Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.5% → 100.0%

Time: 8.1s

Alternatives: 8

Speedup: 25.0×

Specification

\[i > 0\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\\ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t\_0}}{t\_0 - 1} \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* (* 2.0 i) (* 2.0 i))))
   (/ (/ (* (* i i) (* i i)) t_0) (- t_0 1.0))))

C

double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (2.0d0 * i) * (2.0d0 * i)
    code = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0d0)
end function

Java

public static double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Python

def code(i):
	t_0 = (2.0 * i) * (2.0 * i)
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0)

Julia

function code(i)
	t_0 = Float64(Float64(2.0 * i) * Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 - 1.0))
end

MATLAB

function tmp = code(i)
	t_0 = (2.0 * i) * (2.0 * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
end

Wolfram

code[i_] := Block[{t$95$0 = N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 27.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\\ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t\_0}}{t\_0 - 1} \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (let* ((t_0 (* (* 2.0 i) (* 2.0 i))))
   (/ (/ (* (* i i) (* i i)) t_0) (- t_0 1.0))))

C

double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (2.0d0 * i) * (2.0d0 * i)
    code = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0d0)
end function

Java

public static double code(double i) {
	double t_0 = (2.0 * i) * (2.0 * i);
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
}

Python

def code(i):
	t_0 = (2.0 * i) * (2.0 * i)
	return (((i * i) * (i * i)) / t_0) / (t_0 - 1.0)

Julia

function code(i)
	t_0 = Float64(Float64(2.0 * i) * Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 - 1.0))
end

MATLAB

function tmp = code(i)
	t_0 = (2.0 * i) * (2.0 * i);
	tmp = (((i * i) * (i * i)) / t_0) / (t_0 - 1.0);
end

Wolfram

code[i_] := Block[{t$95$0 = N[(N[(2.0 * i), $MachinePrecision] * N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 50000000:\\ \;\;\;\;\frac{\left(i \cdot \frac{i}{4}\right) \cdot \frac{i}{i}}{\left(\mathsf{fma}\left(i, 2, 1\right) + -1\right) \cdot \left(i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 50000000.0)
   (/
    (* (* i (/ i 4.0)) (/ i i))
    (+ (* (+ (fma i 2.0 1.0) -1.0) (* i 2.0)) -1.0))
   0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 50000000.0) {
		tmp = ((i * (i / 4.0)) * (i / i)) / (((fma(i, 2.0, 1.0) + -1.0) * (i * 2.0)) + -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(i)
	tmp = 0.0
	if (i <= 50000000.0)
		tmp = Float64(Float64(Float64(i * Float64(i / 4.0)) * Float64(i / i)) / Float64(Float64(Float64(fma(i, 2.0, 1.0) + -1.0) * Float64(i * 2.0)) + -1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[i_] := If[LessEqual[i, 50000000.0], N[(N[(N[(i * N[(i / 4.0), $MachinePrecision]), $MachinePrecision] * N[(i / i), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(i * 2.0 + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] * N[(i * 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 5e7

   1. Initial program 40.1%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 40.0%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(i \cdot i\right) \cdot i\right) \cdot i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      2. unpow3 39.9%

         \[\leadsto \frac{\frac{\color{blue}{{i}^{3}} \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      3. associate-*r* 39.9%

         \[\leadsto \frac{\frac{{i}^{3} \cdot i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot 2\right) \cdot i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      4. times-frac 87.7%

         \[\leadsto \frac{\color{blue}{\frac{{i}^{3}}{\left(2 \cdot i\right) \cdot 2} \cdot \frac{i}{i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      5. *-commutative 87.7%

         \[\leadsto \frac{\frac{{i}^{3}}{\color{blue}{\left(i \cdot 2\right)} \cdot 2} \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      6. associate-*l* 87.7%

         \[\leadsto \frac{\frac{{i}^{3}}{\color{blue}{i \cdot \left(2 \cdot 2\right)}} \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      7. metadata-eval 87.7%

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

   4. Applied egg-rr 87.7%

      \[\leadsto \frac{\color{blue}{\frac{{i}^{3}}{i \cdot 4} \cdot \frac{i}{i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   5. Step-by-step derivation

      1. cube-mult 87.8%

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

      2. *-commutative 87.8%

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

      3. times-frac 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. sqrt-prod 81.8%

