Octave 3.8, jcobi/4, as called

Average Error: 46.5 → 0.5

Time: 3.9s

Precision: binary64

Cost: 708

\[i > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq 0.00012387664648386409:\\ \;\;\;\;\left(i \cdot i\right) \cdot \left(-0.25 - i \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \end{array} \]

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 0.00012387664648386409)
   (* (* i i) (- -0.25 (* i i)))
   (+ 0.0625 (/ (/ 0.015625 i) 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) {
	double tmp;
	if (i <= 0.00012387664648386409) {
		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
    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 <= 0.00012387664648386409d0) 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) {
	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 <= 0.00012387664648386409) {
		tmp = (i * i) * (-0.25 - (i * i));
	} else {
		tmp = 0.0625 + ((0.015625 / i) / i);
	}
	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 <= 0.00012387664648386409:
		tmp = (i * i) * (-0.25 - (i * i))
	else:
		tmp = 0.0625 + ((0.015625 / i) / i)
	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 <= 0.00012387664648386409)
		tmp = Float64(Float64(i * i) * Float64(-0.25 - Float64(i * i)));
	else
		tmp = Float64(0.0625 + Float64(Float64(0.015625 / i) / i));
	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 <= 0.00012387664648386409)
		tmp = (i * i) * (-0.25 - (i * i));
	else
		tmp = 0.0625 + ((0.015625 / i) / i);
	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, 0.00012387664648386409], N[(N[(i * i), $MachinePrecision] * N[(-0.25 - N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(N[(0.015625 / i), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.2387664648386409e-4

   1. Initial program 45.8

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

   2. Simplified 0.7

      \[\leadsto \color{blue}{\frac{0.25}{4 + \frac{-1}{i \cdot i}}} \]
      Proof
      (/.f64 1/4 (+.f64 4 (/.f64 -1 (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= metadata-eval (/.f64 1 4)) (+.f64 4 (/.f64 -1 (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= *-inverses_binary64 (/.f64 (*.f64 i i) (*.f64 i i))) 4) (+.f64 4 (/.f64 -1 (*.f64 i i)))): 112 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 (*.f64 i i) (*.f64 i i)) (Rewrite<= metadata-eval (*.f64 2 2))) (+.f64 4 (/.f64 -1 (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 i i) (*.f64 (*.f64 i i) (*.f64 2 2)))) (+.f64 4 (/.f64 -1 (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 2) (*.f64 i i)))) (+.f64 4 (/.f64 -1 (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 2 i) (*.f64 2 i)))) (+.f64 4 (/.f64 -1 (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (+.f64 (Rewrite<= metadata-eval (*.f64 2 2)) (/.f64 -1 (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (+.f64 (*.f64 2 2) (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (+.f64 (*.f64 2 2) (/.f64 (neg.f64 (Rewrite<= *-inverses_binary64 (/.f64 i i))) (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (+.f64 (*.f64 2 2) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (/.f64 i i) (*.f64 i i)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (+.f64 (*.f64 2 2) (neg.f64 (Rewrite<= associate-/r*_binary64 (/.f64 i (*.f64 i (*.f64 i i))))))): 21 points increase in error, 1 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (+.f64 (*.f64 2 2) (neg.f64 (/.f64 i (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 i i) i)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 2 2) (/.f64 i (*.f64 (*.f64 i i) i))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (-.f64 (Rewrite=> metadata-eval 4) (/.f64 i (*.f64 (*.f64 i i) i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (-.f64 (Rewrite<= metadata-eval (/.f64 4 1)) (/.f64 i (*.f64 (*.f64 i i) i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (-.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 2)) 1) (/.f64 i (*.f64 (*.f64 i i) i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (-.f64 (/.f64 (*.f64 2 2) (Rewrite<= *-inverses_binary64 (/.f64 (*.f64 i i) (*.f64 i i)))) (/.f64 i (*.f64 (*.f64 i i) i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 2 2) (*.f64 i i)) (*.f64 i i))) (/.f64 i (*.f64 (*.f64 i i) i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (-.f64 (/.f64 (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 2 i) (*.f64 2 i))) (*.f64 i i)) (/.f64 i (*.f64 (*.f64 i i) i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (-.f64 (/.f64 (*.f64 (*.f64 2 i) (*.f64 2 i)) (*.f64 i i)) (/.f64 i (Rewrite=> associate-*l*_binary64 (*.f64 i (*.f64 i i)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (-.f64 (/.f64 (*.f64 (*.f64 2 i) (*.f64 2 i)) (*.f64 i i)) (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 i i) (*.f64 i i))))): 1 points increase in error, 21 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 (*.f64 2 i) (*.f64 2 i)) (/.f64 i i)) (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (/.f64 (-.f64 (*.f64 (*.f64 2 i) (*.f64 2 i)) (Rewrite=> *-inverses_binary64 1)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 (*.f64 i i) (*.f64 (*.f64 2 i) (*.f64 2 i))) (*.f64 i i)) (-.f64 (*.f64 (*.f64 2 i) (*.f64 2 i)) 1))): 1 points increase in error, 11 points decrease in error
      (/.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 i i) (*.f64 i i)) (*.f64 (*.f64 2 i) (*.f64 2 i)))) (-.f64 (*.f64 (*.f64 2 i) (*.f64 2 i)) 1)): 69 points increase in error, 0 points decrease in error

