Kahan p13 Example 1

Average Error: 0.0 → 0.1

Time: 21.2s

Precision: binary64

Cost: 2248

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+155}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 200000000:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-5 + \frac{8}{t}}{-6 + \frac{8}{t}}\\ \end{array} \]

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 200000000.0)
       (/ (+ 1.0 t_1) (+ 2.0 t_1))
       (/ (+ -5.0 (/ 8.0 t)) (+ -6.0 (/ 8.0 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 <= 200000000.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = (-5.0 + (8.0 / t)) / (-6.0 + (8.0 / 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 <= 200000000.0d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = ((-5.0d0) + (8.0d0 / t)) / ((-6.0d0) + (8.0d0 / 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 <= 200000000.0) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = (-5.0 + (8.0 / t)) / (-6.0 + (8.0 / 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 <= 200000000.0:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = (-5.0 + (8.0 / t)) / (-6.0 + (8.0 / 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 <= 200000000.0)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(-5.0 + Float64(8.0 / t)) / Float64(-6.0 + Float64(8.0 / 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 <= 200000000.0)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = (-5.0 + (8.0 / t)) / (-6.0 + (8.0 / 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, 200000000.0], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(-5.0 + N[(8.0 / t), $MachinePrecision]), $MachinePrecision] / N[(-6.0 + N[(8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.00000000000000001e155

   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 0.1

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

   3. Taylor expanded in t around inf 0

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

   if -1.00000000000000001e155 < t < 2e8

   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 0.2

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}} \]
      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/ [=>] 0.2	\[ \frac{1 + \frac{\color{blue}{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{1 + t}}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-commutative [=>] 0.2	\[ \frac{1 + \frac{\frac{\color{blue}{\left(t \cdot 2\right)} \cdot \left(2 \cdot t\right)}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l* [=>] 0.2	\[ \frac{1 + \frac{\frac{\color{blue}{t \cdot \left(2 \cdot \left(2 \cdot t\right)\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r* [=>] 0.2	\[ \frac{1 + \frac{\frac{t \cdot \color{blue}{\left(\left(2 \cdot 2\right) \cdot t\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-commutative [<=] 0.2	\[ \frac{1 + \frac{\frac{t \cdot \color{blue}{\left(t \cdot \left(2 \cdot 2\right)\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      metadata-eval [=>] 0.2	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot \color{blue}{4}\right)}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r/ [=>] 0.2	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \color{blue}{\frac{\frac{2 \cdot t}{1 + t} \cdot \left(2 \cdot t\right)}{1 + t}}} \]

   if 2e8 < t

   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. Simplified 33.8

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

      [Start] 0.1	\[ \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r/ [=>] 0.2	\[ \frac{1 + \color{blue}{\frac{\frac{2 \cdot t}{1 + t} \cdot \left(2 \cdot t\right)}{1 + t}}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l/ [=>] 33.9	\[ \frac{1 + \frac{\color{blue}{\frac{\left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}{1 + t}}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-commutative [=>] 33.9	\[ \frac{1 + \frac{\frac{\color{blue}{\left(t \cdot 2\right)} \cdot \left(2 \cdot t\right)}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*l* [=>] 33.9	\[ \frac{1 + \frac{\frac{\color{blue}{t \cdot \left(2 \cdot \left(2 \cdot t\right)\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r* [=>] 33.9	\[ \frac{1 + \frac{\frac{t \cdot \color{blue}{\left(\left(2 \cdot 2\right) \cdot t\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      *-commutative [<=] 33.9	\[ \frac{1 + \frac{\frac{t \cdot \color{blue}{\left(t \cdot \left(2 \cdot 2\right)\right)}}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      metadata-eval [=>] 33.9	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot \color{blue}{4}\right)}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
      associate-*r/ [=>] 33.9	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \color{blue}{\frac{\frac{2 \cdot t}{1 + t} \cdot \left(2 \cdot t\right)}{1 + t}}} \]

   3. Taylor expanded in t around inf 33.9

      \[\leadsto \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \color{blue}{\left(4 - 8 \cdot \frac{1}{t}\right)}} \]

