Octave 3.8, jcobi/4, as called

Percentage Accurate: 27.7% → 99.5%

Time: 4.7s

Alternatives: 5

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: 27.7% 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: 99.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\left(i \cdot i\right) \cdot \left(-0.25 - i \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5) (* (* i i) (- -0.25 (* i i))) (+ 0.0625 (/ 0.015625 (* i i)))))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = (i * i) * (-0.25 - (i * i));
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 0.5d0) then
        tmp = (i * i) * ((-0.25d0) - (i * i))
    else
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = (i * i) * (-0.25 - (i * i));
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = (i * i) * (-0.25 - (i * i))
	else:
		tmp = 0.0625 + (0.015625 / (i * i))
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(Float64(i * i) * Float64(-0.25 - Float64(i * i)));
	else
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = (i * i) * (-0.25 - (i * i));
	else
		tmp = 0.0625 + (0.015625 / (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(N[(i * i), $MachinePrecision] * N[(-0.25 - N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 28.6%

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

      1. associate-/l* 56.8%

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

      2. swap-sqr 56.8%

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

      3. metadata-eval 56.8%

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

      4. sub-neg 56.8%

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

      5. swap-sqr 56.8%

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

      6. metadata-eval 56.8%

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

      7. metadata-eval 56.8%

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

   3. Simplified 56.8%

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

      1. frac-2neg 56.8%

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

      2. div-inv 56.8%

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

      3. div-inv 56.8%

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

      4. clear-num 56.8%

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

      5. distribute-rgt-neg-in 56.8%

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

      6. *-commutative 56.8%

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

      7. associate-/r* 56.8%

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

      8. *-inverses 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-commutative 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in i around 0 99.7%

      \[\leadsto \left(\left(i \cdot i\right) \cdot -0.25\right) \cdot \color{blue}{\left(1 + 4 \cdot {i}^{2}\right)} \]
   7. Step-by-step derivation

      1. unpow2 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in i around 0 99.7%

      \[\leadsto \color{blue}{-1 \cdot {i}^{4} + -0.25 \cdot {i}^{2}} \]
   10. Step-by-step derivation

       1. +-commutative 99.7%

          \[\leadsto \color{blue}{-0.25 \cdot {i}^{2} + -1 \cdot {i}^{4}} \]

       2. unpow2 99.7%

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

       3. fma-def 99.7%

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

       4. mul-1-neg 99.7%

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

       5. fma-neg 99.7%

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

       6. metadata-eval 99.7%

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

       7. pow-sqr 99.7%

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

       8. unpow2 99.7%

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

       9. unpow2 99.7%

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

       10. distribute-rgt-out-- 99.7%

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

   11. Simplified 99.7%

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

   if 0.5 < i

   1. Initial program 20.1%

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

      1. times-frac 45.9%

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

      2. associate-/l* 45.9%

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

      3. associate-/l* 46.0%

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

      4. associate-/l/ 46.0%

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

      5. associate-/r/ 46.0%

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

      6. associate-/l* 46.0%

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

      7. *-inverses 46.0%

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

      8. metadata-eval 46.0%

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

      9. associate-*l* 46.0%

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

      10. *-commutative 46.0%

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

      11. associate-/r* 46.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{i}{{i}^{3}}}} \]

   4. Taylor expanded in i around inf 99.5%

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 99.5%

         \[\leadsto 0.0625 + 0.015625 \cdot \frac{1}{\color{blue}{i \cdot i}} \]

      2. associate-*r/ 99.5%

         \[\leadsto 0.0625 + \color{blue}{\frac{0.015625 \cdot 1}{i \cdot i}} \]

      3. metadata-eval 99.5%

         \[\leadsto 0.0625 + \frac{\color{blue}{0.015625}}{i \cdot i} \]

   6. Simplified 99.5%

      \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;\left(i \cdot i\right) \cdot \left(-0.25 - i \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \]

Alternative 2: 99.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i)
 :precision binary64
 (if (<= i 0.5) (* i (* i -0.25)) (+ 0.0625 (/ 0.015625 (* i i)))))

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (i * -0.25);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 0.5d0) then
        tmp = i * (i * (-0.25d0))
    else
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    end if
    code = tmp
end function

Java

public static double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (i * -0.25);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Python

def code(i):
	tmp = 0
	if i <= 0.5:
		tmp = i * (i * -0.25)
	else:
		tmp = 0.0625 + (0.015625 / (i * i))
	return tmp

Julia

function code(i)
	tmp = 0.0
	if (i <= 0.5)
		tmp = Float64(i * Float64(i * -0.25));
	else
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i)
	tmp = 0.0;
	if (i <= 0.5)
		tmp = i * (i * -0.25);
	else
		tmp = 0.0625 + (0.015625 / (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_] := If[LessEqual[i, 0.5], N[(i * N[(i * -0.25), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 28.6%

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

      1. times-frac 99.9%

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

      2. associate-/l* 99.7%

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

      3. associate-/l* 99.8%

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

      4. associate-/l/ 99.8%

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

      5. associate-/r/ 98.6%

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

      6. associate-/l* 98.6%

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

      7. *-inverses 98.6%

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

      8. metadata-eval 98.6%

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

      9. associate-*l* 98.6%

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

      10. *-commutative 98.6%

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

      11. associate-/r* 98.8%

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

   3. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{i}{{i}^{3}}}} \]

   4. Taylor expanded in i around 0 98.9%

      \[\leadsto \color{blue}{-0.25 \cdot {i}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*l* 98.9%

