Kahan p13 Example 1

Average Accuracy: 99.9% → 99.8%

Time: 11.6s

Precision: binary64

Cost: 2248

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+155}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 20000000:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

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 (/ (/ (* t (* t 4.0)) (+ 1.0 t)) (+ 1.0 t))))
   (if (<= t -1e+155)
     0.8333333333333334
     (if (<= t 20000000.0)
       (/ (+ 1.0 t_1) (+ 2.0 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 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t);
	double tmp;
	if (t <= -1e+155) {
		tmp = 0.8333333333333334;
	} else if (t <= 20000000.0) {
		tmp = (1.0 + t_1) / (2.0 + 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 = ((t * (t * 4.0d0)) / (1.0d0 + t)) / (1.0d0 + t)
    if (t <= (-1d+155)) then
        tmp = 0.8333333333333334d0
    else if (t <= 20000000.0d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + 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 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t);
	double tmp;
	if (t <= -1e+155) {
		tmp = 0.8333333333333334;
	} else if (t <= 20000000.0) {
		tmp = (1.0 + t_1) / (2.0 + 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 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t)
	tmp = 0
	if t <= -1e+155:
		tmp = 0.8333333333333334
	elif t <= 20000000.0:
		tmp = (1.0 + t_1) / (2.0 + 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(Float64(Float64(t * Float64(t * 4.0)) / Float64(1.0 + t)) / Float64(1.0 + t))
	tmp = 0.0
	if (t <= -1e+155)
		tmp = 0.8333333333333334;
	elseif (t <= 20000000.0)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + 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 = ((t * (t * 4.0)) / (1.0 + t)) / (1.0 + t);
	tmp = 0.0;
	if (t <= -1e+155)
		tmp = 0.8333333333333334;
	elseif (t <= 20000000.0)
		tmp = (1.0 + t_1) / (2.0 + 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[(N[(N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e+155], 0.8333333333333334, If[LessEqual[t, 20000000.0], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.00000000000000001e155

   1. Initial program 99.9%

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

   2. Simplified 100.0%

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

      [Start] 99.9	\[ \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-/l* [=>] 99.9	\[ \frac{1 + \color{blue}{\frac{2}{\frac{1 + t}{t}}} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-/l* [=>] 99.9	\[ \frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \color{blue}{\frac{2}{\frac{1 + t}{t}}}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-/l* [=>] 99.9	\[ \frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \frac{2}{\frac{1 + t}{t}}}{2 + \color{blue}{\frac{2}{\frac{1 + t}{t}}} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-/l* [=>] 100.0	\[ \frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \frac{2}{\frac{1 + t}{t}}}{2 + \frac{2}{\frac{1 + t}{t}} \cdot \color{blue}{\frac{2}{\frac{1 + t}{t}}}} \]

   3. Taylor expanded in t around inf 100.0%

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

   if -1.00000000000000001e155 < t < 2e7

   1. Initial program 100.0%

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

   2. Simplified 99.7%

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

      [Start] 100.0	\[ \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r/ [=>] 100.0	\[ \frac{1 + \color{blue}{\frac{\frac{2 \cdot t}{1 + t} \cdot \left(2 \cdot t\right)}{1 + t}}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l/ [=>] 99.7	\[ \frac{1 + \frac{\color{blue}{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{1 + t}}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-commutative [=>] 99.7	\[ \frac{1 + \frac{\frac{\color{blue}{\left(t \cdot 2\right)} \cdot \left(2 \cdot t\right)}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l* [=>] 99.7	\[ \frac{1 + \frac{\frac{\color{blue}{t \cdot \left(2 \cdot \left(2 \cdot t\right)\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r* [=>] 99.7	\[ \frac{1 + \frac{\frac{t \cdot \color{blue}{\left(\left(2 \cdot 2\right) \cdot t\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-commutative [<=] 99.7	\[ \frac{1 + \frac{\frac{t \cdot \color{blue}{\left(t \cdot \left(2 \cdot 2\right)\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      metadata-eval [=>] 99.7	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot \color{blue}{4}\right)}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r/ [=>] 99.7	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \color{blue}{\frac{\frac{2 \cdot t}{1 + t} \cdot \left(2 \cdot t\right)}{1 + t}}} \]

   if 2e7 < t

   1. Initial program 99.8%

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

   2. Simplified 100.0%

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

      [Start] 99.8	\[ \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-/l* [=>] 99.8	\[ \frac{1 + \color{blue}{\frac{2}{\frac{1 + t}{t}}} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-/l* [=>] 99.8	\[ \frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \color{blue}{\frac{2}{\frac{1 + t}{t}}}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-/l* [=>] 99.8	\[ \frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \frac{2}{\frac{1 + t}{t}}}{2 + \color{blue}{\frac{2}{\frac{1 + t}{t}}} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-/l* [=>] 100.0	\[ \frac{1 + \frac{2}{\frac{1 + t}{t}} \cdot \frac{2}{\frac{1 + t}{t}}}{2 + \frac{2}{\frac{1 + t}{t}} \cdot \color{blue}{\frac{2}{\frac{1 + t}{t}}}} \]

   3. Taylor expanded in t around inf 100.0%

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

   4. Simplified 100.0%

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

      [Start] 100.0	\[ 0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t} \]
      associate-*r/ [=>] 100.0	\[ 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} \]
      metadata-eval [=>] 100.0	\[ 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222}}{t} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+155}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 20000000:\\ \;\;\;\;\frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	2240

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

Alternative 2
Accuracy	99.3%
Cost	1481

\[\begin{array}{l} \mathbf{if}\;t \leq -0.62 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;\left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(t \cdot t\right) \cdot \left(4 + t \cdot -8\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	1481

\[\begin{array}{l} \mathbf{if}\;t \leq -0.6 \lor \neg \left(t \leq 0.25\right):\\ \;\;\;\;\left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - \frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{-2 + t \cdot \left(t \cdot -4\right)}\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.235\right):\\ \;\;\;\;\left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 5
Accuracy	99.2%
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 6
Accuracy	98.6%
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 7
Accuracy	98.4%
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 8
Accuracy	59.2%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023139 
(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))))))