Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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. 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. add-sqr-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%

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

5. Applied egg-rr 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(1 - v \cdot v\right)}^{2} \cdot \left(0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}\right)\right)} \]

   2. expm1-udef 98.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{{\left(1 - v \cdot v\right)}^{2} \cdot \left(0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}\right)} - 1} \]

   3. sqrt-prod 98.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{\left(1 - v \cdot v\right)}^{2}} \cdot \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}}\right)} - 1 \]

   4. unpow2 98.4%

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

   5. sqrt-prod 98.4%

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

   6. add-sqr-sqrt 98.4%

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

9. Applied egg-rr 98.4%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\frac{1}{\sqrt{0.125}}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (fma (* v v) -2.5 1.0) (/ 1.0 (sqrt 0.125))))

C

double code(double v) {
	return fma((v * v), -2.5, 1.0) / (1.0 / sqrt(0.125));
}

Julia

function code(v)
	return Float64(fma(Float64(v * v), -2.5, 1.0) / Float64(1.0 / sqrt(0.125)))
end

Wolfram

code[v_] := N[(N[(N[(v * v), $MachinePrecision] * -2.5 + 1.0), $MachinePrecision] / N[(1.0 / N[Sqrt[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-*l* 100.0%

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

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. Taylor expanded in v around 0 99.8%

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

   1. unpow2 99.8%

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

6. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. clear-num 99.8%

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

   3. un-div-inv 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. fma-def 99.8%

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

   7. clear-num 99.8%

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

   8. add-sqr-sqrt 98.2%

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

   9. sqrt-unprod 99.8%

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

   10. frac-times 99.8%

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

   11. add-sqr-sqrt 99.8%

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

   12. metadata-eval 99.8%

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\frac{1}{\sqrt{\frac{2}{\color{blue}{16}}}}} \]

   13. metadata-eval 99.8%

       \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\frac{1}{\sqrt{\color{blue}{0.125}}}} \]

8. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\frac{1}{\sqrt{0.125}}}} \]

9. Final simplification 99.8%

   \[\leadsto \frac{\mathsf{fma}\left(v \cdot v, -2.5, 1\right)}{\frac{1}{\sqrt{0.125}}} \]

Alternative 3: 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%

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

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. Taylor expanded in v around 0 99.8%

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

   1. unpow2 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

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. associate-*l* 100.0%

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

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. 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. add-sqr-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%

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

5. Applied egg-rr 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(1 - v \cdot v\right)}^{2} \cdot \left(0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}\right)\right)} \]

   2. expm1-udef 98.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{{\left(1 - v \cdot v\right)}^{2} \cdot \left(0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}\right)} - 1} \]

   3. sqrt-prod 98.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{\left(1 - v \cdot v\right)}^{2}} \cdot \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}}\right)} - 1 \]

   4. unpow2 98.4%

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

   5. sqrt-prod 98.4%

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

   6. add-sqr-sqrt 98.4%

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

9. Applied egg-rr 98.4%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in v around 0 99.7%

    \[\leadsto \left(1 - v \cdot v\right) \cdot \color{blue}{\left(-1.5 \cdot \left({v}^{2} \cdot \sqrt{0.125}\right) + \sqrt{0.125}\right)} \]
13. Step-by-step derivation

    1. associate-*r* 99.7%

       \[\leadsto \left(1 - v \cdot v\right) \cdot \left(\color{blue}{\left(-1.5 \cdot {v}^{2}\right) \cdot \sqrt{0.125}} + \sqrt{0.125}\right) \]

    2. distribute-lft1-in 99.7%

       \[\leadsto \left(1 - v \cdot v\right) \cdot \color{blue}{\left(\left(-1.5 \cdot {v}^{2} + 1\right) \cdot \sqrt{0.125}\right)} \]

    3. *-commutative 99.7%

       \[\leadsto \left(1 - v \cdot v\right) \cdot \left(\left(\color{blue}{{v}^{2} \cdot -1.5} + 1\right) \cdot \sqrt{0.125}\right) \]

    4. unpow2 99.7%

       \[\leadsto \left(1 - v \cdot v\right) \cdot \left(\left(\color{blue}{\left(v \cdot v\right)} \cdot -1.5 + 1\right) \cdot \sqrt{0.125}\right) \]

14. Simplified 99.7%

    \[\leadsto \left(1 - v \cdot v\right) \cdot \color{blue}{\left(\left(\left(v \cdot v\right) \cdot -1.5 + 1\right) \cdot \sqrt{0.125}\right)} \]

15. Taylor expanded in v around 0 98.9%

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

16. Final simplification 98.9%

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

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. associate-*l* 100.0%

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

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. 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. add-sqr-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%

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

5. Applied egg-rr 100.0%

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

   1. associate-*r* 100.0%

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

   2. *-commutative 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in v around 0 98.8%

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

9. Final simplification 98.8%

   \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023203 
(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))))