Octave 3.8, jcobi/4, as called

Average Error: 46.5 → 0.0

Time: 9.9s

Precision: binary64

Cost: 836

\[i > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq 5000000:\\ \;\;\;\;i \cdot \frac{i}{-4 + \left(i \cdot i\right) \cdot 16}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

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 5000000.0) (* i (/ i (+ -4.0 (* (* i i) 16.0)))) 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 <= 5000000.0) {
		tmp = i * (i / (-4.0 + ((i * i) * 16.0)));
	} 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 <= 5000000.0d0) then
        tmp = i * (i / ((-4.0d0) + ((i * i) * 16.0d0)))
    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 <= 5000000.0) {
		tmp = i * (i / (-4.0 + ((i * i) * 16.0)));
	} 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 <= 5000000.0:
		tmp = i * (i / (-4.0 + ((i * i) * 16.0)))
	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 <= 5000000.0)
		tmp = Float64(i * Float64(i / Float64(-4.0 + Float64(Float64(i * i) * 16.0))));
	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 <= 5000000.0)
		tmp = i * (i / (-4.0 + ((i * i) * 16.0)));
	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, 5000000.0], N[(i * N[(i / N[(-4.0 + N[(N[(i * i), $MachinePrecision] * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 5e6

   1. Initial program 44.4

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

   2. Simplified 0.1

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

      [Start] 44.4	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational.json-simplify-16 [=>] 44.4	\[ \color{blue}{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}} \]
      rational.json-simplify-3 [=>] 44.4	\[ \frac{\color{blue}{i \cdot \left(\left(i \cdot i\right) \cdot i\right)}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)} \]
      rational.json-simplify-39 [=>] 44.4	\[ \frac{\color{blue}{\left(\left(i \cdot i\right) \cdot i\right) \cdot i}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)} \]
      rational.json-simplify-19 [=>] 39.0	\[ \color{blue}{i \cdot \frac{\left(i \cdot i\right) \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}} \]
      rational.json-simplify-15 [=>] 39.0	\[ i \cdot \color{blue}{\frac{\frac{\left(i \cdot i\right) \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around 0 0.0

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

   4. Applied egg-rr 0.1

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

   5. Applied egg-rr 0.0

      \[\leadsto \color{blue}{0.25 \cdot \frac{i \cdot i}{4 \cdot \left(i \cdot i\right) + -1} + 0} \]

   6. Simplified 0.0

      \[\leadsto \color{blue}{i \cdot \frac{i}{-4 + \left(i \cdot i\right) \cdot 16}} \]
      Proof

      [Start] 0.0	\[ 0.25 \cdot \frac{i \cdot i}{4 \cdot \left(i \cdot i\right) + -1} + 0 \]
      rational.json-simplify-65 [=>] 0.0	\[ \color{blue}{0.25 \cdot \frac{i \cdot i}{4 \cdot \left(i \cdot i\right) + -1}} \]
      rational.json-simplify-18 [=>] 0.6	\[ \color{blue}{\frac{0.25}{\frac{4 \cdot \left(i \cdot i\right) + -1}{i \cdot i}}} \]
      rational.json-simplify-1 [=>] 0.0	\[ \color{blue}{\frac{i \cdot i}{\frac{4 \cdot \left(i \cdot i\right) + -1}{0.25}}} \]
      rational.json-simplify-20 [<=] 0.0	\[ \color{blue}{i \cdot \frac{i}{\frac{4 \cdot \left(i \cdot i\right) + -1}{0.25}}} \]
      rational.json-simplify-41 [=>] 0.0	\[ i \cdot \frac{i}{\frac{\color{blue}{-1 + 4 \cdot \left(i \cdot i\right)}}{0.25}} \]
      rational.json-simplify-37 [=>] 0.0	\[ i \cdot \frac{i}{\color{blue}{\frac{-1}{0.25} + \frac{4 \cdot \left(i \cdot i\right)}{0.25}}} \]
      metadata-eval [=>] 0.0	\[ i \cdot \frac{i}{\color{blue}{-4} + \frac{4 \cdot \left(i \cdot i\right)}{0.25}} \]
      rational.json-simplify-19 [=>] 0.0	\[ i \cdot \frac{i}{-4 + \color{blue}{\left(i \cdot i\right) \cdot \frac{4}{0.25}}} \]
      metadata-eval [=>] 0.0	\[ i \cdot \frac{i}{-4 + \left(i \cdot i\right) \cdot \color{blue}{16}} \]

   if 5e6 < i

   1. Initial program 48.7

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

   2. Simplified 31.4

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

      [Start] 48.7	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
      rational.json-simplify-16 [=>] 48.8	\[ \color{blue}{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}} \]
      rational.json-simplify-3 [=>] 48.9	\[ \frac{\color{blue}{i \cdot \left(\left(i \cdot i\right) \cdot i\right)}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)} \]
      rational.json-simplify-39 [=>] 48.9	\[ \frac{\color{blue}{\left(\left(i \cdot i\right) \cdot i\right) \cdot i}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)} \]
      rational.json-simplify-19 [=>] 48.7	\[ \color{blue}{i \cdot \frac{\left(i \cdot i\right) \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)\right) \cdot \left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right)}} \]
      rational.json-simplify-15 [=>] 43.5	\[ i \cdot \color{blue}{\frac{\frac{\left(i \cdot i\right) \cdot i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5000000:\\ \;\;\;\;i \cdot \frac{i}{-4 + \left(i \cdot i\right) \cdot 16}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	708

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

Alternative 2
Error	0.1
Cost	704

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

Alternative 3
Error	0.6
Cost	452

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

Alternative 4
Error	31.3
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023066 
(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)))