Kahan p13 Example 1

Average Error: 0.0 → 0.0

Time: 36.2s

Precision: binary64

Cost: 1984

Math

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

↓

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

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

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

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

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

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

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

3. Applied egg-rr 0.0

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

Alternatives

Alternative 1
Error	0.6
Cost	2120

\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot 4\\ t_2 := \frac{-8}{t} - 4\\ \mathbf{if}\;t \leq -0.47:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{2 + \begin{array}{l} \mathbf{if}\;t_2 \ne 0:\\ \;\;\;\;\frac{\frac{\frac{64}{t}}{t} - 16}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-8}{t} + 4\\ \end{array}}\\ \mathbf{elif}\;t \leq 2.05:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 \cdot \left(\frac{8}{t} - 6\right) - \left(6 + \frac{-8}{t}\right) \cdot \frac{-8}{t}}{\left(\frac{96}{t} + \frac{-64}{t \cdot t}\right) - 36}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	1864

\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot 4\\ t_2 := \frac{-8}{t} - 4\\ \mathbf{if}\;t \leq -0.47:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{2 + \begin{array}{l} \mathbf{if}\;t_2 \ne 0:\\ \;\;\;\;\frac{\frac{\frac{64}{t}}{t} - 16}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-8}{t} + 4\\ \end{array}}\\ \mathbf{elif}\;t \leq 2.05:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{t} + -5}{\frac{8}{t} - 6}\\ \end{array} \]

Alternative 3
Error	0.6
Cost	1224

\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot 4\\ \mathbf{if}\;t \leq -0.47:\\ \;\;\;\;\frac{5}{6 + \frac{-8}{t}} - \frac{-8}{-6 \cdot t + 8}\\ \mathbf{elif}\;t \leq 2.05:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{t} + -5}{\frac{8}{t} - 6}\\ \end{array} \]

Alternative 4
Error	0.6
Cost	1092

\[\begin{array}{l} \mathbf{if}\;t \leq -0.56:\\ \;\;\;\;\frac{5}{6 + \frac{-8}{t}} - \frac{-8}{-6 \cdot t + 8}\\ \mathbf{elif}\;t \leq 1.66:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{t} + -5}{\frac{8}{t} - 6}\\ \end{array} \]

Alternative 5
Error	0.6
Cost	968

\[\begin{array}{l} t_1 := \frac{\frac{8}{t} + -5}{\frac{8}{t} - 6}\\ \mathbf{if}\;t \leq -0.56:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.66:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	0.9
Cost	584

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

Alternative 7
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 8
Error	26.4
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023010 
(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))))))