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

   6. Simplified 98.9%

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

   if 0.5 < i

   1. Initial program 20.1%

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

      1. times-frac 45.9%

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

      2. associate-/l* 45.9%

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

      3. associate-/l* 46.0%

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

      4. associate-/l/ 46.0%

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

      5. associate-/r/ 46.0%

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

      6. associate-/l* 46.0%

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

      7. *-inverses 46.0%

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

      8. metadata-eval 46.0%

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

      9. associate-*l* 46.0%

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

      10. *-commutative 46.0%

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

      11. associate-/r* 46.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{i}{{i}^{3}}}} \]

   4. Taylor expanded in i around inf 99.5%

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 99.5%

         \[\leadsto 0.0625 + 0.015625 \cdot \frac{1}{\color{blue}{i \cdot i}} \]

      2. associate-*r/ 99.5%

         \[\leadsto 0.0625 + \color{blue}{\frac{0.015625 \cdot 1}{i \cdot i}} \]

      3. metadata-eval 99.5%

         \[\leadsto 0.0625 + \frac{\color{blue}{0.015625}}{i \cdot i} \]

   6. Simplified 99.5%

      \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 0.5:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \]

Alternative 3: 99.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{4 - \frac{1}{i \cdot i}} \end{array} \]

FPCore

(FPCore (i) :precision binary64 (/ 0.25 (- 4.0 (/ 1.0 (* i i)))))

C

double code(double i) {
	return 0.25 / (4.0 - (1.0 / (i * i)));
}

Fortran

real(8) function code(i)
    real(8), intent (in) :: i
    code = 0.25d0 / (4.0d0 - (1.0d0 / (i * i)))
end function

Java

public static double code(double i) {
	return 0.25 / (4.0 - (1.0 / (i * i)));
}

Python

def code(i):
	return 0.25 / (4.0 - (1.0 / (i * i)))

Julia

function code(i)
	return Float64(0.25 / Float64(4.0 - Float64(1.0 / Float64(i * i))))
end

MATLAB

function tmp = code(i)
	tmp = 0.25 / (4.0 - (1.0 / (i * i)));
end

Wolfram

code[i_] := N[(0.25 / N[(4.0 - N[(1.0 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.5%

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

   1. times-frac 73.7%

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

   2. associate-/l* 73.6%

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

   3. associate-/l* 73.7%

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

   4. associate-/l/ 73.7%

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

   5. associate-/r/ 73.1%

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

   6. associate-/l* 73.1%

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

   7. *-inverses 73.1%

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

   8. metadata-eval 73.1%

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

   9. associate-*l* 73.1%

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

   10. *-commutative 73.1%

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

   11. associate-/r* 73.2%

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

3. Simplified 90.7%

   \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{i}{{i}^{3}}}} \]

4. Taylor expanded in i around 0 99.4%

   \[\leadsto \frac{0.25}{4 - \color{blue}{\frac{1}{{i}^{2}}}} \]
5. Step-by-step derivation

   1. unpow2 99.4%

      \[\leadsto \frac{0.25}{4 - \frac{1}{\color{blue}{i \cdot i}}} \]

6. Simplified 99.4%

   \[\leadsto \frac{0.25}{4 - \color{blue}{\frac{1}{i \cdot i}}} \]

7. Final simplification 99.4%

   \[\leadsto \frac{0.25}{4 - \frac{1}{i \cdot i}} \]

Alternative 4: 99.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

double code(double i) {
	double tmp;
	if (i <= 0.5) {
		tmp = i * (i * -0.25);
	} 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 = i * (i * (-0.25d0))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if i < 0.5

   1. Initial program 28.6%

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

      1. times-frac 99.9%

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

      2. associate-/l* 99.7%

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

      3. associate-/l* 99.8%

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

      4. associate-/l/ 99.8%

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

      5. associate-/r/ 98.6%

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

      6. associate-/l* 98.6%

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

      7. *-inverses 98.6%

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

      8. metadata-eval 98.6%

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

      9. associate-*l* 98.6%

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

      10. *-commutative 98.6%

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

      11. associate-/r* 98.8%

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

   3. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{i}{{i}^{3}}}} \]

   4. Taylor expanded in i around 0 98.9%

      \[\leadsto \color{blue}{-0.25 \cdot {i}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*l* 98.9%

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

   6. Simplified 98.9%

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

   if 0.5 < i

   1. Initial program 20.1%

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

      1. times-frac 45.9%

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

      2. associate-/l* 45.9%

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

      3. associate-/l* 46.0%

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

      4. associate-/l/ 46.0%

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

      5. associate-/r/ 46.0%

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

      6. associate-/l* 46.0%

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

      7. *-inverses 46.0%

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

      8. metadata-eval 46.0%

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

      9. associate-*l* 46.0%

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

      10. *-commutative 46.0%

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

      11. associate-/r* 46.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{i}{{i}^{3}}}} \]

   4. Taylor expanded in i around inf 99.4%

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

4. Final simplification 99.2%

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

Alternative 5: 51.1% 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 24.5%

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

   1. times-frac 73.7%

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

   2. associate-/l* 73.6%

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

   3. associate-/l* 73.7%

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

   4. associate-/l/ 73.7%

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

   5. associate-/r/ 73.1%

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

   6. associate-/l* 73.1%

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

   7. *-inverses 73.1%

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

   8. metadata-eval 73.1%

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

   9. associate-*l* 73.1%

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

   10. *-commutative 73.1%

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

   11. associate-/r* 73.2%

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

3. Simplified 90.7%

   \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{i}{{i}^{3}}}} \]

4. Taylor expanded in i around inf 49.5%

   \[\leadsto \color{blue}{0.0625} \]

5. Final simplification 49.5%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023201 
(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)))