Octave 3.8, jcobi/4, as called

Average Error: 46.4 → 0.1

Time: 1.7s

Precision: binary64

\[i > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := \left(i \cdot 2\right) \cdot \left(i \cdot 2\right)\\ \mathbf{if}\;\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{t_0}}{t_0 + -1} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(i, i \cdot -0.25, \mathsf{fma}\left(-4, {i}^{6}, -16 \cdot {i}^{8}\right) - {i}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{4 - \frac{i}{{i}^{3}}}\\ \end{array} \]

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
 (let* ((t_0 (* (* i 2.0) (* i 2.0))))
   (if (<= (/ (/ (* (* i i) (* i i)) t_0) (+ t_0 -1.0)) 0.0)
     (fma
      i
      (* i -0.25)
      (- (fma -4.0 (pow i 6.0) (* -16.0 (pow i 8.0))) (pow i 4.0)))
     (/ 0.25 (- 4.0 (/ i (pow i 3.0)))))))

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 t_0 = (i * 2.0) * (i * 2.0);
	double tmp;
	if (((((i * i) * (i * i)) / t_0) / (t_0 + -1.0)) <= 0.0) {
		tmp = fma(i, (i * -0.25), (fma(-4.0, pow(i, 6.0), (-16.0 * pow(i, 8.0))) - pow(i, 4.0)));
	} else {
		tmp = 0.25 / (4.0 - (i / pow(i, 3.0)));
	}
	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)
	t_0 = Float64(Float64(i * 2.0) * Float64(i * 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(i * i) * Float64(i * i)) / t_0) / Float64(t_0 + -1.0)) <= 0.0)
		tmp = fma(i, Float64(i * -0.25), Float64(fma(-4.0, (i ^ 6.0), Float64(-16.0 * (i ^ 8.0))) - (i ^ 4.0)));
	else
		tmp = Float64(0.25 / Float64(4.0 - Float64(i / (i ^ 3.0))));
	end
	return 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_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] * N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(i * i), $MachinePrecision] * N[(i * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(i * N[(i * -0.25), $MachinePrecision] + N[(N[(-4.0 * N[Power[i, 6.0], $MachinePrecision] + N[(-16.0 * N[Power[i, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[i, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 / N[(4.0 - N[(i / N[Power[i, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i i) (*.f64 i i)) (*.f64 (*.f64 2 i) (*.f64 2 i))) (-.f64 (*.f64 (*.f64 2 i) (*.f64 2 i)) 1)) < -0.0

   1. Initial program 28.9

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

   2. Simplified 19.5

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

   3. Taylor expanded in i around 0 0.3

      \[\leadsto \color{blue}{-16 \cdot {i}^{8} + \left(-1 \cdot {i}^{4} + \left(-0.25 \cdot {i}^{2} + -4 \cdot {i}^{6}\right)\right)} \]

   4. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, i \cdot -0.25, \mathsf{fma}\left(-4, {i}^{6}, -16 \cdot {i}^{8}\right) - {i}^{4}\right)} \]

   if -0.0 < (/.f64 (/.f64 (*.f64 (*.f64 i i) (*.f64 i i)) (*.f64 (*.f64 2 i) (*.f64 2 i))) (-.f64 (*.f64 (*.f64 2 i) (*.f64 2 i)) 1))

   1. Initial program 52.7

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{0.25}{4 - \frac{i}{{i}^{3}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right)}}{\left(i \cdot 2\right) \cdot \left(i \cdot 2\right) + -1} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(i, i \cdot -0.25, \mathsf{fma}\left(-4, {i}^{6}, -16 \cdot {i}^{8}\right) - {i}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{4 - \frac{i}{{i}^{3}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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)))