Octave 3.8, jcobi/4, as called

Average Accuracy: 26.7% → 99.7%

Time: 5.8s

Precision: binary64

Cost: 576

\[i > 0\]

Math

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

↓

\[\frac{i}{i \cdot 16 + \frac{-4}{i}} \]

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 (/ i (+ (* i 16.0) (/ -4.0 i))))

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) {
	return i / ((i * 16.0) + (-4.0 / i));
}

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
    code = i / ((i * 16.0d0) + ((-4.0d0) / i))
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) {
	return i / ((i * 16.0) + (-4.0 / i));
}

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):
	return i / ((i * 16.0) + (-4.0 / i))

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)
	return Float64(i / Float64(Float64(i * 16.0) + Float64(-4.0 / i)))
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 = code(i)
	tmp = i / ((i * 16.0) + (-4.0 / i));
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_] := N[(i / N[(N[(i * 16.0), $MachinePrecision] + N[(-4.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.7%

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

2. Simplified 35.0%

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

   [Start] 26.7	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   associate-/r* [=>] 50.5	\[ \frac{\color{blue}{\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{2 \cdot i}}{2 \cdot i}}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
   associate-/l/ [=>] 50.5	\[ \color{blue}{\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{2 \cdot i}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot i\right)}} \]
   *-commutative [=>] 50.5	\[ \frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\color{blue}{i \cdot 2}}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot i\right)} \]
   times-frac [=>] 58.9	\[ \frac{\color{blue}{\frac{i \cdot i}{i} \cdot \frac{i \cdot i}{2}}}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot i\right)} \]
   times-frac [=>] 74.7	\[ \color{blue}{\frac{\frac{i \cdot i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \cdot \frac{\frac{i \cdot i}{2}}{2 \cdot i}} \]
   associate-/r* [<=] 74.7	\[ \frac{\frac{i \cdot i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \cdot \color{blue}{\frac{i \cdot i}{2 \cdot \left(2 \cdot i\right)}} \]
   *-commutative [<=] 74.7	\[ \frac{\frac{i \cdot i}{i}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \cdot \frac{i \cdot i}{\color{blue}{\left(2 \cdot i\right) \cdot 2}} \]

3. Applied egg-rr 51.0%

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

4. Simplified 75.5%

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

   [Start] 51.0	\[ e^{\mathsf{log1p}\left(\frac{\frac{i \cdot i}{4}}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)} - 1 \]
   expm1-def [=>] 74.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{i \cdot i}{4}}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}\right)\right)} \]
   expm1-log1p [=>] 74.9	\[ \color{blue}{\frac{\frac{i \cdot i}{4}}{\mathsf{fma}\left(i, i \cdot 4, -1\right)}} \]
   associate-/l/ [=>] 74.8	\[ \color{blue}{\frac{i \cdot i}{\mathsf{fma}\left(i, i \cdot 4, -1\right) \cdot 4}} \]
   associate-/l* [=>] 75.5	\[ \color{blue}{\frac{i}{\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right) \cdot 4}{i}}} \]
   associate-*l/ [<=] 75.5	\[ \frac{i}{\color{blue}{\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i} \cdot 4}} \]
   associate-/r* [=>] 75.5	\[ \color{blue}{\frac{\frac{i}{\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i}}}{4}} \]
   metadata-eval [<=] 75.5	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i}}}{\color{blue}{\frac{1}{0.25}}} \]
   metadata-eval [<=] 75.5	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i}}}{\frac{1}{\color{blue}{\frac{1}{4}}}} \]
   *-inverses [<=] 75.5	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i}}}{\frac{1}{\frac{\color{blue}{\frac{i}{i}}}{4}}} \]
   associate-/r* [<=] 75.5	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i}}}{\frac{1}{\color{blue}{\frac{i}{i \cdot 4}}}} \]
   associate-/l* [<=] 75.5	\[ \color{blue}{\frac{\frac{i}{\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i}} \cdot \frac{i}{i \cdot 4}}{1}} \]
   times-frac [<=] 74.8	\[ \frac{\color{blue}{\frac{i \cdot i}{\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i} \cdot \left(i \cdot 4\right)}}}{1} \]
   associate-/r* [<=] 74.8	\[ \color{blue}{\frac{i \cdot i}{\left(\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i} \cdot \left(i \cdot 4\right)\right) \cdot 1}} \]
   *-inverses [<=] 74.8	\[ \frac{i \cdot i}{\left(\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i} \cdot \left(i \cdot 4\right)\right) \cdot \color{blue}{\frac{i}{i}}} \]
   associate-/l* [=>] 75.5	\[ \color{blue}{\frac{i}{\frac{\left(\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i} \cdot \left(i \cdot 4\right)\right) \cdot \frac{i}{i}}{i}}} \]
   *-inverses [=>] 75.5	\[ \frac{i}{\frac{\left(\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i} \cdot \left(i \cdot 4\right)\right) \cdot \color{blue}{1}}{i}} \]
   associate-/l* [=>] 75.5	\[ \frac{i}{\color{blue}{\frac{\frac{\mathsf{fma}\left(i, i \cdot 4, -1\right)}{i} \cdot \left(i \cdot 4\right)}{\frac{i}{1}}}} \]

5. Taylor expanded in i around 0 99.7%

   \[\leadsto \frac{i}{\color{blue}{16 \cdot i - 4 \cdot \frac{1}{i}}} \]

6. Simplified 99.7%

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

   [Start] 99.7	\[ \frac{i}{16 \cdot i - 4 \cdot \frac{1}{i}} \]
   *-commutative [=>] 99.7	\[ \frac{i}{\color{blue}{i \cdot 16} - 4 \cdot \frac{1}{i}} \]
   cancel-sign-sub-inv [=>] 99.7	\[ \frac{i}{\color{blue}{i \cdot 16 + \left(-4\right) \cdot \frac{1}{i}}} \]
   metadata-eval [=>] 99.7	\[ \frac{i}{i \cdot 16 + \color{blue}{-4} \cdot \frac{1}{i}} \]
   associate-*r/ [=>] 99.7	\[ \frac{i}{i \cdot 16 + \color{blue}{\frac{-4 \cdot 1}{i}}} \]
   metadata-eval [=>] 99.7	\[ \frac{i}{i \cdot 16 + \frac{\color{blue}{-4}}{i}} \]

7. Final simplification 99.7%

   \[\leadsto \frac{i}{i \cdot 16 + \frac{-4}{i}} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	580

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

Alternative 2
Accuracy	99.5%
Cost	576

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

Alternative 3
Accuracy	99.0%
Cost	452

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

Alternative 4
Accuracy	50.3%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023125 
(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)))