Kahan p13 Example 1

Average Error: 0.11% → 0.05%

Time: 12.3s

Precision: binary64

Cost: 2376

Math

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

↓

\[\begin{array}{l} t_1 := 2 + \frac{\frac{\left(t \cdot t\right) \cdot 4}{t + 1}}{t + 1}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 20000:\\ \;\;\;\;\frac{t_1 + -1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

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 (+ 2.0 (/ (/ (* (* t t) 4.0) (+ t 1.0)) (+ t 1.0)))))
   (if (<= t -1e+154)
     0.8333333333333334
     (if (<= t 20000.0)
       (/ (+ t_1 -1.0) t_1)
       (+ 0.8333333333333334 (/ -0.2222222222222222 t))))))

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 = 2.0 + ((((t * t) * 4.0) / (t + 1.0)) / (t + 1.0));
	double tmp;
	if (t <= -1e+154) {
		tmp = 0.8333333333333334;
	} else if (t <= 20000.0) {
		tmp = (t_1 + -1.0) / t_1;
	} else {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	}
	return tmp;
}

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) :: tmp
    t_1 = 2.0d0 + ((((t * t) * 4.0d0) / (t + 1.0d0)) / (t + 1.0d0))
    if (t <= (-1d+154)) then
        tmp = 0.8333333333333334d0
    else if (t <= 20000.0d0) then
        tmp = (t_1 + (-1.0d0)) / t_1
    else
        tmp = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    end if
    code = tmp
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 = 2.0 + ((((t * t) * 4.0) / (t + 1.0)) / (t + 1.0));
	double tmp;
	if (t <= -1e+154) {
		tmp = 0.8333333333333334;
	} else if (t <= 20000.0) {
		tmp = (t_1 + -1.0) / t_1;
	} else {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	}
	return tmp;
}

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 = 2.0 + ((((t * t) * 4.0) / (t + 1.0)) / (t + 1.0))
	tmp = 0
	if t <= -1e+154:
		tmp = 0.8333333333333334
	elif t <= 20000.0:
		tmp = (t_1 + -1.0) / t_1
	else:
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	return tmp

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(2.0 + Float64(Float64(Float64(Float64(t * t) * 4.0) / Float64(t + 1.0)) / Float64(t + 1.0)))
	tmp = 0.0
	if (t <= -1e+154)
		tmp = 0.8333333333333334;
	elseif (t <= 20000.0)
		tmp = Float64(Float64(t_1 + -1.0) / t_1);
	else
		tmp = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t));
	end
	return tmp
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_2 = code(t)
	t_1 = 2.0 + ((((t * t) * 4.0) / (t + 1.0)) / (t + 1.0));
	tmp = 0.0;
	if (t <= -1e+154)
		tmp = 0.8333333333333334;
	elseif (t <= 20000.0)
		tmp = (t_1 + -1.0) / t_1;
	else
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	end
	tmp_2 = tmp;
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[(2.0 + N[(N[(N[(N[(t * t), $MachinePrecision] * 4.0), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e+154], 0.8333333333333334, If[LessEqual[t, 20000.0], N[(N[(t$95$1 + -1.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.00000000000000004e154

   1. Initial program 0.1

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

   2. Taylor expanded in t around inf 0

      \[\leadsto \color{blue}{0.8333333333333334} \]

   if -1.00000000000000004e154 < t < 2e4

   1. Initial program 0.02

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

   2. Applied egg-rr 0.09

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

   3. Applied egg-rr 0.02

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

   4. Applied egg-rr 0.06

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

   5. Applied egg-rr 0.03

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

   if 2e4 < t

   1. Initial program 0.35

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

   2. Taylor expanded in t around inf 0.14

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]

   3. Simplified 0.14

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
      Proof

      [Start] 0.14	\[ 0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t} \]
      associate-*r/ [=>] 0.14	\[ 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} \]
      metadata-eval [=>] 0.14	\[ 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222}}{t} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.05

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 20000:\\ \;\;\;\;\frac{\left(2 + \frac{\frac{\left(t \cdot t\right) \cdot 4}{t + 1}}{t + 1}\right) + -1}{2 + \frac{\frac{\left(t \cdot t\right) \cdot 4}{t + 1}}{t + 1}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.17%
Cost	2368

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

Alternative 2
Error	0.06%
Cost	2248

\[\begin{array}{l} t_1 := \frac{\frac{t \cdot \left(t \cdot 4\right)}{t + 1}}{t + 1}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 20000:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

Alternative 3
Error	0.11%
Cost	2240

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

Alternative 4
Error	0.54%
Cost	1736

\[\begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;t_1 + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;\frac{1 + \left(1 + \left(t \cdot \frac{4}{\frac{t + 1}{t}} + -1\right)\right)}{2 + \left(t \cdot t\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.54%
Cost	1480

\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot 4\\ t_2 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;t_2 + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;\frac{1 + \frac{t_1}{t + 1}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	0.63%
Cost	836

\[\begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;t_1 + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	0.69%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.55\right):\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 8
Error	1.26%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 9
Error	1.42%
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -0.335:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 10
Error	40.63%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023090 
(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))))))