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

      6. sqr-neg 81.8%

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

      7. distribute-rgt-neg-out 81.8%

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

      8. distribute-rgt-neg-out 81.8%

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

      9. sqrt-unprod 41.7%

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

      10. add-sqr-sqrt 43.6%

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

      11. distribute-rgt-neg-out 43.6%

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

      12. neg-sub0 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{\color{blue}{0 - i \cdot i}}{i}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      13. metadata-eval 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{\color{blue}{0 \cdot 0} - i \cdot i}{i}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      14. add-sqr-sqrt 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{i} \cdot \sqrt{i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      15. sqrt-prod 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{i \cdot i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      16. sqr-neg 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\sqrt{\color{blue}{\left(-i\right) \cdot \left(-i\right)}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      17. sqrt-unprod 0.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{-i} \cdot \sqrt{-i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      18. add-sqr-sqrt 100.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{-i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      19. neg-sub0 100.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{0 - i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      20. sub-neg 100.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{0 + \left(-i\right)}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      21. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{-i} \cdot \sqrt{-i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      22. sqrt-unprod 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{\left(-i\right) \cdot \left(-i\right)}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      23. sqr-neg 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \sqrt{\color{blue}{i \cdot i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      24. sqrt-prod 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{i} \cdot \sqrt{i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      25. add-sqr-sqrt 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   6. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \frac{\left(\frac{i}{4} \cdot i\right) \cdot \frac{i}{i}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot i\right)\right)} \cdot \left(2 \cdot i\right) - 1} \]

      2. expm1-undefine 99.9%

         \[\leadsto \frac{\left(\frac{i}{4} \cdot i\right) \cdot \frac{i}{i}}{\color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot i\right)} - 1\right)} \cdot \left(2 \cdot i\right) - 1} \]

      3. log1p-undefine 99.9%

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

      4. +-commutative 99.9%

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

      5. add-exp-log 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-define 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 5e7 < i

   1. Initial program 25.6%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   3. Simplified 47.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 100.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 50000000:\\ \;\;\;\;\frac{\left(i \cdot \frac{i}{4}\right) \cdot \frac{i}{i}}{\left(\mathsf{fma}\left(i, 2, 1\right) + -1\right) \cdot \left(i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(i \cdot \left(\frac{1}{\mathsf{fma}\left(i, 2, 1\right)} \cdot \frac{i}{\mathsf{fma}\left(i, 2, -1\right)}\right)\right) \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (* 0.25 (* i (* (/ 1.0 (fma i 2.0 1.0)) (/ i (fma i 2.0 -1.0))))))

C

double code(double i) {
	return 0.25 * (i * ((1.0 / fma(i, 2.0, 1.0)) * (i / fma(i, 2.0, -1.0))));
}

Julia

function code(i)
	return Float64(0.25 * Float64(i * Float64(Float64(1.0 / fma(i, 2.0, 1.0)) * Float64(i / fma(i, 2.0, -1.0)))))
end

Wolfram

code[i_] := N[(0.25 * N[(i * N[(N[(1.0 / N[(i * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.4%

   \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

3. Simplified 71.8%

   \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 71.8%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{\color{blue}{1 \cdot i}}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right) \]

   2. fma-undefine 71.8%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{1 \cdot i}{\color{blue}{i \cdot \left(i \cdot 4\right) + -1}}\right) \]

   3. associate-*r* 71.8%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{1 \cdot i}{\color{blue}{\left(i \cdot i\right) \cdot 4} + -1}\right) \]

   4. *-commutative 71.8%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{1 \cdot i}{\color{blue}{4 \cdot \left(i \cdot i\right)} + -1}\right) \]

   5. metadata-eval 71.8%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{1 \cdot i}{\color{blue}{\left(2 \cdot 2\right)} \cdot \left(i \cdot i\right) + -1}\right) \]

   6. swap-sqr 71.8%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{1 \cdot i}{\color{blue}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)} + -1}\right) \]

   7. difference-of-sqr--1 71.8%

      \[\leadsto 0.25 \cdot \left(i \cdot \frac{1 \cdot i}{\color{blue}{\left(2 \cdot i + 1\right) \cdot \left(2 \cdot i - 1\right)}}\right) \]

   8. times-frac 99.8%

      \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(\frac{1}{2 \cdot i + 1} \cdot \frac{i}{2 \cdot i - 1}\right)}\right) \]

