Kahan p13 Example 2

Average Error: 0.0 → 0.0

Time: 13.3s

Precision: binary64

Cost: 2240

Math

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

↓

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

FPCore

(FPCore (t)
 :precision binary64
 (/
  (+
   1.0
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
  (+
   2.0
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))

↓

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ 2.0 (+ 1.0 t)))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))));
}

↓

double code(double t) {
	double t_1 = 2.0 - (2.0 / (1.0 + t));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (1.0d0 + ((2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))) * (2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))))) / (2.0d0 + ((2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))) * (2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t))))))
end function

↓

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - (2.0d0 / (1.0d0 + t))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))));
}

↓

public static double code(double t) {
	double t_1 = 2.0 - (2.0 / (1.0 + t));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))))

↓

def code(t):
	t_1 = 2.0 - (2.0 / (1.0 + t))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	return Float64(Float64(1.0 + Float64(Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) * Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))))) / Float64(2.0 + Float64(Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) * Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))))))
end

↓

function code(t)
	t_1 = Float64(2.0 - Float64(2.0 / Float64(1.0 + t)))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	tmp = (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))));
end

↓

function tmp = code(t)
	t_1 = 2.0 - (2.0 / (1.0 + t));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

code[t_] := N[(N[(1.0 + N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_] := Block[{t$95$1 = N[(2.0 - N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\frac{1 + \left(2 - \frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}\right) \cdot \left(2 - \frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}\right)}{2 + \left(2 - \frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}\right) \cdot \left(2 - \frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}\right)}} \]
   Proof
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t))) (-.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t))))) (+.f64 2 (*.f64 (-.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t))) (-.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (-.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t))))) (+.f64 2 (*.f64 (-.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t))) (-.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t)))))): 1 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))))) (+.f64 2 (*.f64 (-.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t))) (-.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 2 (*.f64 (-.f64 2 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t))))) (-.f64 2 (/.f64 2 (*.f64 (+.f64 1 (/.f64 1 t)) t)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 1 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))) (+.f64 2 (*.f64 (-.f64 2 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))) (-.f64 2 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 2 t) (+.f64 1 (/.f64 1 t)))))))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in t around 0 0.0

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

4. Simplified 0.0

   \[\leadsto \frac{1 + \left(2 - \frac{2}{\color{blue}{t + 1}}\right) \cdot \left(2 - \frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}\right)}{2 + \left(2 - \frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}\right) \cdot \left(2 - \frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}\right)} \]
   Proof
   (+.f64 t 1): 0 points increase in error, 0 points decrease in error
   (Rewrite=> +-commutative_binary64 (+.f64 1 t)): 0 points increase in error, 0 points decrease in error

5. Taylor expanded in t around 0 0.0

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

6. Simplified 0.0

   \[\leadsto \frac{1 + \left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{\color{blue}{t + 1}}\right)}{2 + \left(2 - \frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}\right) \cdot \left(2 - \frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}\right)} \]
   Proof
   (+.f64 t 1): 0 points increase in error, 0 points decrease in error
   (Rewrite=> +-commutative_binary64 (+.f64 1 t)): 0 points increase in error, 0 points decrease in error

7. Taylor expanded in t around 0 0.0

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

8. Simplified 0.0

   \[\leadsto \frac{1 + \left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)}{2 + \left(2 - \frac{2}{\color{blue}{t + 1}}\right) \cdot \left(2 - \frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}\right)} \]
   Proof
   (+.f64 t 1): 0 points increase in error, 0 points decrease in error
   (Rewrite=> +-commutative_binary64 (+.f64 1 t)): 0 points increase in error, 0 points decrease in error

9. Taylor expanded in t around 0 0.0

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

10. Simplified 0.0

    \[\leadsto \frac{1 + \left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)}{2 + \left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{\color{blue}{t + 1}}\right)} \]
    Proof
    (+.f64 t 1): 0 points increase in error, 0 points decrease in error
    (Rewrite=> +-commutative_binary64 (+.f64 1 t)): 0 points increase in error, 0 points decrease in error

11. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.4
Cost	1480

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t} + \frac{1}{\left(1.2 + \frac{0.32}{t}\right) + \frac{0.08533333333333333}{t \cdot t}}\\ \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.38:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.33:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.037037037037037035}{t \cdot t} + \left(1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	1096

\[\begin{array}{l} t_1 := \frac{0.037037037037037035}{t \cdot t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;t_1 + \left(-0.16666666666666666 + \left(1 + \frac{-0.2222222222222222}{t}\right)\right)\\ \mathbf{elif}\;t \leq 0.33:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	840

\[\begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.33:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 6
Error	0.5
Cost	584

\[\begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.78:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.55:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	0.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 8
Error	26.3
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022330 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))