Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 6

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[Sqrt[N[(v * N[(v * -3.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[(1.0 - N[(v * v), $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. *-commutative 100.0%

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

   2. sqr-neg 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-*l* 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(v * N[(v * -3.0), $MachinePrecision] + 1.0), $MachinePrecision] * 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. Step-by-step derivation

   1. associate-*r* 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 99.9%

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

   7. sub-neg 99.9%

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

   8. +-commutative 99.9%

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

   9. *-commutative 99.9%

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

   10. distribute-rgt-neg-in 99.9%

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

   11. distribute-lft-neg-in 99.9%

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

   12. metadata-eval 99.9%

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

   13. *-commutative 99.9%

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

   14. fma-udef 99.9%

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

   15. frac-times 99.9%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(\sqrt{2} + \left(v \cdot v\right) \cdot \left(\sqrt{2} \cdot -2.5\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* 0.25 (+ (sqrt 2.0) (* (* v v) (* (sqrt 2.0) -2.5)))))

C

double code(double v) {
	return 0.25 * (sqrt(2.0) + ((v * v) * (sqrt(2.0) * -2.5)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = 0.25d0 * (sqrt(2.0d0) + ((v * v) * (sqrt(2.0d0) * (-2.5d0))))
end function

Java

public static double code(double v) {
	return 0.25 * (Math.sqrt(2.0) + ((v * v) * (Math.sqrt(2.0) * -2.5)));
}

Python

def code(v):
	return 0.25 * (math.sqrt(2.0) + ((v * v) * (math.sqrt(2.0) * -2.5)))

Julia

function code(v)
	return Float64(0.25 * Float64(sqrt(2.0) + Float64(Float64(v * v) * Float64(sqrt(2.0) * -2.5))))
end

MATLAB

function tmp = code(v)
	tmp = 0.25 * (sqrt(2.0) + ((v * v) * (sqrt(2.0) * -2.5)));
end

Wolfram

code[v_] := N[(0.25 * N[(N[Sqrt[2.0], $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -2.5), $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. *-commutative 100.0%

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

   2. sqr-neg 100.0%

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

   3. sqr-neg 100.0%

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

   4. associate-*l* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.6%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} + 0.25 \cdot \left({v}^{2} \cdot \left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)} \]
5. Step-by-step derivation

   1. distribute-lft-out 99.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{2} + {v}^{2} \cdot \left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)} \]

   2. unpow2 99.6%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} + \color{blue}{\left(v \cdot v\right)} \cdot \left(-1.5 \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right) \]

   3. distribute-rgt-out 99.6%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} + \left(v \cdot v\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-1.5 + -1\right)\right)}\right) \]

   4. metadata-eval 99.6%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} + \left(v \cdot v\right) \cdot \left(\sqrt{2} \cdot \color{blue}{-2.5}\right)\right) \]

6. Simplified 99.6%

   \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{2} + \left(v \cdot v\right) \cdot \left(\sqrt{2} \cdot -2.5\right)\right)} \]

7. Final simplification 99.6%

   \[\leadsto 0.25 \cdot \left(\sqrt{2} + \left(v \cdot v\right) \cdot \left(\sqrt{2} \cdot -2.5\right)\right) \]

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 + N[(N[(v * v), $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-*r* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.6%

   \[\leadsto \color{blue}{\left(-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \sqrt{2}\right)} \cdot \left(1 - v \cdot v\right) \]
5. Step-by-step derivation

   1. associate-*r* 99.6%

      \[\leadsto \left(\color{blue}{\left(-0.375 \cdot {v}^{2}\right) \cdot \sqrt{2}} + 0.25 \cdot \sqrt{2}\right) \cdot \left(1 - v \cdot v\right) \]

   2. distribute-rgt-out 99.6%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(-0.375 \cdot {v}^{2} + 0.25\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   3. unpow2 99.6%

      \[\leadsto \left(\sqrt{2} \cdot \left(-0.375 \cdot \color{blue}{\left(v \cdot v\right)} + 0.25\right)\right) \cdot \left(1 - v \cdot v\right) \]

   4. associate-*r* 99.6%

      \[\leadsto \left(\sqrt{2} \cdot \left(\color{blue}{\left(-0.375 \cdot v\right) \cdot v} + 0.25\right)\right) \cdot \left(1 - v \cdot v\right) \]

6. Simplified 99.6%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\left(-0.375 \cdot v\right) \cdot v + 0.25\right)\right)} \cdot \left(1 - v \cdot v\right) \]

7. Taylor expanded in v around 0 99.6%

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

   1. unpow2 99.6%

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

   2. associate-*r* 99.6%

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

   3. distribute-rgt-out 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 5: 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. Step-by-step derivation

   1. associate-*r* 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 99.9%

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

   7. sub-neg 99.9%

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

   8. +-commutative 99.9%

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

   9. *-commutative 99.9%

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

   10. distribute-rgt-neg-in 99.9%

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

   11. distribute-lft-neg-in 99.9%

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

   12. metadata-eval 99.9%

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

   13. *-commutative 99.9%

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

   14. fma-udef 99.9%

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

   15. frac-times 99.9%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in v around 0 98.9%

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

7. Final simplification 98.9%

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

Alternative 6: 99.0% 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. associate-*r* 100.0%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 99.9%

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

   7. sub-neg 99.9%

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

   8. +-commutative 99.9%

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

   9. *-commutative 99.9%

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

   10. distribute-rgt-neg-in 99.9%

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

   11. distribute-lft-neg-in 99.9%

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

   12. metadata-eval 99.9%

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

   13. *-commutative 99.9%

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

   14. fma-udef 99.9%

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

   15. frac-times 99.9%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in v around 0 98.9%

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

7. Taylor expanded in v around 0 98.9%

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

8. Final simplification 98.9%

   \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023278 
(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))))