Kahan p13 Example 3

Average Error: 0.0 → 0.0

Time: 9.4s

Precision: binary64

Cost: 1216

Math

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

↓

\[1 + \frac{-1}{\frac{1}{-1 - t} \cdot \left(8 + \frac{-4}{1 + t}\right) + 6} \]

FPCore

(FPCore (t)
 :precision binary64
 (-
  1.0
  (/
   1.0
   (+
    2.0
    (*
     (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
     (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))

↓

(FPCore (t)
 :precision binary64
 (+ 1.0 (/ -1.0 (+ (* (/ 1.0 (- -1.0 t)) (+ 8.0 (/ -4.0 (+ 1.0 t)))) 6.0))))

C

double code(double t) {
	return 1.0 - (1.0 / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))));
}

↓

double code(double t) {
	return 1.0 + (-1.0 / (((1.0 / (-1.0 - t)) * (8.0 + (-4.0 / (1.0 + t)))) + 6.0));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 - (1.0d0 / (2.0d0 + ((2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))) * (2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))))))
end function

↓

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 + ((-1.0d0) / (((1.0d0 / ((-1.0d0) - t)) * (8.0d0 + ((-4.0d0) / (1.0d0 + t)))) + 6.0d0))
end function

Java

public static double code(double t) {
	return 1.0 - (1.0 / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))));
}

↓

public static double code(double t) {
	return 1.0 + (-1.0 / (((1.0 / (-1.0 - t)) * (8.0 + (-4.0 / (1.0 + t)))) + 6.0));
}

Python

def code(t):
	return 1.0 - (1.0 / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))))

↓

def code(t):
	return 1.0 + (-1.0 / (((1.0 / (-1.0 - t)) * (8.0 + (-4.0 / (1.0 + t)))) + 6.0))

Julia

function code(t)
	return Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) * Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))))))
end

↓

function code(t)
	return Float64(1.0 + Float64(-1.0 / Float64(Float64(Float64(1.0 / Float64(-1.0 - t)) * Float64(8.0 + Float64(-4.0 / Float64(1.0 + t)))) + 6.0)))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 - (1.0 / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))));
end

↓

function tmp = code(t)
	tmp = 1.0 + (-1.0 / (((1.0 / (-1.0 - t)) * (8.0 + (-4.0 / (1.0 + t)))) + 6.0));
end

Wolfram

code[t_] := N[(1.0 - N[(1.0 / N[(2.0 + N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_] := N[(1.0 + N[(-1.0 / N[(N[(N[(1.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * N[(8.0 + N[(-4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{1 + \frac{-1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}} \]
   Proof

   [Start] 0.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   sub-neg [=>] 0.0	\[ \color{blue}{1 + \left(-\frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\right)} \]
   distribute-neg-frac [=>] 0.0	\[ 1 + \color{blue}{\frac{-1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}} \]
   metadata-eval [=>] 0.0	\[ 1 + \frac{\color{blue}{-1}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{-1}{\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2}} \]

3. Applied egg-rr 0.0

   \[\leadsto 1 + \frac{-1}{\color{blue}{\left(8 - \frac{4}{1 + t}\right) \cdot \frac{1}{-1 - t}} + 6} \]

4. Simplified 0.0

   \[\leadsto 1 + \frac{-1}{\color{blue}{\frac{1}{-1 - t} \cdot \left(8 - \frac{4}{t + 1}\right)} + 6} \]
   Proof

   [Start] 0.0	\[ 1 + \frac{-1}{\left(8 - \frac{4}{1 + t}\right) \cdot \frac{1}{-1 - t} + 6} \]
   *-commutative [<=] 0.0	\[ 1 + \frac{-1}{\color{blue}{\frac{1}{-1 - t} \cdot \left(8 - \frac{4}{1 + t}\right)} + 6} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{-1}{\frac{1}{-1 - t} \cdot \left(8 - \frac{4}{\color{blue}{t + 1}}\right) + 6} \]

5. Final simplification 0.0

   \[\leadsto 1 + \frac{-1}{\frac{1}{-1 - t} \cdot \left(8 + \frac{-4}{1 + t}\right) + 6} \]

Alternatives

Alternative 1
Error	0.5
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.33\right):\\ \;\;\;\;1 + \left(\frac{0.037037037037037035}{t \cdot t} + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 2
Error	0.0
Cost	1088

\[1 + \frac{-1}{6 + \frac{\frac{4}{1 + t} + -8}{1 + t}} \]

Alternative 3
Error	0.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -0.84 \lor \neg \left(t \leq 0.39\right):\\ \;\;\;\;\left(0.8333333333333334 + \frac{0.05925925925925926}{t \cdot t}\right) + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 4
Error	0.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 5
Error	0.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -0.85:\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 6
Error	0.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 7
Error	0.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 8
Error	1.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -0.335:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 9
Error	26.4
Cost	64

\[0.8333333333333334 \]

Reproduce

herbie shell --seed 2023031 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))