Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. associate-/l* 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. frac-2neg 100.0%

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

   4. associate-*r* 100.0%

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

   5. sqrt-unprod 100.0%

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

   6. sub-neg 100.0%

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

   7. +-commutative 100.0%

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

   8. *-commutative 100.0%

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

   9. distribute-rgt-neg-in 100.0%

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

   10. fma-def 100.0%

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

   11. metadata-eval 100.0%

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

   12. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

   1. distribute-rgt-neg-in 100.0%

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

   2. associate-/l* 100.0%

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

   3. fma-udef 100.0%

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

   4. associate-*l* 100.0%

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

   5. fma-def 100.0%

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

   6. sub-neg 100.0%

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

   7. distribute-rgt-neg-out 100.0%

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

   8. distribute-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

   10. distribute-rgt-neg-out 100.0%

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

   11. remove-double-neg 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

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(Float64(v * v), -3.0, 1.0) * 0.125)))
end

Wolfram

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

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. fma-def 100.0%

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

   12. metadata-eval 100.0%

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

   13. frac-times 100.0%

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

   14. add-sqr-sqrt 100.0%

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

   15. metadata-eval 100.0%

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

   16. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

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%

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

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 5: 99.5% accurate, 2.0× speedup

Math

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

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(sqrt(2.0) / Float64(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[Sqrt[2.0], $MachinePrecision] / N[(4.0 / N[(1.0 + N[(N[(v * v), $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. associate-*r* 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. associate-*l/ 100.0%

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

   3. associate-/l* 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in v around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. unpow2 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 6: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l/ 100.0%

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

   3. sqrt-unprod 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/l* 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in v around 0 99.2%

   \[\leadsto \frac{1 - v \cdot v}{\color{blue}{\frac{4}{\sqrt{2}}}} \]

7. Final simplification 99.2%

   \[\leadsto \frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}} \]

Alternative 7: 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. add-sqr-sqrt 98.4%

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

   2. sqrt-unprod 100.0%

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

   3. *-commutative 100.0%

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

   4. *-commutative 100.0%

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

   5. swap-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. fma-def 100.0%

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

   12. metadata-eval 100.0%

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

   13. frac-times 100.0%

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

   14. add-sqr-sqrt 100.0%

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

   15. metadata-eval 100.0%

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

   16. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in v around 0 99.2%

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

5. Final simplification 99.2%

   \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023229 
(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))))