqr-sqrt 99.0%

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

   12. sqrt-unprod 99.0%

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

   13. swap-sqr 99.0%

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

   14. metadata-eval 99.0%

       \[\leadsto \sqrt{0.125} + \sqrt{\color{blue}{0.0625} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \left(-v \cdot v\right) \]

   15. add-sqr-sqrt 99.0%

       \[\leadsto \sqrt{0.125} + \sqrt{0.0625 \cdot \color{blue}{2}} \cdot \left(-v \cdot v\right) \]

   16. metadata-eval 99.0%

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

   17. distribute-rgt-neg-in 99.0%

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

6. Applied egg-rr 99.0%

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

   1. *-commutative 99.0%

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

   2. distribute-rgt1-in 99.0%

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

   3. +-commutative 99.0%

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

   4. *-commutative 99.0%

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

   5. cancel-sign-sub-inv 99.0%

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

   6. *-commutative 99.0%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

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

Alternative 5: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.125} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (sqrt 0.125))

C

double code(double v) {
	return sqrt(0.125);
}

Fortran

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

Java

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

Python

def code(v):
	return math.sqrt(0.125)

Julia

function code(v)
	return sqrt(0.125)
end

MATLAB

function tmp = code(v)
	tmp = sqrt(0.125);
end

Wolfram

code[v_] := N[Sqrt[0.125], $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. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. metadata-eval 100.0%

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

   12. associate-*r* 100.0%

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

   13. fma-udef 100.0%

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

   14. frac-times 100.0%

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

   15. add-sqr-sqrt 100.0%

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

   16. metadata-eval 100.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot \frac{2}{\color{blue}{16}}} \cdot \left(1 - v \cdot v\right) \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]

4. Taylor expanded in v around 0 98.9%

   \[\leadsto \color{blue}{\sqrt{0.125}} \]

5. Final simplification 98.9%

   \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023223 
(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))))