Octave 3.8, jcobi/4, as called

Average Error: 46.6 → 0.1

Time: 5.9s

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 46.6

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

2. Simplified 46.7

   \[\leadsto \color{blue}{\frac{i \cdot \left(i \cdot \left(i \cdot i\right)\right)}{\left(4 \cdot \left(i \cdot i\right) + -1\right) \cdot \left(4 \cdot \left(i \cdot i\right)\right)}} \]
   Proof
   (/.f64 (*.f64 i (*.f64 i (*.f64 i i))) (*.f64 (+.f64 (*.f64 4 (*.f64 i i)) -1) (*.f64 4 (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 i i) (*.f64 i i))) (*.f64 (+.f64 (*.f64 4 (*.f64 i i)) -1) (*.f64 4 (*.f64 i i)))): 0 points increase in error, 12 points decrease in error
   (/.f64 (*.f64 (*.f64 i i) (*.f64 i i)) (*.f64 (+.f64 (*.f64 (Rewrite<= metadata-eval (*.f64 2 2)) (*.f64 i i)) -1) (*.f64 4 (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (*.f64 i i) (*.f64 i i)) (*.f64 (+.f64 (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 2 i) (*.f64 2 i))) -1) (*.f64 4 (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (*.f64 i i) (*.f64 i i)) (*.f64 (Rewrite=> difference-of-sqr--1_binary64 (*.f64 (+.f64 (*.f64 2 i) 1) (-.f64 (*.f64 2 i) 1))) (*.f64 4 (*.f64 i i)))): 1 points increase in error, 2 points decrease in error
   (/.f64 (*.f64 (*.f64 i i) (*.f64 i i)) (*.f64 (Rewrite<= difference-of-sqr-1_binary64 (-.f64 (*.f64 (*.f64 2 i) (*.f64 2 i)) 1)) (*.f64 4 (*.f64 i i)))): 2 points increase in error, 1 points decrease in error
   (/.f64 (*.f64 (*.f64 i i) (*.f64 i i)) (*.f64 (-.f64 (*.f64 (*.f64 2 i) (*.f64 2 i)) 1) (*.f64 (Rewrite<= metadata-eval (*.f64 2 2)) (*.f64 i i)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (*.f64 i i) (*.f64 i i)) (*.f64 (-.f64 (*.f64 (*.f64 2 i) (*.f64 2 i)) 1) (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 2 i) (*.f64 2 i))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l/_binary64 (/.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))): 3 points increase in error, 3 points decrease in error

3. Applied egg-rr 41.4

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

4. Simplified 15.6

   \[\leadsto \color{blue}{\frac{i}{\frac{i \cdot \left(i \cdot 16\right) + -4}{i}}} \]
   Proof
   (/.f64 i (/.f64 (+.f64 (*.f64 i (*.f64 i 16)) -4) i)): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (+.f64 (*.f64 i (*.f64 i (Rewrite<= metadata-eval (*.f64 4 4)))) -4) i)): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (+.f64 (*.f64 i (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 i 4) 4))) -4) i)): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 i (*.f64 i 4)) 4)) -4) i)): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (+.f64 (*.f64 (*.f64 i (*.f64 i 4)) 4) (Rewrite<= metadata-eval (*.f64 -1 4))) i)): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 4 (+.f64 (*.f64 i (*.f64 i 4)) -1))) i)): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (*.f64 4 (Rewrite<= fma-udef_binary64 (fma.f64 i (*.f64 i 4) -1))) i)): 0 points increase in error, 1 points decrease in error
   (/.f64 i (/.f64 (*.f64 4 (fma.f64 i (*.f64 i 4) -1)) (Rewrite<= /-rgt-identity_binary64 (/.f64 i 1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (*.f64 4 (fma.f64 i (*.f64 i 4) -1)) (/.f64 i (Rewrite<= *-inverses_binary64 (/.f64 (*.f64 i i) (*.f64 i i)))))): 114 points increase in error, 0 points decrease in error
   (/.f64 i (/.f64 (*.f64 4 (fma.f64 i (*.f64 i 4) -1)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 i (*.f64 i i)) (*.f64 i i))))): 54 points increase in error, 1 points decrease in error
   (/.f64 i (/.f64 (*.f64 4 (fma.f64 i (*.f64 i 4) -1)) (/.f64 (Rewrite<= cube-mult_binary64 (pow.f64 i 3)) (*.f64 i i)))): 11 points increase in error, 1 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 i (/.f64 (pow.f64 i 3) (*.f64 i i))) (*.f64 4 (fma.f64 i (*.f64 i 4) -1)))): 1 points increase in error, 18 points decrease in error
   (/.f64 (*.f64 i (/.f64 (pow.f64 i 3) (*.f64 i i))) (Rewrite=> *-commutative_binary64 (*.f64 (fma.f64 i (*.f64 i 4) -1) 4))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> times-frac_binary64 (*.f64 (/.f64 i (fma.f64 i (*.f64 i 4) -1)) (/.f64 (/.f64 (pow.f64 i 3) (*.f64 i i)) 4))): 13 points increase in error, 1 points decrease in error
   (*.f64 (/.f64 i (fma.f64 i (*.f64 i 4) -1)) (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 i 3) (*.f64 (*.f64 i i) 4)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 i (fma.f64 i (*.f64 i 4) -1)) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (pow.f64 i 3) 4) (*.f64 i i)))): 2 points increase in error, 0 points decrease in error

5. Taylor expanded in i around 0 0.1

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

6. Simplified 0.1

   \[\leadsto \frac{i}{\color{blue}{i \cdot 16 - \frac{4}{i}}} \]
   Proof
   (-.f64 (*.f64 i 16) (/.f64 4 i)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 16 i)) (/.f64 4 i)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 16 i) (/.f64 (Rewrite<= metadata-eval (*.f64 4 1)) i)): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 16 i) (Rewrite<= associate-*r/_binary64 (*.f64 4 (/.f64 1 i)))): 0 points increase in error, 0 points decrease in error

7. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.5
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
Error	0.3
Cost	576

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

Alternative 3
Error	0.7
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
Error	31.0
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022338 
(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)))