Kahan p13 Example 1

Average Error: 0.0 → 0.0

Time: 13.1s

Precision: binary64

Cost: 2248

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{4 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}\\ \mathbf{if}\;t \leq -2.3403016542663982 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1443494.361181734:\\ \;\;\;\;\frac{1 + t_1}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(1 + \left(\frac{-0.2222222222222222}{t} + -1\right)\right)\\ \end{array} \]

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 (/ (/ (* 4.0 (* t t)) (+ 1.0 t)) (+ 1.0 t))))
   (if (<= t -2.3403016542663982e+154)
     0.8333333333333334
     (if (<= t 1443494.361181734)
       (/ (+ 1.0 t_1) (+ 2.0 t_1))
       (+ 0.8333333333333334 (+ 1.0 (+ (/ -0.2222222222222222 t) -1.0)))))))

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 = ((4.0 * (t * t)) / (1.0 + t)) / (1.0 + t);
	double tmp;
	if (t <= -2.3403016542663982e+154) {
		tmp = 0.8333333333333334;
	} else if (t <= 1443494.361181734) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 + (1.0 + ((-0.2222222222222222 / t) + -1.0));
	}
	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 = ((4.0d0 * (t * t)) / (1.0d0 + t)) / (1.0d0 + t)
    if (t <= (-2.3403016542663982d+154)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1443494.361181734d0) then
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    else
        tmp = 0.8333333333333334d0 + (1.0d0 + (((-0.2222222222222222d0) / t) + (-1.0d0)))
    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 = ((4.0 * (t * t)) / (1.0 + t)) / (1.0 + t);
	double tmp;
	if (t <= -2.3403016542663982e+154) {
		tmp = 0.8333333333333334;
	} else if (t <= 1443494.361181734) {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	} else {
		tmp = 0.8333333333333334 + (1.0 + ((-0.2222222222222222 / t) + -1.0));
	}
	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 = ((4.0 * (t * t)) / (1.0 + t)) / (1.0 + t)
	tmp = 0
	if t <= -2.3403016542663982e+154:
		tmp = 0.8333333333333334
	elif t <= 1443494.361181734:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	else:
		tmp = 0.8333333333333334 + (1.0 + ((-0.2222222222222222 / t) + -1.0))
	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(4.0 * Float64(t * t)) / Float64(1.0 + t)) / Float64(1.0 + t))
	tmp = 0.0
	if (t <= -2.3403016542663982e+154)
		tmp = 0.8333333333333334;
	elseif (t <= 1443494.361181734)
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	else
		tmp = Float64(0.8333333333333334 + Float64(1.0 + Float64(Float64(-0.2222222222222222 / t) + -1.0)));
	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 = ((4.0 * (t * t)) / (1.0 + t)) / (1.0 + t);
	tmp = 0.0;
	if (t <= -2.3403016542663982e+154)
		tmp = 0.8333333333333334;
	elseif (t <= 1443494.361181734)
		tmp = (1.0 + t_1) / (2.0 + t_1);
	else
		tmp = 0.8333333333333334 + (1.0 + ((-0.2222222222222222 / t) + -1.0));
	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[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.3403016542663982e+154], 0.8333333333333334, If[LessEqual[t, 1443494.361181734], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(1.0 + N[(N[(-0.2222222222222222 / t), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.34030165426639823e154

   1. Initial program 0.2

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

   2. Taylor expanded in t around inf 0

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

   3. Simplified 0

      \[\leadsto \color{blue}{0.8333333333333334 + \frac{-0.2222222222222222}{t}} \]
      Proof
      (+.f64 5/6 (/.f64 -2/9 t)): 0 points increase in error, 0 points decrease in error
      (+.f64 5/6 (/.f64 (Rewrite<= metadata-eval (neg.f64 2/9)) t)): 0 points increase in error, 0 points decrease in error
      (+.f64 5/6 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2/9 t)))): 0 points increase in error, 0 points decrease in error
      (+.f64 5/6 (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2/9 1)) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 5/6 (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2/9 (/.f64 1 t))))): 17 points increase in error, 16 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 5/6 (*.f64 2/9 (/.f64 1 t)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 0

      \[\leadsto 0.8333333333333334 + \color{blue}{\left(\left(1 + \frac{-0.2222222222222222}{t}\right) - 1\right)} \]

   5. Applied egg-rr 0

      \[\leadsto 0.8333333333333334 + \color{blue}{\left(\left(\frac{-0.2222222222222222}{t} + -1\right) + 1\right)} \]

