Kahan p13 Example 2

Average Error: 0.0 → 0.0

Time: 10.7s

Precision: binary64

Cost: 1216

Math

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

↓

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

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
 (+ 1.0 (/ 1.0 (+ -6.0 (* (/ 1.0 (- -1.0 t)) (+ (/ 4.0 (+ 1.0 t)) -8.0))))))

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

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
    code = 1.0d0 + (1.0d0 / ((-6.0d0) + ((1.0d0 / ((-1.0d0) - t)) * ((4.0d0 / (1.0d0 + t)) + (-8.0d0)))))
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) {
	return 1.0 + (1.0 / (-6.0 + ((1.0 / (-1.0 - t)) * ((4.0 / (1.0 + t)) + -8.0))));
}

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

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)
	return Float64(1.0 + Float64(1.0 / Float64(-6.0 + Float64(Float64(1.0 / Float64(-1.0 - t)) * Float64(Float64(4.0 / Float64(1.0 + t)) + -8.0)))))
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)
	tmp = 1.0 + (1.0 / (-6.0 + ((1.0 / (-1.0 - t)) * ((4.0 / (1.0 + t)) + -8.0))));
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_] := N[(1.0 + N[(1.0 / N[(-6.0 + N[(N[(1.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * N[(N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + -8.0), $MachinePrecision]), $MachinePrecision]), $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. Simplified 0.0

   \[\leadsto \color{blue}{\frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}} \]
   Proof

   [Start] 0.0	\[ \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]

3. Applied egg-rr 0.0

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

4. Applied egg-rr 0.0

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

5. Applied egg-rr 0.0

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

6. Simplified 0.0

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

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

7. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	1088

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

Alternative 2
Error	0.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.24\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + t \cdot t\\ \end{array} \]

Alternative 3
Error	0.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;1 + \left(\frac{-0.2222222222222222}{t} + \left(-0.16666666666666666 + \frac{\frac{0.037037037037037035}{t}}{t}\right)\right)\\ \mathbf{elif}\;t \leq 0.24:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 + t \cdot t\\ \end{array} \]

Alternative 5
Error	0.9
Cost	584

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

Alternative 6
Error	0.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;1 + \left(-0.16666666666666666 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 7
Error	1.0
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 8
Error	25.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023066 
(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))))))))