Kahan p13 Example 3

Average Error: 0.0 → 0.0

Time: 7.6s

Precision: binary64

Cost: 1600

Math

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

↓

\[\begin{array}{l} t_1 := \frac{4}{2 + \left(t + t\right)} - 2\\ 1 - \frac{1}{2 + t_1 \cdot t_1} \end{array} \]

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 (- (/ 4.0 (+ 2.0 (+ t t))) 2.0)))
   (- 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 = (4.0 / (2.0 + (t + t))) - 2.0;
	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 = (4.0d0 / (2.0d0 + (t + t))) - 2.0d0
    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 = (4.0 / (2.0 + (t + t))) - 2.0;
	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 = (4.0 / (2.0 + (t + t))) - 2.0
	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(Float64(4.0 / Float64(2.0 + Float64(t + t))) - 2.0)
	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 = (4.0 / (2.0 + (t + t))) - 2.0;
	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[(N[(4.0 / N[(2.0 + N[(t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $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. Simplified 0.0

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

   [Start] 0.0	\[ 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   rational.json-simplify-21 [=>] 0.0	\[ 1 - \frac{1}{2 + \color{blue}{\left|\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right|}} \]
   rational.json-simplify-39 [=>] 0.0	\[ 1 - \frac{1}{2 + \color{blue}{\left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right| \cdot \left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right|}} \]
   rational.json-simplify-58 [=>] 0.0	\[ 1 - \frac{1}{2 + \color{blue}{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right|} \cdot \left|2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right|} \]
   rational.json-simplify-58 [=>] 0.0	\[ 1 - \frac{1}{2 + \left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right| \cdot \color{blue}{\left|\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right|}} \]
   rational.json-simplify-38 [=>] 0.0	\[ 1 - \frac{1}{2 + \color{blue}{\left|\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right)\right|}} \]
   rational.json-simplify-21 [<=] 0.0	\[ 1 - \frac{1}{2 + \color{blue}{\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right)}} \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.5
Cost	1736

\[\begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -1.1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.86:\\ \;\;\;\;1 - \frac{1}{2 + \left(-1 + \left(1 - \left(t + t\right) \cdot \left(\frac{4}{2 + \left(t + t\right)} + -2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.5
Cost	1480

\[\begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -1.1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.86:\\ \;\;\;\;1 - \frac{1}{2 + \left(\frac{4}{2 + \left(t + t\right)} - 2\right) \cdot \left(-2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.5
Cost	968

\[\begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;1 - \frac{2}{4 + t \cdot \left(t \cdot 8\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.6
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 5
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 6
Error	26.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023067 
(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)))))))))