   6. Taylor expanded in t around inf 0

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

   if -2.34030165426639823e154 < t < 1443494.3611817339

   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 0.1

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

   3. Applied egg-rr 0.1

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

   if 1443494.3611817339 < 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
      (+.f64 5/6 (/.f64 -2/9 t)): 0 points increase in error, 0 points decrease in error
      (+.f64 5/6 (/.f64 (Rewrite<= metadata-eval (neg.f64 2/9)) t)): 0 points increase in error, 0 points decrease in error
      (+.f64 5/6 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2/9 t)))): 0 points increase in error, 0 points decrease in error
      (+.f64 5/6 (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2/9 1)) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 5/6 (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2/9 (/.f64 1 t))))): 17 points increase in error, 16 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 5/6 (*.f64 2/9 (/.f64 1 t)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 0.0

      \[\leadsto 0.8333333333333334 + \color{blue}{\left(\left(1 + \frac{-0.2222222222222222}{t}\right) - 1\right)} \]

   5. Applied egg-rr 0.0

      \[\leadsto 0.8333333333333334 + \color{blue}{\left(\left(\frac{-0.2222222222222222}{t} + -1\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3403016542663982 \cdot 10^{+154}:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1443494.361181734:\\ \;\;\;\;\frac{1 + \frac{\frac{4 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}{2 + \frac{\frac{4 \cdot \left(t \cdot t\right)}{1 + t}}{1 + t}}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(1 + \left(\frac{-0.2222222222222222}{t} + -1\right)\right)\\ \end{array} \]

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	0.4
Cost	2120

\[\begin{array}{l} \mathbf{if}\;t \leq -53.43526821078497:\\ \;\;\;\;\left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{elif}\;t \leq 0.0017100900223120843:\\ \;\;\;\;\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \left(t \cdot \left(2 + t \cdot -2\right)\right)}{2 + t \cdot \left(t \cdot \left(4 + t \cdot -8\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1.2 + \left(\frac{0.32}{t} + \frac{0.032}{t \cdot t}\right)}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	1992

\[\begin{array}{l} \mathbf{if}\;t \leq -53.43526821078497:\\ \;\;\;\;\left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{elif}\;t \leq 0.0017100900223120843:\\ \;\;\;\;\frac{1 + \left(t \cdot t\right) \cdot \left(4 + t \cdot \left(-8 + t \cdot 12\right)\right)}{2 + t \cdot \left(t \cdot \left(4 + t \cdot -8\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1.2 + \left(\frac{0.32}{t} + \frac{0.032}{t \cdot t}\right)}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	1736

\[\begin{array}{l} t_1 := 4 + t \cdot -8\\ \mathbf{if}\;t \leq -53.43526821078497:\\ \;\;\;\;\left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{elif}\;t \leq 0.0017100900223120843:\\ \;\;\;\;\frac{1 + \left(t \cdot t\right) \cdot t_1}{2 + t \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1.2 + \left(\frac{0.32}{t} + \frac{0.032}{t \cdot t}\right)}\\ \end{array} \]

Alternative 5
Error	0.5
Cost	1480

\[\begin{array}{l} \mathbf{if}\;t \leq -53.43526821078497:\\ \;\;\;\;\left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{elif}\;t \leq 0.0017100900223120843:\\ \;\;\;\;\frac{1 + \left(t \cdot t\right) \cdot \left(4 + t \cdot -8\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1.2 + \left(\frac{0.32}{t} + \frac{0.032}{t \cdot t}\right)}\\ \end{array} \]

Alternative 6
Error	0.6
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq -53.43526821078497:\\ \;\;\;\;\left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{elif}\;t \leq 0.0017100900223120843:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1.2 + \left(\frac{0.32}{t} + \frac{0.032}{t \cdot t}\right)}\\ \end{array} \]

Alternative 7
Error	0.6
Cost	968

\[\begin{array}{l} t_1 := \left(0.8333333333333334 + \frac{-0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{if}\;t \leq -53.43526821078497:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.0017100900223120843:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	0.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;t \leq -53.43526821078497:\\ \;\;\;\;\frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{elif}\;t \leq 0.0017100900223120843:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(1 + \left(\frac{-0.2222222222222222}{t} + -1\right)\right)\\ \end{array} \]

Alternative 9
Error	0.7
Cost	712

\[\begin{array}{l} t_1 := \frac{1}{1.2 + \frac{0.32}{t}}\\ \mathbf{if}\;t \leq -53.43526821078497:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.0017100900223120843:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	0.7
Cost	584

\[\begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -53.43526821078497:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.0017100900223120843:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	0.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -53.43526821078497:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.0017100900223120843:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 12
Error	26.0
Cost	64

\[0.8333333333333334 \]

Reproduce

herbie shell --seed 2022295 
(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))))))