Kahan p13 Example 3

Average Error: 0.0 → 0.0

Time: 2.2min

Precision: binary64

Cost: 2240

Math

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

↓

\[\begin{array}{l} t_1 := t + \frac{t}{t}\\ t_2 := \frac{-8 + \frac{4}{t_1}}{t_1}\\ \frac{t_2 - -5}{6 + t_2} \end{array} \]

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 (+ t (/ t t))) (t_2 (/ (+ -8.0 (/ 4.0 t_1)) t_1)))
   (/ (- t_2 -5.0) (+ 6.0 t_2))))

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 = t + (t / t);
	double t_2 = (-8.0 + (4.0 / t_1)) / t_1;
	return (t_2 - -5.0) / (6.0 + t_2);
}

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
    real(8) :: t_2
    t_1 = t + (t / t)
    t_2 = ((-8.0d0) + (4.0d0 / t_1)) / t_1
    code = (t_2 - (-5.0d0)) / (6.0d0 + t_2)
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 = t + (t / t);
	double t_2 = (-8.0 + (4.0 / t_1)) / t_1;
	return (t_2 - -5.0) / (6.0 + t_2);
}

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 = t + (t / t)
	t_2 = (-8.0 + (4.0 / t_1)) / t_1
	return (t_2 - -5.0) / (6.0 + t_2)

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(t + Float64(t / t))
	t_2 = Float64(Float64(-8.0 + Float64(4.0 / t_1)) / t_1)
	return Float64(Float64(t_2 - -5.0) / Float64(6.0 + t_2))
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 = t + (t / t);
	t_2 = (-8.0 + (4.0 / t_1)) / t_1;
	tmp = (t_2 - -5.0) / (6.0 + t_2);
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[(t + N[(t / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-8.0 + N[(4.0 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, N[(N[(t$95$2 - -5.0), $MachinePrecision] / N[(6.0 + t$95$2), $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

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

3. Applied egg-rr 0.0

   \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\sqrt[3]{{\left(4 + \frac{-4}{t + \frac{t}{t}}\right)}^{2}} + \left(\sqrt[3]{{\left(\frac{-4 + \frac{4}{t + \frac{t}{t}}}{t + \frac{t}{t}}\right)}^{2}} - \sqrt[3]{4 + \frac{-4}{t + \frac{t}{t}}} \cdot \sqrt[3]{\frac{-4 + \frac{4}{t + \frac{t}{t}}}{t + \frac{t}{t}}}\right)\right) \cdot \left(\sqrt[3]{4 + \frac{-4}{t + \frac{t}{t}}} + \sqrt[3]{\frac{-4 + \frac{4}{t + \frac{t}{t}}}{t + \frac{t}{t}}}\right)}} \]

4. Simplified 0.0

   \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(4 + \frac{-4 + \left(-4 + \frac{4}{t + \frac{t}{t}}\right)}{t + \frac{t}{t}}\right)}} \]
   Proof

5. Applied egg-rr 0.0

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

6. Applied egg-rr 0.0

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

Alternatives

Alternative 1
Error	0.6
Cost	1352

\[\begin{array}{l} t_1 := t + \frac{t}{t}\\ t_2 := \frac{-1}{6 + \frac{-8}{t_1}} - -1\\ \mathbf{if}\;t \leq -0.66:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.25:\\ \;\;\;\;1 - \frac{1}{2 + t \cdot \left(4 + \frac{-4}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	0.5
Cost	1352

\[\begin{array}{l} t_1 := t + \frac{t}{t}\\ t_2 := \frac{-1}{6 + \frac{\frac{4}{t} + -8}{t_1}} - -1\\ \mathbf{if}\;t \leq -0.65:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;1 - \frac{1}{2 + t \cdot \left(4 + \frac{-4}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	0.0
Cost	1344

\[\begin{array}{l} t_1 := t + \frac{t}{t}\\ \frac{-1}{6 + \frac{-8 + \frac{4}{t_1}}{t_1}} - -1 \end{array} \]

Alternative 4
Error	1.4
Cost	1216

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

Alternative 5
Error	0.7
Cost	1096

\[\begin{array}{l} t_1 := \frac{-1}{6 + \frac{-8}{t + \frac{t}{t}}} - -1\\ \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;1 - \frac{1}{2 + t \cdot \left(4 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	0.7
Cost	968

\[\begin{array}{l} t_1 := 1 - \frac{0.16666666666666666}{\frac{t - 1.3333333333333333}{t}}\\ \mathbf{if}\;t \leq -0.47:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2:\\ \;\;\;\;1 - \frac{1}{2 + t \cdot \left(4 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	0.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -0.47:\\ \;\;\;\;1 - \frac{\frac{12}{\frac{-1.3333333333333333 + t}{t}}}{72}\\ \mathbf{elif}\;t \leq 2:\\ \;\;\;\;1 - \frac{1}{2 + t \cdot \left(4 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.16666666666666666}{\frac{t - 1.3333333333333333}{t}}\\ \end{array} \]

Alternative 8
Error	0.7
Cost	840

\[\begin{array}{l} t_1 := 1 - \frac{0.16666666666666666}{\frac{t - 1.3333333333333333}{t}}\\ \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.65:\\ \;\;\;\;t \cdot t - -0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	1.0
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 10
Error	1.1
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 11
Error	26.2
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023033 
(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)))))))))