   9. *-commutative 99.8%

      \[\leadsto 0.25 \cdot \left(i \cdot \left(\frac{1}{\color{blue}{i \cdot 2} + 1} \cdot \frac{i}{2 \cdot i - 1}\right)\right) \]

   10. fma-define 99.8%

       \[\leadsto 0.25 \cdot \left(i \cdot \left(\frac{1}{\color{blue}{\mathsf{fma}\left(i, 2, 1\right)}} \cdot \frac{i}{2 \cdot i - 1}\right)\right) \]

   11. *-commutative 99.8%

       \[\leadsto 0.25 \cdot \left(i \cdot \left(\frac{1}{\mathsf{fma}\left(i, 2, 1\right)} \cdot \frac{i}{\color{blue}{i \cdot 2} - 1}\right)\right) \]

   12. fma-neg 99.8%

       \[\leadsto 0.25 \cdot \left(i \cdot \left(\frac{1}{\mathsf{fma}\left(i, 2, 1\right)} \cdot \frac{i}{\color{blue}{\mathsf{fma}\left(i, 2, -1\right)}}\right)\right) \]

   13. metadata-eval 99.8%

       \[\leadsto 0.25 \cdot \left(i \cdot \left(\frac{1}{\mathsf{fma}\left(i, 2, 1\right)} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \color{blue}{-1}\right)}\right)\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(\frac{1}{\mathsf{fma}\left(i, 2, 1\right)} \cdot \frac{i}{\mathsf{fma}\left(i, 2, -1\right)}\right)}\right) \]
7. Add Preprocessing

Alternative 3: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{i \cdot \left(i \cdot \frac{0.25}{\mathsf{fma}\left(i, 2, -1\right)}\right)}{\mathsf{fma}\left(i, 2, 1\right)} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (/ (* i (* i (/ 0.25 (fma i 2.0 -1.0)))) (fma i 2.0 1.0)))

C

double code(double i) {
	return (i * (i * (0.25 / fma(i, 2.0, -1.0)))) / fma(i, 2.0, 1.0);
}

Julia

function code(i)
	return Float64(Float64(i * Float64(i * Float64(0.25 / fma(i, 2.0, -1.0)))) / fma(i, 2.0, 1.0))
end

Wolfram

code[i_] := N[(N[(i * N[(i * N[(0.25 / N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.4%

   \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* 32.3%

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(i \cdot i\right) \cdot i\right) \cdot i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   2. unpow3 32.2%

      \[\leadsto \frac{\frac{\color{blue}{{i}^{3}} \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   3. associate-*r* 32.2%

      \[\leadsto \frac{\frac{{i}^{3} \cdot i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot 2\right) \cdot i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   4. times-frac 56.9%

      \[\leadsto \frac{\color{blue}{\frac{{i}^{3}}{\left(2 \cdot i\right) \cdot 2} \cdot \frac{i}{i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   5. *-commutative 56.9%

      \[\leadsto \frac{\frac{{i}^{3}}{\color{blue}{\left(i \cdot 2\right)} \cdot 2} \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   6. associate-*l* 56.9%

      \[\leadsto \frac{\frac{{i}^{3}}{\color{blue}{i \cdot \left(2 \cdot 2\right)}} \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   7. metadata-eval 56.9%

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

4. Applied egg-rr 56.9%

   \[\leadsto \frac{\color{blue}{\frac{{i}^{3}}{i \cdot 4} \cdot \frac{i}{i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
5. Step-by-step derivation

