Kahan p13 Example 1

Average Error: 0.0 → 0.0

Time: 9.7s

Precision: binary64

Cost: 2376

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 10000000:\\ \;\;\;\;\frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{1 + \left(1 + \frac{\frac{\left(t \cdot t\right) \cdot -4}{1 + t}}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

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
 (if (<= t -4e+154)
   0.8333333333333334
   (if (<= t 10000000.0)
     (/
      (+ 1.0 (/ (/ (* t (* t 4.0)) (+ 1.0 t)) (+ 1.0 t)))
      (+ 1.0 (+ 1.0 (/ (/ (* (* t t) -4.0) (+ 1.0 t)) (- -1.0 t)))))
     (+ 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 tmp;
	if (t <= -4e+154) {
		tmp = 0.8333333333333334;
	} else if (t <= 10000000.0) {
		tmp = (1.0 + (((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t))) / (1.0 + (1.0 + ((((t * t) * -4.0) / (1.0 + t)) / (-1.0 - t))));
	} 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) :: tmp
    if (t <= (-4d+154)) then
        tmp = 0.8333333333333334d0
    else if (t <= 10000000.0d0) then
        tmp = (1.0d0 + (((t * (t * 4.0d0)) / (1.0d0 + t)) / (1.0d0 + t))) / (1.0d0 + (1.0d0 + ((((t * t) * (-4.0d0)) / (1.0d0 + t)) / ((-1.0d0) - t))))
    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 tmp;
	if (t <= -4e+154) {
		tmp = 0.8333333333333334;
	} else if (t <= 10000000.0) {
		tmp = (1.0 + (((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t))) / (1.0 + (1.0 + ((((t * t) * -4.0) / (1.0 + t)) / (-1.0 - t))));
	} 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):
	tmp = 0
	if t <= -4e+154:
		tmp = 0.8333333333333334
	elif t <= 10000000.0:
		tmp = (1.0 + (((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t))) / (1.0 + (1.0 + ((((t * t) * -4.0) / (1.0 + t)) / (-1.0 - t))))
	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)
	tmp = 0.0
	if (t <= -4e+154)
		tmp = 0.8333333333333334;
	elseif (t <= 10000000.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(t * Float64(t * 4.0)) / Float64(1.0 + t)) / Float64(1.0 + t))) / Float64(1.0 + Float64(1.0 + Float64(Float64(Float64(Float64(t * t) * -4.0) / Float64(1.0 + t)) / Float64(-1.0 - t)))));
	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)
	tmp = 0.0;
	if (t <= -4e+154)
		tmp = 0.8333333333333334;
	elseif (t <= 10000000.0)
		tmp = (1.0 + (((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t))) / (1.0 + (1.0 + ((((t * t) * -4.0) / (1.0 + t)) / (-1.0 - t))));
	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_] := If[LessEqual[t, -4e+154], 0.8333333333333334, If[LessEqual[t, 10000000.0], N[(N[(1.0 + N[(N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.0 + N[(N[(N[(N[(t * t), $MachinePrecision] * -4.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.00000000000000015e154

   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 -4.00000000000000015e154 < t < 1e7

   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

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}} \]
      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-*r/ [=>] 0.0	\[ \frac{1 + \color{blue}{\frac{\frac{2 \cdot t}{1 + t} \cdot \left(2 \cdot t\right)}{1 + t}}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l/ [=>] 0.1	\[ \frac{1 + \frac{\color{blue}{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{1 + t}}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-commutative [=>] 0.1	\[ \frac{1 + \frac{\frac{\color{blue}{\left(t \cdot 2\right)} \cdot \left(2 \cdot t\right)}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l* [=>] 0.1	\[ \frac{1 + \frac{\frac{\color{blue}{t \cdot \left(2 \cdot \left(2 \cdot t\right)\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r* [=>] 0.1	\[ \frac{1 + \frac{\frac{t \cdot \color{blue}{\left(\left(2 \cdot 2\right) \cdot t\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-commutative [<=] 0.1	\[ \frac{1 + \frac{\frac{t \cdot \color{blue}{\left(t \cdot \left(2 \cdot 2\right)\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      metadata-eval [=>] 0.1	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot \color{blue}{4}\right)}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r/ [=>] 0.1	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \color{blue}{\frac{\frac{2 \cdot t}{1 + t} \cdot \left(2 \cdot t\right)}{1 + t}}} \]
      associate-*l/ [=>] 0.0	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\color{blue}{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{1 + t}}}{1 + t}} \]
      *-commutative [=>] 0.0	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{\color{blue}{\left(t \cdot 2\right)} \cdot \left(2 \cdot t\right)}{1 + t}}{1 + t}} \]
      associate-*l* [=>] 0.0	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{\color{blue}{t \cdot \left(2 \cdot \left(2 \cdot t\right)\right)}}{1 + t}}{1 + t}} \]
      associate-*r* [=>] 0.0	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \color{blue}{\left(\left(2 \cdot 2\right) \cdot t\right)}}{1 + t}}{1 + t}} \]
      *-commutative [<=] 0.0	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \color{blue}{\left(t \cdot \left(2 \cdot 2\right)\right)}}{1 + t}}{1 + t}} \]
      metadata-eval [=>] 0.0	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot \color{blue}{4}\right)}{1 + t}}{1 + t}} \]

   3. Applied egg-rr 0.1

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

   4. Applied egg-rr 0.0

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

   if 1e7 < t

   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. Taylor expanded in t around inf 0.0

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

   3. Simplified 0.0

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

      [Start] 0.0	\[ 0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t} \]
      associate-*r/ [=>] 0.0	\[ 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} \]
      metadata-eval [=>] 0.0	\[ 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222}}{t} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 10000000:\\ \;\;\;\;\frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{1 + \left(1 + \frac{\frac{\left(t \cdot t\right) \cdot -4}{1 + t}}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	2248

\[\begin{array}{l} t_1 := \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 10000000:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	2240

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

Alternative 3
Error	0.3
Cost	1480

\[\begin{array}{l} t_1 := 4 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;\frac{5 + \frac{-8}{t}}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;\frac{1 + \frac{t_1}{1 + t}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	968

\[\begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.33:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{0.037037037037037035}{t \cdot t}\\ \end{array} \]

Alternative 5
Error	0.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -0.55:\\ \;\;\;\;\frac{5 + \frac{-8}{t}}{6 + \frac{-8}{t}}\\ \mathbf{elif}\;t \leq 0.33:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \end{array} \]

Alternative 6
Error	0.4
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 7
Error	0.8
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 8
Error	0.9
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 9
Error	26.1
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022356 
(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))))))