Kahan p13 Example 3

Average Error: 0.0 → 0.0

Time: 11.6s

Precision: binary64

Cost: 1344

Math

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

↓

\[\begin{array}{l} t_1 := \frac{2}{1 + \frac{1}{t}}\\ 1 - \frac{1}{2 + t_1 \cdot t_1} \end{array} \]

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
 (let* ((t_1 (/ 2.0 (+ 1.0 (/ 1.0 t))))) (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1))))))

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) {
	double t_1 = 2.0 / (1.0 + (1.0 / t));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

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
    real(8) :: t_1
    t_1 = 2.0d0 / (1.0d0 + (1.0d0 / t))
    code = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
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) {
	double t_1 = 2.0 / (1.0 + (1.0 / t));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

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):
	t_1 = 2.0 / (1.0 + (1.0 / t))
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)))

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)
	t_1 = Float64(2.0 / Float64(1.0 + Float64(1.0 / t)))
	return Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1))))
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)
	t_1 = 2.0 / (1.0 + (1.0 / t));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
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_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $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. Applied egg-rr 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ 1 - \frac{1}{2 + \left(\left(\frac{2}{t + 1} + -2\right) \cdot \left(\frac{2}{t + 1} + -2\right) + 0\right)} \]
   rational_best-simplify-3 [=>] 0.0	\[ 1 - \frac{1}{2 + \color{blue}{\left(\frac{2}{t + 1} + -2\right) \cdot \left(\frac{2}{t + 1} + -2\right)}} \]
   metadata-eval [<=] 0.0	\[ 1 - \frac{1}{2 + \left(\frac{2}{t + 1} + -2\right) \cdot \left(\frac{2}{t + 1} + \color{blue}{\left(0 - 2\right)}\right)} \]
   rational_best-simplify-49 [<=] 0.0	\[ 1 - \frac{1}{2 + \left(\frac{2}{t + 1} + -2\right) \cdot \color{blue}{\left(0 - \left(2 - \frac{2}{t + 1}\right)\right)}} \]
   rational_best-simplify-10 [<=] 0.0	\[ 1 - \frac{1}{2 + \left(\frac{2}{t + 1} + -2\right) \cdot \color{blue}{\left(-\left(2 - \frac{2}{t + 1}\right)\right)}} \]
   rational_best-simplify-8 [<=] 0.0	\[ 1 - \frac{1}{2 + \left(\frac{2}{t + 1} + -2\right) \cdot \color{blue}{\frac{2 - \frac{2}{t + 1}}{-1}}} \]
   rational_best-simplify-47 [<=] 0.0	\[ 1 - \frac{1}{2 + \color{blue}{\frac{\left(2 - \frac{2}{t + 1}\right) \cdot \left(\frac{2}{t + 1} + -2\right)}{-1}}} \]
   rational_best-simplify-8 [=>] 0.0	\[ 1 - \frac{1}{2 + \color{blue}{\left(-\left(2 - \frac{2}{t + 1}\right) \cdot \left(\frac{2}{t + 1} + -2\right)\right)}} \]
   rational_best-simplify-13 [=>] 0.0	\[ 1 - \frac{1}{2 + \color{blue}{\left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(\frac{2}{t + 1} + -2\right)\right) \cdot -1}} \]
   rational_best-simplify-2 [=>] 0.0	\[ 1 - \frac{1}{2 + \color{blue}{-1 \cdot \left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(\frac{2}{t + 1} + -2\right)\right)}} \]
   rational_best-simplify-5 [<=] 0.0	\[ 1 - \frac{1}{2 + -1 \cdot \color{blue}{\left(1 \cdot \left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(\frac{2}{t + 1} + -2\right)\right)\right)}} \]
   rational_best-simplify-44 [=>] 0.0	\[ 1 - \frac{1}{2 + -1 \cdot \color{blue}{\left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(1 \cdot \left(\frac{2}{t + 1} + -2\right)\right)\right)}} \]
   rational_best-simplify-2 [<=] 0.0	\[ 1 - \frac{1}{2 + -1 \cdot \left(\left(2 - \frac{2}{t + 1}\right) \cdot \color{blue}{\left(\left(\frac{2}{t + 1} + -2\right) \cdot 1\right)}\right)} \]
   rational_best-simplify-44 [=>] 0.0	\[ 1 - \frac{1}{2 + \color{blue}{\left(2 - \frac{2}{t + 1}\right) \cdot \left(-1 \cdot \left(\left(\frac{2}{t + 1} + -2\right) \cdot 1\right)\right)}} \]

4. Final simplification 0.0

   \[\leadsto 1 - \frac{1}{2 + \frac{2}{1 + \frac{1}{t}} \cdot \frac{2}{1 + \frac{1}{t}}} \]

Alternatives

Alternative 1
Error	0.5
Cost	1352

\[\begin{array}{l} t_1 := 1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -1.1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.88:\\ \;\;\;\;\left(2 + \frac{-1}{2 + \frac{t}{1 + \frac{1}{t}} \cdot 4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.5
Cost	1224

\[\begin{array}{l} t_1 := 1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -1.1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.88:\\ \;\;\;\;\frac{-1}{2 + \frac{t}{1 + \frac{1}{t}} \cdot 4} - -1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.5
Cost	968

\[\begin{array}{l} t_1 := 1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;1 - \frac{1}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.7
Cost	712

\[\begin{array}{l} t_1 := 1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.7
Cost	584

\[\begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	1.0
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 7
Error	25.9
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023096 
(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)))))))))