   1. cube-mult 57.0%

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

   2. *-commutative 57.0%

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

   3. times-frac 71.1%

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

   4. add-sqr-sqrt 71.1%

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

   5. sqrt-prod 52.0%

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

   6. sqr-neg 52.0%

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

   7. distribute-rgt-neg-out 52.0%

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

   8. distribute-rgt-neg-out 52.0%

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

   9. sqrt-unprod 19.5%

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

   10. add-sqr-sqrt 20.8%

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

   11. distribute-rgt-neg-out 20.8%

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

   12. neg-sub0 20.8%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{\color{blue}{0 - i \cdot i}}{i}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   13. metadata-eval 20.8%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{\color{blue}{0 \cdot 0} - i \cdot i}{i}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   14. add-sqr-sqrt 20.8%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{i} \cdot \sqrt{i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   15. sqrt-prod 1.3%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{i \cdot i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   16. sqr-neg 1.3%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\sqrt{\color{blue}{\left(-i\right) \cdot \left(-i\right)}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   17. sqrt-unprod 0.0%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{-i} \cdot \sqrt{-i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   18. add-sqr-sqrt 71.1%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{-i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   19. neg-sub0 71.1%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{0 - i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   20. sub-neg 71.1%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{0 + \left(-i\right)}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   21. add-sqr-sqrt 0.0%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{-i} \cdot \sqrt{-i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   22. sqrt-unprod 1.3%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{\left(-i\right) \cdot \left(-i\right)}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   23. sqr-neg 1.3%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \sqrt{\color{blue}{i \cdot i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   24. sqrt-prod 20.8%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{i} \cdot \sqrt{i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   25. add-sqr-sqrt 20.8%

       \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

6. Applied egg-rr 71.1%

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

   1. expm1-log1p-u 69.2%

      \[\leadsto \frac{\left(\frac{i}{4} \cdot i\right) \cdot \frac{i}{i}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot i\right)\right)} \cdot \left(2 \cdot i\right) - 1} \]

   2. expm1-undefine 69.2%

      \[\leadsto \frac{\left(\frac{i}{4} \cdot i\right) \cdot \frac{i}{i}}{\color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot i\right)} - 1\right)} \cdot \left(2 \cdot i\right) - 1} \]

   3. log1p-undefine 69.2%

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

   4. +-commutative 69.2%

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

   5. add-exp-log 71.1%

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

   6. *-commutative 71.1%

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

   7. fma-define 71.1%

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

8. Applied egg-rr 71.1%

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

   1. *-inverses 71.1%

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

   2. *-un-lft-identity 71.1%

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

   3. *-rgt-identity 71.1%

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

   4. *-un-lft-identity 71.1%

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

   5. *-commutative 71.1%

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

   6. fma-neg 71.1%

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

   7. add-exp-log 69.2%

      \[\leadsto \frac{i \cdot \frac{i}{4}}{\mathsf{fma}\left(\color{blue}{e^{\log \left(\mathsf{fma}\left(i, 2, 1\right)\right)}} - 1, 2 \cdot i, -1\right)} \]

   8. fma-undefine 69.2%

      \[\leadsto \frac{i \cdot \frac{i}{4}}{\mathsf{fma}\left(e^{\log \color{blue}{\left(i \cdot 2 + 1\right)}} - 1, 2 \cdot i, -1\right)} \]

   9. *-commutative 69.2%

      \[\leadsto \frac{i \cdot \frac{i}{4}}{\mathsf{fma}\left(e^{\log \left(\color{blue}{2 \cdot i} + 1\right)} - 1, 2 \cdot i, -1\right)} \]

   10. +-commutative 69.2%

       \[\leadsto \frac{i \cdot \frac{i}{4}}{\mathsf{fma}\left(e^{\log \color{blue}{\left(1 + 2 \cdot i\right)}} - 1, 2 \cdot i, -1\right)} \]

   11. log1p-undefine 69.2%

       \[\leadsto \frac{i \cdot \frac{i}{4}}{\mathsf{fma}\left(e^{\color{blue}{\mathsf{log1p}\left(2 \cdot i\right)}} - 1, 2 \cdot i, -1\right)} \]

   12. expm1-undefine 69.2%

       \[\leadsto \frac{i \cdot \frac{i}{4}}{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot i\right)\right)}, 2 \cdot i, -1\right)} \]

   13. expm1-log1p-u 71.1%

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

   14. fma-neg 71.1%

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

   15. difference-of-sqr-1 71.1%

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

   16. *-commutative 71.1%

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

   17. fma-undefine 71.1%

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

10. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(i, 2, 1\right)} \cdot \frac{i \cdot 0.25}{\mathsf{fma}\left(i, 2, -1\right)}} \]
11. Step-by-step derivation

    1. associate-*l/ 100.0%

       \[\leadsto \color{blue}{\frac{i \cdot \frac{i \cdot 0.25}{\mathsf{fma}\left(i, 2, -1\right)}}{\mathsf{fma}\left(i, 2, 1\right)}} \]

    2. associate-/l* 99.8%

       \[\leadsto \frac{i \cdot \color{blue}{\left(i \cdot \frac{0.25}{\mathsf{fma}\left(i, 2, -1\right)}\right)}}{\mathsf{fma}\left(i, 2, 1\right)} \]