   4. Simplified 33.9

      \[\leadsto \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \color{blue}{\left(4 - \frac{8}{t}\right)}} \]
      Proof

      [Start] 33.9	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \left(4 - 8 \cdot \frac{1}{t}\right)} \]
      associate-*r/ [=>] 33.9	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \left(4 - \color{blue}{\frac{8 \cdot 1}{t}}\right)} \]
      metadata-eval [=>] 33.9	\[ \frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \left(4 - \frac{\color{blue}{8}}{t}\right)} \]

   5. Taylor expanded in t around inf 0.0

      \[\leadsto \frac{1 + \color{blue}{\left(4 - 8 \cdot \frac{1}{t}\right)}}{2 + \left(4 - \frac{8}{t}\right)} \]

   6. Simplified 0.0

      \[\leadsto \frac{1 + \color{blue}{\left(4 - \frac{8}{t}\right)}}{2 + \left(4 - \frac{8}{t}\right)} \]
      Proof

      [Start] 0.0	\[ \frac{1 + \left(4 - 8 \cdot \frac{1}{t}\right)}{2 + \left(4 - \frac{8}{t}\right)} \]
      associate-*r/ [=>] 0.0	\[ \frac{1 + \left(4 - \color{blue}{\frac{8 \cdot 1}{t}}\right)}{2 + \left(4 - \frac{8}{t}\right)} \]
      metadata-eval [=>] 0.0	\[ \frac{1 + \left(4 - \frac{\color{blue}{8}}{t}\right)}{2 + \left(4 - \frac{8}{t}\right)} \]

   7. Applied egg-rr 1.0

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-5 + \frac{8}{t}}{-6 - \frac{-8}{t}}\right)} - 1} \]

   8. Simplified 0.0

      \[\leadsto \color{blue}{\frac{-5 - \frac{-8}{t}}{\frac{8}{t} + -6}} \]
      Proof

      [Start] 1.0	\[ e^{\mathsf{log1p}\left(\frac{-5 + \frac{8}{t}}{-6 - \frac{-8}{t}}\right)} - 1 \]
      expm1-def [=>] 1.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-5 + \frac{8}{t}}{-6 - \frac{-8}{t}}\right)\right)} \]
      expm1-log1p [=>] 0.0	\[ \color{blue}{\frac{-5 + \frac{8}{t}}{-6 - \frac{-8}{t}}} \]
      metadata-eval [<=] 0.0	\[ \frac{-5 + \frac{\color{blue}{--8}}{t}}{-6 - \frac{-8}{t}} \]
      distribute-neg-frac [<=] 0.0	\[ \frac{-5 + \color{blue}{\left(-\frac{-8}{t}\right)}}{-6 - \frac{-8}{t}} \]
      unsub-neg [=>] 0.0	\[ \frac{\color{blue}{-5 - \frac{-8}{t}}}{-6 - \frac{-8}{t}} \]
      sub-neg [=>] 0.0	\[ \frac{-5 - \frac{-8}{t}}{\color{blue}{-6 + \left(-\frac{-8}{t}\right)}} \]
      +-commutative [=>] 0.0	\[ \frac{-5 - \frac{-8}{t}}{\color{blue}{\left(-\frac{-8}{t}\right) + -6}} \]
      distribute-neg-frac [=>] 0.0	\[ \frac{-5 - \frac{-8}{t}}{\color{blue}{\frac{--8}{t}} + -6} \]
      metadata-eval [=>] 0.0	\[ \frac{-5 - \frac{-8}{t}}{\frac{\color{blue}{8}}{t} + -6} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+155}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 200000000:\\ \;\;\;\;\frac{1 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{t \cdot \left(t \cdot 4\right)}{1 + t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-5 + \frac{8}{t}}{-6 + \frac{8}{t}}\\ \end{array} \]

Alternatives

Alternative 1
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 2
Error	0.8
Cost	1732

\[\begin{array}{l} t_1 := 4 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{1 + \frac{t_1}{1 + t}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -0.81 \lor \neg \left(t \leq 0.235\right):\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 4
Error	0.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -0.78 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 5
Error	0.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 6
Error	0.8
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 7
Error	26.2
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023038 
(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))))))