Kahan p13 Example 1

Average Error: 0.0 → 0.0

Time: 9.1s

Precision: binary64

Cost: 1984

Math

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

↓

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

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

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

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

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

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(Float64(1.0 + t) / t)
	t_2 = Float64(4.0 / Float64(t_1 * t_1))
	return Float64(Float64(1.0 + t_2) / Float64(t_2 + 2.0))
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 = (1.0 + t) / t;
	t_2 = 4.0 / (t_1 * t_1);
	tmp = (1.0 + t_2) / (t_2 + 2.0);
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[(N[(1.0 + t), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(t$95$2 + 2.0), $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. Applied egg-rr 0.0

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

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.7
Cost	2120

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ t_2 := \frac{1 + t}{t}\\ \mathbf{if}\;t \leq -940663.7488860491:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.39211478482209317:\\ \;\;\;\;\frac{1 + \frac{4}{t_2 \cdot t_2}}{2 + \frac{4}{\frac{1}{t \cdot t} + \frac{2}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.6
Cost	1992

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ t_2 := \frac{1 + t}{t}\\ \mathbf{if}\;t \leq -940663.7488860491:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.39211478482209317:\\ \;\;\;\;\frac{1 + \frac{4}{t_2 \cdot t_2}}{2 + t \cdot \left(t \cdot \left(4 + t \cdot -8\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.6
Cost	1224

\[\begin{array}{l} t_1 := 4 \cdot \left(t \cdot t\right)\\ t_2 := \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -940663.7488860491:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 0.39211478482209317:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	0.7
Cost	968

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{if}\;t \leq -940663.7488860491:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.07561304932366755:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.8
Cost	584

\[\begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -940663.7488860491:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.39211478482209317:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	0.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -940663.7488860491:\\ \;\;\;\;1 + \left(\frac{-0.2222222222222222}{t} + -0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 0.39211478482209317:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

Alternative 7
Error	1.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -940663.7488860491:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.39211478482209317:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 8
Error	26.3
Cost	64

\[0.8333333333333334 \]

Reproduce

herbie shell --seed 2022228 
(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))))))