12. Simplified 99.8%

    \[\leadsto \color{blue}{\frac{i \cdot \left(i \cdot \frac{0.25}{\mathsf{fma}\left(i, 2, -1\right)}\right)}{\mathsf{fma}\left(i, 2, 1\right)}} \]
13. Add Preprocessing

Alternative 4: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 50000000:\\ \;\;\;\;\frac{\frac{i \cdot \left(-i\right)}{-4}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 50000000.0)
   (/ (/ (* i (- i)) -4.0) (+ (* (* i 2.0) (* i 2.0)) -1.0))
   0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 50000000.0) {
		tmp = ((i * -i) / -4.0) / (((i * 2.0) * (i * 2.0)) + -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 50000000.0d0) then
        tmp = ((i * -i) / (-4.0d0)) / (((i * 2.0d0) * (i * 2.0d0)) + (-1.0d0))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 50000000.0) {
		tmp = ((i * -i) / -4.0) / (((i * 2.0) * (i * 2.0)) + -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 50000000.0:
		tmp = ((i * -i) / -4.0) / (((i * 2.0) * (i * 2.0)) + -1.0)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 50000000.0)
		tmp = Float64(Float64(Float64(i * Float64(-i)) / -4.0) / Float64(Float64(Float64(i * 2.0) * Float64(i * 2.0)) + -1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 50000000.0)
		tmp = ((i * -i) / -4.0) / (((i * 2.0) * (i * 2.0)) + -1.0);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 50000000.0], N[(N[(N[(i * (-i)), $MachinePrecision] / -4.0), $MachinePrecision] / N[(N[(N[(i * 2.0), $MachinePrecision] * N[(i * 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 5e7

   1. Initial program 40.1%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 40.0%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(i \cdot i\right) \cdot i\right) \cdot i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      2. unpow3 39.9%

         \[\leadsto \frac{\frac{\color{blue}{{i}^{3}} \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      3. associate-*r* 39.9%

         \[\leadsto \frac{\frac{{i}^{3} \cdot i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot 2\right) \cdot i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      4. times-frac 87.7%

         \[\leadsto \frac{\color{blue}{\frac{{i}^{3}}{\left(2 \cdot i\right) \cdot 2} \cdot \frac{i}{i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      5. *-commutative 87.7%

         \[\leadsto \frac{\frac{{i}^{3}}{\color{blue}{\left(i \cdot 2\right)} \cdot 2} \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      6. associate-*l* 87.7%

         \[\leadsto \frac{\frac{{i}^{3}}{\color{blue}{i \cdot \left(2 \cdot 2\right)}} \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      7. metadata-eval 87.7%

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

   4. Applied egg-rr 87.7%

      \[\leadsto \frac{\color{blue}{\frac{{i}^{3}}{i \cdot 4} \cdot \frac{i}{i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   5. Step-by-step derivation

      1. cube-mult 87.8%

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

      2. *-commutative 87.8%

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

      3. times-frac 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. sqrt-prod 81.8%

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

      6. sqr-neg 81.8%

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

      7. distribute-rgt-neg-out 81.8%

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

      8. distribute-rgt-neg-out 81.8%

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

      9. sqrt-unprod 41.7%

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

      10. add-sqr-sqrt 43.6%

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

      11. distribute-rgt-neg-out 43.6%

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

      12. neg-sub0 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{\color{blue}{0 - i \cdot i}}{i}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      13. metadata-eval 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{\color{blue}{0 \cdot 0} - i \cdot i}{i}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      14. add-sqr-sqrt 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{i} \cdot \sqrt{i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      15. sqrt-prod 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{i \cdot i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      16. sqr-neg 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\sqrt{\color{blue}{\left(-i\right) \cdot \left(-i\right)}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      17. sqrt-unprod 0.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{-i} \cdot \sqrt{-i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      18. add-sqr-sqrt 100.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{-i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      19. neg-sub0 100.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{0 - i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      20. sub-neg 100.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{0 + \left(-i\right)}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      21. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{-i} \cdot \sqrt{-i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      22. sqrt-unprod 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{\left(-i\right) \cdot \left(-i\right)}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      23. sqr-neg 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \sqrt{\color{blue}{i \cdot i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      24. sqrt-prod 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{i} \cdot \sqrt{i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      25. add-sqr-sqrt 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in i around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. frac-2neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 5e7 < i

