Octave 3.8, jcobi/4, as called

Average Error: 46.6 → 0.0

Time: 5.5s

Precision: binary64

Cost: 964

\[i > 0\]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Split input into 2 regimes

2. if i < 5e7

   1. Initial program 44.0

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

   2. Simplified 9.5

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

      [Start] 44.0	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]

   3. Applied egg-rr 9.6

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

   4. Simplified 0.0

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

      [Start] 9.6	\[ \frac{\frac{\frac{i \cdot i}{\frac{4}{i}}}{i} + 0}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-3 [<=] 9.6	\[ \frac{\color{blue}{0 + \frac{\frac{i \cdot i}{\frac{4}{i}}}{i}}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-6 [=>] 9.6	\[ \frac{\color{blue}{\frac{\frac{i \cdot i}{\frac{4}{i}}}{i}}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-49 [=>] 0.1	\[ \frac{\color{blue}{\frac{\frac{i \cdot i}{i}}{\frac{4}{i}}}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-89 [<=] 0.0	\[ \frac{\color{blue}{\frac{i \cdot i}{4} - \frac{0}{i}}}{i \cdot \left(i \cdot 4\right) + -1} \]
      metadata-eval [<=] 0.0	\[ \frac{\frac{i \cdot i}{4} - \frac{\color{blue}{\frac{0}{-1}}}{i}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-49 [<=] 0.0	\[ \frac{\frac{i \cdot i}{4} - \color{blue}{\frac{\frac{0}{i}}{-1}}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-13 [<=] 0.0	\[ \frac{\frac{i \cdot i}{4} - \color{blue}{\left(-\frac{0}{i}\right)}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-14 [=>] 0.0	\[ \frac{\frac{i \cdot i}{4} - \color{blue}{\left(0 - \frac{0}{i}\right)}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-51 [=>] 0.0	\[ \frac{\color{blue}{\frac{0}{i} - \left(0 - \frac{i \cdot i}{4}\right)}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-14 [<=] 0.0	\[ \frac{\frac{0}{i} - \color{blue}{\left(-\frac{i \cdot i}{4}\right)}}{i \cdot \left(i \cdot 4\right) + -1} \]
      metadata-eval [<=] 0.0	\[ \frac{\frac{\color{blue}{\frac{0}{-1}}}{i} - \left(-\frac{i \cdot i}{4}\right)}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-49 [<=] 0.0	\[ \frac{\color{blue}{\frac{\frac{0}{i}}{-1}} - \left(-\frac{i \cdot i}{4}\right)}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-13 [<=] 0.0	\[ \frac{\color{blue}{\left(-\frac{0}{i}\right)} - \left(-\frac{i \cdot i}{4}\right)}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-11 [=>] 0.0	\[ \frac{\color{blue}{\frac{0}{i} \cdot -1} - \left(-\frac{i \cdot i}{4}\right)}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-59 [<=] 0.0	\[ \frac{\color{blue}{\frac{i \cdot i}{4} + \frac{0}{i} \cdot -1}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-11 [<=] 0.0	\[ \frac{\frac{i \cdot i}{4} + \color{blue}{\left(-\frac{0}{i}\right)}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-13 [=>] 0.0	\[ \frac{\frac{i \cdot i}{4} + \color{blue}{\frac{\frac{0}{i}}{-1}}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-49 [=>] 0.0	\[ \frac{\frac{i \cdot i}{4} + \color{blue}{\frac{\frac{0}{-1}}{i}}}{i \cdot \left(i \cdot 4\right) + -1} \]
      metadata-eval [=>] 0.0	\[ \frac{\frac{i \cdot i}{4} + \frac{\color{blue}{0}}{i}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-3 [=>] 0.0	\[ \frac{\color{blue}{\frac{0}{i} + \frac{i \cdot i}{4}}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-108 [=>] 0.0	\[ \frac{\frac{0}{i} + \color{blue}{\frac{i \cdot i + i \cdot i}{4 + 4}}}{i \cdot \left(i \cdot 4\right) + -1} \]
      metadata-eval [=>] 0.0	\[ \frac{\frac{0}{i} + \frac{i \cdot i + i \cdot i}{\color{blue}{8}}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-65 [=>] 0.0	\[ \frac{\frac{0}{i} + \color{blue}{\left(\frac{i \cdot i}{8} + \frac{i \cdot i}{8}\right)}}{i \cdot \left(i \cdot 4\right) + -1} \]
      rational_best-simplify-47 [<=] 0.0	\[ \frac{\color{blue}{\frac{i \cdot i}{8} + \left(\frac{i \cdot i}{8} + \frac{0}{i}\right)}}{i \cdot \left(i \cdot 4\right) + -1} \]

   if 5e7 < i

   1. Initial program 49.3

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

   2. Simplified 33.8

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

      [Start] 49.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-50 [=>] 49.3	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\color{blue}{i \cdot \left(2 \cdot \left(2 \cdot i\right)\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-84 [=>] 33.8	\[ \frac{\color{blue}{\frac{i \cdot i}{i} \cdot \frac{i \cdot i}{2 \cdot \left(2 \cdot i\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-50 [=>] 33.8	\[ \frac{\frac{i \cdot i}{i} \cdot \frac{i \cdot i}{\color{blue}{i \cdot \left(2 \cdot 2\right)}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      metadata-eval [=>] 33.8	\[ \frac{\frac{i \cdot i}{i} \cdot \frac{i \cdot i}{i \cdot \color{blue}{4}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational_best-simplify-18 [=>] 33.8	\[ \frac{\frac{i \cdot i}{i} \cdot \frac{i \cdot i}{i \cdot 4}}{\color{blue}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) + -1}} \]
      rational_best-simplify-50 [=>] 33.8	\[ \frac{\frac{i \cdot i}{i} \cdot \frac{i \cdot i}{i \cdot 4}}{\color{blue}{i \cdot \left(2 \cdot \left(2 \cdot i\right)\right)} + -1} \]
      rational_best-simplify-50 [=>] 33.8	\[ \frac{\frac{i \cdot i}{i} \cdot \frac{i \cdot i}{i \cdot 4}}{i \cdot \color{blue}{\left(i \cdot \left(2 \cdot 2\right)\right)} + -1} \]
      metadata-eval [=>] 33.8	\[ \frac{\frac{i \cdot i}{i} \cdot \frac{i \cdot i}{i \cdot 4}}{i \cdot \left(i \cdot \color{blue}{4}\right) + -1} \]

   3. Taylor expanded in i around inf 0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	31.6
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023100 
(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)))