Kahan p13 Example 1

Average Error: 0.0 → 0.0

Time: 9.1s

Precision: binary64

Cost: 1216

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{t + 1}\\ \frac{-2}{4 + 8 \cdot \left(t_1 \cdot t_1\right)} + 1 \end{array} \]

FPCore

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

↓

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ t (+ t 1.0)))) (+ (/ -2.0 (+ 4.0 (* 8.0 (* t_1 t_1)))) 1.0)))

C

double code(double t) {
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))));
}

↓

double code(double t) {
	double t_1 = t / (t + 1.0);
	return (-2.0 / (4.0 + (8.0 * (t_1 * t_1)))) + 1.0;
}

Fortran

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

↓

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = t / (t + 1.0d0)
    code = ((-2.0d0) / (4.0d0 + (8.0d0 * (t_1 * t_1)))) + 1.0d0
end function

Java

public static double code(double t) {
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))));
}

↓

public static double code(double t) {
	double t_1 = t / (t + 1.0);
	return (-2.0 / (4.0 + (8.0 * (t_1 * t_1)))) + 1.0;
}

Python

def code(t):
	return (1.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t)))) / (2.0 + (((2.0 * t) / (1.0 + t)) * ((2.0 * t) / (1.0 + t))))

↓

def code(t):
	t_1 = t / (t + 1.0)
	return (-2.0 / (4.0 + (8.0 * (t_1 * t_1)))) + 1.0

Julia

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

↓

function code(t)
	t_1 = Float64(t / Float64(t + 1.0))
	return Float64(Float64(-2.0 / Float64(4.0 + Float64(8.0 * Float64(t_1 * t_1)))) + 1.0)
end

MATLAB

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

↓

function tmp = code(t)
	t_1 = t / (t + 1.0);
	tmp = (-2.0 / (4.0 + (8.0 * (t_1 * t_1)))) + 1.0;
end

Wolfram

code[t_] := N[(N[(1.0 + N[(N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_] := Block[{t$95$1 = N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(-2.0 / N[(4.0 + N[(8.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   rational.json-simplify-2 [=>] 0.0	\[ \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{\color{blue}{t \cdot 2}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   rational.json-simplify-49 [=>] 0.0	\[ \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \color{blue}{\left(2 \cdot \frac{t}{1 + t}\right)}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   rational.json-simplify-2 [=>] 0.0	\[ \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \color{blue}{\left(\frac{t}{1 + t} \cdot 2\right)}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   rational.json-simplify-43 [=>] 0.0	\[ \frac{1 + \color{blue}{\frac{t}{1 + t} \cdot \left(2 \cdot \frac{2 \cdot t}{1 + t}\right)}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   rational.json-simplify-2 [=>] 0.0	\[ \frac{1 + \frac{t}{1 + t} \cdot \left(2 \cdot \frac{\color{blue}{t \cdot 2}}{1 + t}\right)}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   rational.json-simplify-49 [=>] 0.0	\[ \frac{1 + \frac{t}{1 + t} \cdot \left(2 \cdot \color{blue}{\left(2 \cdot \frac{t}{1 + t}\right)}\right)}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   rational.json-simplify-2 [=>] 0.0	\[ \frac{1 + \frac{t}{1 + t} \cdot \left(2 \cdot \color{blue}{\left(\frac{t}{1 + t} \cdot 2\right)}\right)}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   rational.json-simplify-43 [=>] 0.0	\[ \frac{1 + \frac{t}{1 + t} \cdot \color{blue}{\left(\frac{t}{1 + t} \cdot \left(2 \cdot 2\right)\right)}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   metadata-eval [=>] 0.0	\[ \frac{1 + \frac{t}{1 + t} \cdot \left(\frac{t}{1 + t} \cdot \color{blue}{4}\right)}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   rational.json-simplify-2 [=>] 0.0	\[ \frac{1 + \frac{t}{1 + t} \cdot \left(\frac{t}{1 + t} \cdot 4\right)}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{\color{blue}{t \cdot 2}}{1 + t}} \]
   rational.json-simplify-49 [=>] 0.0	\[ \frac{1 + \frac{t}{1 + t} \cdot \left(\frac{t}{1 + t} \cdot 4\right)}{2 + \frac{2 \cdot t}{1 + t} \cdot \color{blue}{\left(2 \cdot \frac{t}{1 + t}\right)}} \]
   rational.json-simplify-2 [=>] 0.0	\[ \frac{1 + \frac{t}{1 + t} \cdot \left(\frac{t}{1 + t} \cdot 4\right)}{2 + \frac{2 \cdot t}{1 + t} \cdot \color{blue}{\left(\frac{t}{1 + t} \cdot 2\right)}} \]
   rational.json-simplify-43 [=>] 0.0	\[ \frac{1 + \frac{t}{1 + t} \cdot \left(\frac{t}{1 + t} \cdot 4\right)}{2 + \color{blue}{\frac{t}{1 + t} \cdot \left(2 \cdot \frac{2 \cdot t}{1 + t}\right)}} \]

3. Applied egg-rr 0.5

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

4. Simplified 0.0

   \[\leadsto \color{blue}{\frac{-2}{4 + 8 \cdot \left(\frac{t}{t + 1} \cdot \frac{t}{t + 1}\right)} + 1} \]
   Proof

   [Start] 0.5	\[ -1 + \left(1 - \frac{-1 - \frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right)}{\frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right) + 2}\right) \]
   rational.json-simplify-1 [=>] 0.5	\[ \color{blue}{\left(1 - \frac{-1 - \frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right)}{\frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right) + 2}\right) + -1} \]
   rational.json-simplify-15 [=>] 0.5	\[ \color{blue}{\left(1 - \frac{-1 - \frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right)}{\frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right) + 2}\right) - 1} \]
   rational.json-simplify-42 [=>] 0.0	\[ \color{blue}{\left(1 - 1\right) - \frac{-1 - \frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right)}{\frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right) + 2}} \]
   metadata-eval [=>] 0.0	\[ \color{blue}{0} - \frac{-1 - \frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right)}{\frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right) + 2} \]
   rational.json-simplify-12 [<=] 0.0	\[ \color{blue}{-\frac{-1 - \frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right)}{\frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right) + 2}} \]
   rational.json-simplify-10 [=>] 0.0	\[ \color{blue}{\frac{\frac{-1 - \frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right)}{\frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right) + 2}}{-1}} \]
   rational.json-simplify-44 [=>] 0.0	\[ \color{blue}{\frac{\frac{-1 - \frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right)}{-1}}{\frac{t}{t + 1} \cdot \left(\frac{t}{t + 1} \cdot 4\right) + 2}} \]

5. Final simplification 0.0

   \[\leadsto \frac{-2}{4 + 8 \cdot \left(\frac{t}{t + 1} \cdot \frac{t}{t + 1}\right)} + 1 \]

Alternatives

Alternative 1
Error	0.6
Cost	584

\[\begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]

Alternative 3
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 4
Error	26.1
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023074 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))