Average Error: 46.9 → 0.3
Time: 5.5s
Precision: binary64
Cost: 576
\[i > 0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \]
↓
\[\frac{0.25}{4 + \frac{-1}{i \cdot i}} \]
(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 (/ 0.25 (+ 4.0 (/ -1.0 (* i i)))))
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 0.25 / (4.0 + (-1.0 / (i * i)));
}

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 = 0.25d0 / (4.0d0 + ((-1.0d0) / (i * i)))
end function

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 0.25 / (4.0 + (-1.0 / (i * i)));
}

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 0.25 / (4.0 + (-1.0 / (i * i)))

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(0.25 / Float64(4.0 + Float64(-1.0 / Float64(i * i))))
end

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 = 0.25 / (4.0 + (-1.0 / (i * i)));
end

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[(0.25 / N[(4.0 + N[(-1.0 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.9

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{0.25}{4 + \frac{-1}{i \cdot i}}} \]
    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)))): 127 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))))))): 29 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 (/.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))))): 0 points increase in error, 29 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)))): 3 points increase in error, 1 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))): 0 points increase in error, 20 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)): 74 points increase in error, 0 points decrease in error

  3. Final simplification0.3

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

Alternatives

Alternative 1
Error0.5
Cost580
\[\begin{array}{l} \mathbf{if}\;i \leq 0.47298990335889374:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \end{array} \]
Alternative 2
Error0.7
Cost452
\[\begin{array}{l} \mathbf{if}\;i \leq 0.47298990335889374:\\ \;\;\;\;i \cdot \left(i \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
Alternative 3
Error31.0
Cost64
\[0.0625 \]

Error

Reproduce

herbie shell --seed 2022306 
(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)))