Kahan p13 Example 2

Average Error: 0.0 → 0.0

Time: 13.0s

Precision: binary64

Cost: 9216

Math

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

↓

\[\begin{array}{l} t_1 := 2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\\ \frac{1 + t_1 \cdot t_1}{2 + \left(\left(1 + {t_1}^{2}\right) + -1\right)} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (/
  (+
   1.0
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
  (+
   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 (/ (/ -2.0 t) (+ 1.0 (/ 1.0 t))))))
   (/ (+ 1.0 (* t_1 t_1)) (+ 2.0 (+ (+ 1.0 (pow t_1 2.0)) -1.0)))))

C

double code(double t) {
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (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 + ((-2.0 / t) / (1.0 + (1.0 / t)));
	return (1.0 + (t_1 * t_1)) / (2.0 + ((1.0 + pow(t_1, 2.0)) + -1.0));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (1.0d0 + ((2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))) * (2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))))) / (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 + (((-2.0d0) / t) / (1.0d0 + (1.0d0 / t)))
    code = (1.0d0 + (t_1 * t_1)) / (2.0d0 + ((1.0d0 + (t_1 ** 2.0d0)) + (-1.0d0)))
end function

Java

public static double code(double t) {
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (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 + ((-2.0 / t) / (1.0 + (1.0 / t)));
	return (1.0 + (t_1 * t_1)) / (2.0 + ((1.0 + Math.pow(t_1, 2.0)) + -1.0));
}

Python

def code(t):
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (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 + ((-2.0 / t) / (1.0 + (1.0 / t)))
	return (1.0 + (t_1 * t_1)) / (2.0 + ((1.0 + math.pow(t_1, 2.0)) + -1.0))

Julia

function code(t)
	return Float64(Float64(1.0 + 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)))))) / 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(Float64(-2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	return Float64(Float64(1.0 + Float64(t_1 * t_1)) / Float64(2.0 + Float64(Float64(1.0 + (t_1 ^ 2.0)) + -1.0)))
end

MATLAB

function tmp = code(t)
	tmp = (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (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 + ((-2.0 / t) / (1.0 + (1.0 / t)));
	tmp = (1.0 + (t_1 * t_1)) / (2.0 + ((1.0 + (t_1 ^ 2.0)) + -1.0));
end

Wolfram

code[t_] := N[(N[(1.0 + 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] / 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]

↓

code[t_] := Block[{t$95$1 = N[(2.0 + N[(N[(-2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	3264

\[\begin{array}{l} t_1 := 2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\\ t_2 := t_1 \cdot t_1\\ \frac{1 + t_2}{2 + t_2} \end{array} \]

Alternative 2
Error	0.7
Cost	2888

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -14562.364739258366:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4437455465371752 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + \left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(t \cdot \left(2 + t \cdot \left(-2 + 2 \cdot t\right)\right)\right)}{2 + \left(t \cdot t\right) \cdot \left(4 + t \cdot \left(-8 + t \cdot 12\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.7
Cost	1864

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -14562.364739258366:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4437455465371752 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + \left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 \cdot t\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.6
Cost	1608

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -14562.364739258366:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4437455465371752 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + \left(2 \cdot t\right) \cdot \left(t \cdot \left(2 + t \cdot -2\right)\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.7
Cost	1480

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -14562.364739258366:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4437455465371752 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + t \cdot \left(t \cdot \left(4 + t \cdot -8\right)\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	0.6
Cost	1224

\[\begin{array}{l} t_1 := t \cdot \left(t \cdot 4\right)\\ t_2 := \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -14562.364739258366:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.4437455465371752 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	0.7
Cost	968

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -14562.364739258366:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4437455465371752 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	0.7
Cost	840

\[\begin{array}{l} t_1 := 0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} + -0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -14562.364739258366:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4437455465371752 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	0.8
Cost	584

\[\begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -14562.364739258366:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4437455465371752 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	1.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -14562.364739258366:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1.4437455465371752 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 11
Error	26.1
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022297 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))