Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(0.125 \cdot \mathsf{fma}\left(-3, {v}^{2}, 1\right)\right) \cdot {\left(1 - {v}^{2}\right)}^{2}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (sqrt (* (* 0.125 (fma -3.0 (pow v 2.0) 1.0)) (pow (- 1.0 (pow v 2.0)) 2.0))))

C

double code(double v) {
	return sqrt(((0.125 * fma(-3.0, pow(v, 2.0), 1.0)) * pow((1.0 - pow(v, 2.0)), 2.0)));
}

Julia

function code(v)
	return sqrt(Float64(Float64(0.125 * fma(-3.0, (v ^ 2.0), 1.0)) * (Float64(1.0 - (v ^ 2.0)) ^ 2.0)))
end

Wolfram

code[v_] := N[Sqrt[N[(N[(0.125 * N[(-3.0 * N[Power[v, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 - N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. add-sqr-sqrt 98.4%

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

   2. sqrt-unprod 100.0%

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

   3. swap-sqr 100.0%

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

   4. frac-times 100.0%

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

   5. rem-square-sqrt 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

   8. swap-sqr 100.0%

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

   9. add-sqr-sqrt 100.0%

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

   10. +-commutative 100.0%

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

   11. fma-define 100.0%

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

   12. pow2 100.0%

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

   13. pow2 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-*r* 100.0%

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

8. Simplified 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

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(sqrt(2.0) / 4.0) * Float64(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[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 + N[(-3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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. 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. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * ((sqrt(2.0) / 4.0) * sqrt((1.0 - ((v * v) * 3.0))));
end

Wolfram

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

3. Final simplification 100.0%

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

Alternative 4: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2}}{4} \cdot \left(1 + \left(v \cdot v\right) \cdot -2.5\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (/ (sqrt 2.0) 4.0) (+ 1.0 (* (* v v) -2.5))))

C

double code(double v) {
	return (sqrt(2.0) / 4.0) * (1.0 + ((v * v) * -2.5));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (sqrt(2.0d0) / 4.0d0) * (1.0d0 + ((v * v) * (-2.5d0)))
end function

Java

public static double code(double v) {
	return (Math.sqrt(2.0) / 4.0) * (1.0 + ((v * v) * -2.5));
}

Python

def code(v):
	return (math.sqrt(2.0) / 4.0) * (1.0 + ((v * v) * -2.5))

Julia

function code(v)
	return Float64(Float64(sqrt(2.0) / 4.0) * Float64(1.0 + Float64(Float64(v * v) * -2.5)))
end

MATLAB

function tmp = code(v)
	tmp = (sqrt(2.0) / 4.0) * (1.0 + ((v * v) * -2.5));
end

Wolfram

code[v_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[(1.0 + N[(N[(v * v), $MachinePrecision] * -2.5), $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 99.6%

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

   1. *-commutative 99.6%

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

7. Simplified 99.6%

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

   1. unpow2 99.6%

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

9. Applied egg-rr 99.6%

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

Alternative 5: 99.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.125 + \left(v \cdot v\right) \cdot -0.625} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (sqrt (+ 0.125 (* (* v v) -0.625))))

C

double code(double v) {
	return sqrt((0.125 + ((v * v) * -0.625)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt((0.125d0 + ((v * v) * (-0.625d0))))
end function

Java

public static double code(double v) {
	return Math.sqrt((0.125 + ((v * v) * -0.625)));
}

Python

def code(v):
	return math.sqrt((0.125 + ((v * v) * -0.625)))

Julia

function code(v)
	return sqrt(Float64(0.125 + Float64(Float64(v * v) * -0.625)))
end

MATLAB

function tmp = code(v)
	tmp = sqrt((0.125 + ((v * v) * -0.625)));
end

Wolfram

code[v_] := N[Sqrt[N[(0.125 + 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-*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. Step-by-step derivation

   1. add-sqr-sqrt 98.4%

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

   2. sqrt-unprod 100.0%

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

   3. swap-sqr 100.0%

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

   4. frac-times 100.0%

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

   5. rem-square-sqrt 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

   8. swap-sqr 100.0%

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

   9. add-sqr-sqrt 100.0%

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

   10. +-commutative 100.0%

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

   11. fma-define 100.0%

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

   12. pow2 100.0%

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

   13. pow2 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-*r* 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in v around 0 99.5%

   \[\leadsto \sqrt{\color{blue}{0.125 + -0.625 \cdot {v}^{2}}} \]
10. Step-by-step derivation

    1. *-commutative 99.5%

       \[\leadsto \sqrt{0.125 + \color{blue}{{v}^{2} \cdot -0.625}} \]

11. Simplified 99.5%

    \[\leadsto \sqrt{\color{blue}{0.125 + {v}^{2} \cdot -0.625}} \]
12. Step-by-step derivation

    1. unpow2 99.6%

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

13. Applied egg-rr 99.5%

    \[\leadsto \sqrt{0.125 + \color{blue}{\left(v \cdot v\right)} \cdot -0.625} \]
14. Add Preprocessing

Alternative 6: 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. 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. Step-by-step derivation

   1. add-sqr-sqrt 98.4%

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

   2. sqrt-unprod 100.0%

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

   3. swap-sqr 100.0%

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

   4. frac-times 100.0%

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

   5. rem-square-sqrt 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

   8. swap-sqr 100.0%

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

   9. add-sqr-sqrt 100.0%

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

   10. +-commutative 100.0%

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

   11. fma-define 100.0%

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

   12. pow2 100.0%

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

   13. pow2 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-*r* 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in v around 0 99.3%

   \[\leadsto \color{blue}{\sqrt{0.125}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(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))))