Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - v \cdot v\\ \sqrt{0.125 \cdot \left(\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot \left(t_0 \cdot t_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (- 1.0 (* v v))))
   (sqrt (* 0.125 (* (fma (* v v) -3.0 1.0) (* t_0 t_0))))))

C

double code(double v) {
	double t_0 = 1.0 - (v * v);
	return sqrt((0.125 * (fma((v * v), -3.0, 1.0) * (t_0 * t_0))));
}

Julia

function code(v)
	t_0 = Float64(1.0 - Float64(v * v))
	return sqrt(Float64(0.125 * Float64(fma(Float64(v * v), -3.0, 1.0) * Float64(t_0 * t_0))))
end

Wolfram

code[v_] := Block[{t$95$0 = N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]}, N[Sqrt[N[(0.125 * N[(N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision] * N[(t$95$0 * t$95$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. unpow2 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 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. expm1-log1p-u 100.0%

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

   2. expm1-udef 98.5%

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

   3. associate-*r* 98.5%

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

   4. sqrt-prod 98.5%

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

   5. *-commutative 98.5%

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

   6. unpow2 98.5%

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

   7. sqrt-prod 98.5%

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

   8. add-sqr-sqrt 98.5%

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

7. Applied egg-rr 98.5%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

9. Simplified 100.0%

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

10. 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 4: 99.6% 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 99.3%

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

   1. unpow2 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 5: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.125 \cdot \left(1 + \left(v \cdot v\right) \cdot -5\right)} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (sqrt (* 0.125 (+ 1.0 (* (* v v) -5.0)))))

C

double code(double v) {
	return sqrt((0.125 * (1.0 + ((v * v) * -5.0))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt((0.125d0 * (1.0d0 + ((v * v) * (-5.0d0)))))
end function

Java

public static double code(double v) {
	return Math.sqrt((0.125 * (1.0 + ((v * v) * -5.0))));
}

Python

def code(v):
	return math.sqrt((0.125 * (1.0 + ((v * v) * -5.0))))

Julia

function code(v)
	return sqrt(Float64(0.125 * Float64(1.0 + Float64(Float64(v * v) * -5.0))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt((0.125 * (1.0 + ((v * v) * -5.0))));
end

Wolfram

code[v_] := N[Sqrt[N[(0.125 * N[(1.0 + N[(N[(v * v), $MachinePrecision] * -5.0), $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. Taylor expanded in v around 0 99.2%

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

   1. *-commutative 99.2%

      \[\leadsto \sqrt{0.125 \cdot \left(1 + \color{blue}{{v}^{2} \cdot -5}\right)} \]

   2. unpow2 99.2%

      \[\leadsto \sqrt{0.125 \cdot \left(1 + \color{blue}{\left(v \cdot v\right)} \cdot -5\right)} \]

8. Simplified 99.2%

   \[\leadsto \sqrt{0.125 \cdot \color{blue}{\left(1 + \left(v \cdot v\right) \cdot -5\right)}} \]

9. Final simplification 99.2%

   \[\leadsto \sqrt{0.125 \cdot \left(1 + \left(v \cdot v\right) \cdot -5\right)} \]

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-*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. Taylor expanded in v around 0 98.5%

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

7. Final simplification 98.5%

   \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023238 
(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))))