Kahan p13 Example 1

Average Error: 0.0 → 0.0

Time: 10.6s

Precision: binary64

Cost: 2112

Math

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

↓

\[\begin{array}{l} t_1 := \frac{1 + t}{t}\\ t_2 := \frac{2}{t_1}\\ \frac{1 + t_2 \cdot t_2}{2 + \frac{4 \cdot \frac{t}{1 + t}}{t_1}} \end{array} \]

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 (/ (+ 1.0 t) t)) (t_2 (/ 2.0 t_1)))
   (/ (+ 1.0 (* t_2 t_2)) (+ 2.0 (/ (* 4.0 (/ t (+ 1.0 t))) t_1)))))

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 = (1.0 + t) / t;
	double t_2 = 2.0 / t_1;
	return (1.0 + (t_2 * t_2)) / (2.0 + ((4.0 * (t / (1.0 + t))) / t_1));
}

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
    real(8) :: t_2
    t_1 = (1.0d0 + t) / t
    t_2 = 2.0d0 / t_1
    code = (1.0d0 + (t_2 * t_2)) / (2.0d0 + ((4.0d0 * (t / (1.0d0 + t))) / t_1))
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 = (1.0 + t) / t;
	double t_2 = 2.0 / t_1;
	return (1.0 + (t_2 * t_2)) / (2.0 + ((4.0 * (t / (1.0 + t))) / t_1));
}

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 = (1.0 + t) / t
	t_2 = 2.0 / t_1
	return (1.0 + (t_2 * t_2)) / (2.0 + ((4.0 * (t / (1.0 + t))) / t_1))

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(Float64(1.0 + t) / t)
	t_2 = Float64(2.0 / t_1)
	return Float64(Float64(1.0 + Float64(t_2 * t_2)) / Float64(2.0 + Float64(Float64(4.0 * Float64(t / Float64(1.0 + t))) / t_1)))
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 = (1.0 + t) / t;
	t_2 = 2.0 / t_1;
	tmp = (1.0 + (t_2 * t_2)) / (2.0 + ((4.0 * (t / (1.0 + t))) / t_1));
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[(N[(1.0 + t), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / t$95$1), $MachinePrecision]}, N[(N[(1.0 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(4.0 * N[(t / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $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

   \[\leadsto \color{blue}{\frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \frac{2}{\frac{1 + t}{t}}}{2 + \frac{2}{\frac{1 + t}{t}} \cdot \frac{2}{\frac{1 + t}{t}}}} \]
   Proof

   [Start] 0.0	\[ \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   associate-/l* [=>] 0.0	\[ \frac{1 + \color{blue}{\frac{2}{\frac{1 + t}{t}}} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   associate-/l* [=>] 0.0	\[ \frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \color{blue}{\frac{2}{\frac{1 + t}{t}}}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   associate-/l* [=>] 0.0	\[ \frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \frac{2}{\frac{1 + t}{t}}}{2 + \color{blue}{\frac{2}{\frac{1 + t}{t}}} \cdot \frac{2 \cdot t}{1 + t}} \]
   associate-/l* [=>] 0.0	\[ \frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \frac{2}{\frac{1 + t}{t}}}{2 + \frac{2}{\frac{1 + t}{t}} \cdot \color{blue}{\frac{2}{\frac{1 + t}{t}}}} \]

3. Applied egg-rr 0.0

   \[\leadsto \frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \frac{2}{\frac{1 + t}{t}}}{2 + \color{blue}{\frac{4 \cdot \frac{t}{1 + t}}{\frac{1 + t}{t}}}} \]

4. Final simplification 0.0

   \[\leadsto \frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \frac{2}{\frac{1 + t}{t}}}{2 + \frac{4 \cdot \frac{t}{1 + t}}{\frac{1 + t}{t}}} \]

Alternatives

Alternative 1
Error	0.1
Cost	2248

\[\begin{array}{l} t_1 := \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 10^{+16}:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1481

\[\begin{array}{l} t_1 := t \cdot \left(t \cdot -4\right)\\ \mathbf{if}\;t \leq -0.59 \lor \neg \left(t \leq 0.55\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{t_1}{1 + t}}{t_1 + -2}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	969

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

Alternative 4
Error	0.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.58\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 5
Error	0.9
Cost	584

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

Alternative 6
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 7
Error	26.1
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023060 
(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))))))