Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-/r/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. neg-sub0 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]

   13. associate-+l- 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]

   14. sub0-neg 100.0%

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

   15. neg-mul-1 100.0%

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

   16. associate-/r* 100.0%

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

   17. metadata-eval 100.0%

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

   18. fma-neg 100.0%

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

   19. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[Exp[N[Log[N[Sqrt[N[(N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision] * 0.125), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $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-exp-log 100.0%

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

   2. add-sqr-sqrt 98.4%

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

   3. sqrt-unprod 100.0%

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

   4. *-commutative 100.0%

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

   5. *-commutative 100.0%

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

   6. swap-sqr 99.9%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

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. 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-/r/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. neg-sub0 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]

   13. associate-+l- 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]

   14. sub0-neg 100.0%

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

   15. neg-mul-1 100.0%

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

   16. associate-/r* 100.0%

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

   17. metadata-eval 100.0%

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

   18. fma-neg 100.0%

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

   19. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125} \cdot \left(1 - v \cdot v\right)\right)} - 1} \]
6. 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%

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

7. Simplified 100.0%

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

8. 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: 99.6% 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 99.7%

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

   1. *-commutative 99.7%

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

   2. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. add-exp-log 100.0%

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

   2. add-sqr-sqrt 98.4%

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

   3. sqrt-unprod 100.0%

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

   4. *-commutative 100.0%

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

   5. *-commutative 100.0%

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

   6. swap-sqr 99.9%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in v around 0 99.0%

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

5. Final simplification 99.0%

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

Alternative 6: 99.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.125} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (sqrt 0.125))

C

double code(double v) {
	return sqrt(0.125);
}

Fortran

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

Java

public static double code(double v) {
	return Math.sqrt(0.125);
}

Python

def code(v):
	return math.sqrt(0.125)

Julia

function code(v)
	return sqrt(0.125)
end

MATLAB

function tmp = code(v)
	tmp = sqrt(0.125);
end

Wolfram

code[v_] := N[Sqrt[0.125], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. add-exp-log 100.0%

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

   2. add-sqr-sqrt 98.4%

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

   3. sqrt-unprod 100.0%

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

   4. *-commutative 100.0%

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

   5. *-commutative 100.0%

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

   6. swap-sqr 99.9%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in v around 0 99.0%

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

   1. add-sqr-sqrt 97.5%

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

   2. pow2 97.5%

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

   3. *-commutative 97.5%

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

6. Applied egg-rr 97.5%

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

7. Taylor expanded in v around 0 99.0%

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

8. Final simplification 99.0%

   \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023207 
(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))))