Kahan p13 Example 3

Average Accuracy: 100.0% → 100.0%

Time: 10.3s

Precision: binary64

Cost: 1344

Math

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

↓

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

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 (+ 2.0 (* (+ 2.0 (/ -2.0 (+ 1.0 t))) (- 2.0 (/ 2.0 (+ 1.0 t))))))))

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 / (2.0 + ((2.0 + (-2.0 / (1.0 + t))) * (2.0 - (2.0 / (1.0 + t))))));
}

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) / (2.0d0 + ((2.0d0 + ((-2.0d0) / (1.0d0 + t))) * (2.0d0 - (2.0d0 / (1.0d0 + t))))))
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 / (2.0 + ((2.0 + (-2.0 / (1.0 + t))) * (2.0 - (2.0 / (1.0 + t))))));
}

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 / (2.0 + ((2.0 + (-2.0 / (1.0 + t))) * (2.0 - (2.0 / (1.0 + t))))))

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(2.0 + Float64(Float64(2.0 + Float64(-2.0 / Float64(1.0 + t))) * Float64(2.0 - Float64(2.0 / Float64(1.0 + t)))))))
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 / (2.0 + ((2.0 + (-2.0 / (1.0 + t))) * (2.0 - (2.0 / (1.0 + t))))));
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[(2.0 + N[(N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}\right)} \]
   Proof

   [Start] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   add-log-exp [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\log \left(e^{\frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right)}\right)} \]
   *-un-lft-identity [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \log \color{blue}{\left(1 \cdot e^{\frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right)}\right)} \]
   log-prod [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\left(\log 1 + \log \left(e^{\frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right)\right)}\right)} \]
   metadata-eval [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \left(\color{blue}{0} + \log \left(e^{\frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right)\right)\right)} \]
   add-log-exp [<=] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \left(0 + \color{blue}{\frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right)\right)} \]
   associate-/l/ [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \left(0 + \color{blue}{\frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}}\right)\right)} \]
   *-commutative [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \left(0 + \frac{2}{\color{blue}{t \cdot \left(1 + \frac{1}{t}\right)}}\right)\right)} \]

3. Simplified 100.0%

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

   [Start] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \left(0 + \frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}\right)\right)} \]
   +-lft-identity [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{2}{t \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]
   distribute-lft-in [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{t \cdot 1 + t \cdot \frac{1}{t}}}\right)} \]
   rgt-mult-inverse [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + \color{blue}{1}}\right)} \]
   *-rgt-identity [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{\color{blue}{t} + 1}\right)} \]

4. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{t + 1}\right)} \]
   sub-neg [=>] 100.0	\[ 1 - \frac{1}{2 + \color{blue}{\left(2 + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)} \cdot \left(2 - \frac{2}{t + 1}\right)} \]
   distribute-neg-frac [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 + \color{blue}{\frac{-\frac{2}{t}}{1 + \frac{1}{t}}}\right) \cdot \left(2 - \frac{2}{t + 1}\right)} \]
   distribute-neg-frac [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 + \frac{\color{blue}{\frac{-2}{t}}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{t + 1}\right)} \]
   metadata-eval [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 + \frac{\frac{\color{blue}{-2}}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{t + 1}\right)} \]

5. Simplified 100.0%

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

   [Start] 100.0	\[ 1 - \frac{1}{2 + \left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{2}{t + 1}\right)} \]
   associate-/r* [<=] 100.0	\[ 1 - \frac{1}{2 + \left(2 + \color{blue}{\frac{-2}{t \cdot \left(1 + \frac{1}{t}\right)}}\right) \cdot \left(2 - \frac{2}{t + 1}\right)} \]
   distribute-lft-in [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 + \frac{-2}{\color{blue}{t \cdot 1 + t \cdot \frac{1}{t}}}\right) \cdot \left(2 - \frac{2}{t + 1}\right)} \]
   rgt-mult-inverse [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 + \frac{-2}{t \cdot 1 + \color{blue}{1}}\right) \cdot \left(2 - \frac{2}{t + 1}\right)} \]
   *-rgt-identity [=>] 100.0	\[ 1 - \frac{1}{2 + \left(2 + \frac{-2}{\color{blue}{t} + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)} \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.44\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + \left(t \cdot t\right) \cdot \left(4 + t \cdot -8\right)}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	1088

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

Alternative 3
Accuracy	99.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.33\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -0.65 \lor \neg \left(t \leq 0.43\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 5
Accuracy	99.2%
Cost	840

\[\begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;-0.16666666666666666 + \left(1 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(1 + \left(-1 - \frac{0.2222222222222222}{t}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	99.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.55\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 7
Accuracy	98.7%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 8
Accuracy	99.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;-0.16666666666666666 + \left(1 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 9
Accuracy	98.5%
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 10
Accuracy	58.7%
Cost	64

\[0.8333333333333334 \]

Reproduce

herbie shell --seed 2023151 
(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)))))))))