   3. Taylor expanded in i around 0 0.0

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

   4. Simplified 0.0

      \[\leadsto \color{blue}{\left(i \cdot i\right) \cdot \left(-0.25 - i \cdot i\right)} \]
      Proof
      (*.f64 (*.f64 i i) (-.f64 -1/4 (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 i 2)) (-.f64 -1/4 (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 i 2) (-.f64 -1/4 (Rewrite<= unpow2_binary64 (pow.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 -1/4 (pow.f64 i 2)) (*.f64 (pow.f64 i 2) (pow.f64 i 2)))): 1 points increase in error, 1 points decrease in error
      (-.f64 (*.f64 -1/4 (pow.f64 i 2)) (Rewrite=> pow-sqr_binary64 (pow.f64 i (*.f64 2 2)))): 12 points increase in error, 11 points decrease in error
      (-.f64 (*.f64 -1/4 (pow.f64 i 2)) (pow.f64 i (Rewrite=> metadata-eval 4))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1/4 (pow.f64 i 2)) (neg.f64 (pow.f64 i 4)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/4 (pow.f64 i 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 i 4)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (pow.f64 i 4)) (*.f64 -1/4 (pow.f64 i 2)))): 0 points increase in error, 0 points decrease in error

   if 1.2387664648386409e-4 < i

   1. Initial program 47.1

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

   2. Taylor expanded in i around inf 1.0

      \[\leadsto \color{blue}{0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}} \]

   3. Simplified 1.0

      \[\leadsto \color{blue}{0.0625 + \frac{\frac{0.015625}{i}}{i}} \]
      Proof
      (+.f64 1/16 (/.f64 (/.f64 1/64 i) i)): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (Rewrite<= associate-/r*_binary64 (/.f64 1/64 (*.f64 i i)))): 9 points increase in error, 13 points decrease in error
      (+.f64 1/16 (/.f64 (Rewrite<= metadata-eval (*.f64 1/64 1)) (*.f64 i i))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (/.f64 (*.f64 1/64 1) (Rewrite<= unpow2_binary64 (pow.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1/16 (Rewrite<= associate-*r/_binary64 (*.f64 1/64 (/.f64 1 (pow.f64 i 2))))): 1 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 0.00012387664648386409:\\ \;\;\;\;\left(i \cdot i\right) \cdot \left(-0.25 - i \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;i \leq 0.00012387664648386409:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	576

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

Alternative 3
Error	0.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;i \leq 0.00012387664648386409:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 4
Error	31.5
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022295 
(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)))