   1. Initial program 25.6%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   3. Simplified 47.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 100.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 50000000:\\ \;\;\;\;\frac{\frac{i \cdot \left(-i\right)}{-4}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 50000000:\\ \;\;\;\;\frac{\frac{i}{\frac{4}{i}}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 50000000.0)
   (/ (/ i (/ 4.0 i)) (+ (* (* i 2.0) (* i 2.0)) -1.0))
   0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 50000000.0) {
		tmp = (i / (4.0 / i)) / (((i * 2.0) * (i * 2.0)) + -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 50000000.0d0) then
        tmp = (i / (4.0d0 / i)) / (((i * 2.0d0) * (i * 2.0d0)) + (-1.0d0))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 50000000.0) {
		tmp = (i / (4.0 / i)) / (((i * 2.0) * (i * 2.0)) + -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 50000000.0:
		tmp = (i / (4.0 / i)) / (((i * 2.0) * (i * 2.0)) + -1.0)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 50000000.0)
		tmp = Float64(Float64(i / Float64(4.0 / i)) / Float64(Float64(Float64(i * 2.0) * Float64(i * 2.0)) + -1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 50000000.0)
		tmp = (i / (4.0 / i)) / (((i * 2.0) * (i * 2.0)) + -1.0);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 50000000.0], N[(N[(i / N[(4.0 / i), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i * 2.0), $MachinePrecision] * N[(i * 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 5e7

   1. Initial program 40.1%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 40.0%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(i \cdot i\right) \cdot i\right) \cdot i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      2. unpow3 39.9%

         \[\leadsto \frac{\frac{\color{blue}{{i}^{3}} \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      3. associate-*r* 39.9%

         \[\leadsto \frac{\frac{{i}^{3} \cdot i}{\color{blue}{\left(\left(2 \cdot i\right) \cdot 2\right) \cdot i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      4. times-frac 87.7%

         \[\leadsto \frac{\color{blue}{\frac{{i}^{3}}{\left(2 \cdot i\right) \cdot 2} \cdot \frac{i}{i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      5. *-commutative 87.7%

         \[\leadsto \frac{\frac{{i}^{3}}{\color{blue}{\left(i \cdot 2\right)} \cdot 2} \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      6. associate-*l* 87.7%

         \[\leadsto \frac{\frac{{i}^{3}}{\color{blue}{i \cdot \left(2 \cdot 2\right)}} \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      7. metadata-eval 87.7%

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

   4. Applied egg-rr 87.7%

      \[\leadsto \frac{\color{blue}{\frac{{i}^{3}}{i \cdot 4} \cdot \frac{i}{i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   5. Step-by-step derivation

      1. cube-mult 87.8%

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

      2. *-commutative 87.8%

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

      3. times-frac 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. sqrt-prod 81.8%

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

      6. sqr-neg 81.8%

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

      7. distribute-rgt-neg-out 81.8%

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

      8. distribute-rgt-neg-out 81.8%

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

      9. sqrt-unprod 41.7%

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

      10. add-sqr-sqrt 43.6%

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

      11. distribute-rgt-neg-out 43.6%

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

      12. neg-sub0 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{\color{blue}{0 - i \cdot i}}{i}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      13. metadata-eval 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{\color{blue}{0 \cdot 0} - i \cdot i}{i}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      14. add-sqr-sqrt 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{i} \cdot \sqrt{i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      15. sqrt-prod 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{i \cdot i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      16. sqr-neg 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\sqrt{\color{blue}{\left(-i\right) \cdot \left(-i\right)}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      17. sqrt-unprod 0.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{\sqrt{-i} \cdot \sqrt{-i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      18. add-sqr-sqrt 100.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{-i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      19. neg-sub0 100.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{0 - i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      20. sub-neg 100.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{\color{blue}{0 + \left(-i\right)}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      21. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{-i} \cdot \sqrt{-i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      22. sqrt-unprod 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{\left(-i\right) \cdot \left(-i\right)}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      23. sqr-neg 2.0%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \sqrt{\color{blue}{i \cdot i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      24. sqrt-prod 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{\sqrt{i} \cdot \sqrt{i}}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

      25. add-sqr-sqrt 43.6%

          \[\leadsto \frac{\left(\frac{i}{4} \cdot \frac{0 \cdot 0 - i \cdot i}{0 + \color{blue}{i}}\right) \cdot \frac{i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in i around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. associate-/r/ 99.7%

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

   9. Applied egg-rr 99.7%

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

   if 5e7 < i

   1. Initial program 25.6%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   3. Simplified 47.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 100.0%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 50000000:\\ \;\;\;\;\frac{\frac{i}{\frac{4}{i}}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;0.25 \cdot \left(i \cdot \left(-i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (if (<= i 0.5) (* 0.25 (* i (- i))) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = 0.25 * (i * -i);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 0.5d0) then
        tmp = 0.25d0 * (i * -i)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = 0.25 * (i * -i);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = 0.25 * (i * -i)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(0.25 * Float64(i * Float64(-i)));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = 0.25 * (i * -i);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(0.25 * N[(i * (-i)), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 36.4%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 98.5%

      \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 98.5%

         \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(-i\right)}\right) \]

   7. Simplified 98.5%

      \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(-i\right)}\right) \]

   if 0.5 < i

   1. Initial program 29.2%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 98.1%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.8:\\ \;\;\;\;0.25 \cdot \left(i \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (if (<= i 0.8) (* 0.25 (* i i)) 0.0625))

C

double code(double i) {
	double tmp;
	if (i <= 0.8) {
		tmp = 0.25 * (i * i);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 0.8d0) then
        tmp = 0.25d0 * (i * i)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 0.8) {
		tmp = 0.25 * (i * i);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.8:
		tmp = 0.25 * (i * i)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.8)
		tmp = Float64(0.25 * Float64(i * i));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.8)
		tmp = 0.25 * (i * i);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.8], N[(0.25 * N[(i * i), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 0.80000000000000004

   1. Initial program 36.4%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around 0 98.5%

      \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 98.5%

         \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(-i\right)}\right) \]

   7. Simplified 98.5%

      \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(-i\right)}\right) \]
   8. Step-by-step derivation

      1. neg-sub0 98.5%

         \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(0 - i\right)}\right) \]

      2. sub-neg 98.5%

         \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(0 + \left(-i\right)\right)}\right) \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto 0.25 \cdot \left(i \cdot \left(0 + \color{blue}{\sqrt{-i} \cdot \sqrt{-i}}\right)\right) \]

      4. sqrt-unprod 46.2%

         \[\leadsto 0.25 \cdot \left(i \cdot \left(0 + \color{blue}{\sqrt{\left(-i\right) \cdot \left(-i\right)}}\right)\right) \]

      5. sqr-neg 46.2%

         \[\leadsto 0.25 \cdot \left(i \cdot \left(0 + \sqrt{\color{blue}{i \cdot i}}\right)\right) \]

      6. sqrt-prod 46.2%

         \[\leadsto 0.25 \cdot \left(i \cdot \left(0 + \color{blue}{\sqrt{i} \cdot \sqrt{i}}\right)\right) \]

      7. add-sqr-sqrt 46.2%

         \[\leadsto 0.25 \cdot \left(i \cdot \left(0 + \color{blue}{i}\right)\right) \]

   9. Applied egg-rr 46.2%

      \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{\left(0 + i\right)}\right) \]
   10. Step-by-step derivation

       1. +-lft-identity 46.2%

          \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{i}\right) \]

   11. Simplified 46.2%

       \[\leadsto 0.25 \cdot \left(i \cdot \color{blue}{i}\right) \]

   if 0.80000000000000004 < i

   1. Initial program 29.2%

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf 98.1%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.2% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (i) :precision binary64 0.0625)

C

double code(double i) {
	return 0.0625;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

public static double code(double i) {
	return 0.0625;
}

Python

def code(i):
	return 0.0625

Julia

function code(i)
	return 0.0625
end

MATLAB

function tmp = code(i)
	tmp = 0.0625;
end

Wolfram

code[i_] := 0.0625

Derivation

1. Initial program 32.4%

   \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

3. Simplified 71.8%

   \[\leadsto \color{blue}{0.25 \cdot \left(i \cdot \frac{i}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} \]
4. Add Preprocessing

5. Taylor expanded in i around inf 56.0%

   \[\leadsto \color{blue}{0.0625} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024129 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (> i 0.0)
  (/ (/ (* (* i i) (* i i)) (* (* 2.0 i) (* 2.0 i))) (- (* (* 2.0 i) (* 2.0 i)) 1.0)))