Kahan p13 Example 2

Average Error: 0.0 → 0.0

Time: 23.0s

Precision: binary64

Cost: 3136

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t_1 \cdot \left(t_1 + -4\right)\\ \frac{1 + \left(t_2 - -4\right)}{t_2 - -6} \end{array} \]

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 t) (+ 1.0 (/ 1.0 t)))) (t_2 (* t_1 (+ t_1 -4.0))))
   (/ (+ 1.0 (- t_2 -4.0)) (- t_2 -6.0))))

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 / t) / (1.0 + (1.0 / t));
	double t_2 = t_1 * (t_1 + -4.0);
	return (1.0 + (t_2 - -4.0)) / (t_2 - -6.0);
}

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 / t) / (1.0d0 + (1.0d0 / t))
    t_2 = t_1 * (t_1 + (-4.0d0))
    code = (1.0d0 + (t_2 - (-4.0d0))) / (t_2 - (-6.0d0))
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 / t) / (1.0 + (1.0 / t));
	double t_2 = t_1 * (t_1 + -4.0);
	return (1.0 + (t_2 - -4.0)) / (t_2 - -6.0);
}

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 / t) / (1.0 + (1.0 / t))
	t_2 = t_1 * (t_1 + -4.0)
	return (1.0 + (t_2 - -4.0)) / (t_2 - -6.0)

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(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))
	t_2 = Float64(t_1 * Float64(t_1 + -4.0))
	return Float64(Float64(1.0 + Float64(t_2 - -4.0)) / Float64(t_2 - -6.0))
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 / t) / (1.0 + (1.0 / t));
	t_2 = t_1 * (t_1 + -4.0);
	tmp = (1.0 + (t_2 - -4.0)) / (t_2 - -6.0);
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[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$1 + -4.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(t$95$2 - -4.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - -6.0), $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. Applied egg-rr 0.0

   \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -2\right) - 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) - -6}} \]

3. Simplified 0.0

   \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -6}} \]
   Proof

   [Start] 0.0	\[ \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -2\right) - 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) - -6} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\left(\color{blue}{\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}} - 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) - -6} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\left(\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} - \color{blue}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot 2}\right) - -6} \]
   rational_best_oopsla_all_46_json_45_simplify-102 [=>] 0.0	\[ \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -2\right) - 2\right)} - -6} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.0	\[ \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\color{blue}{\left(-2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} - 2\right) - -6} \]
   rational_best_oopsla_all_46_json_45_simplify-107 [=>] 0.0	\[ \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \color{blue}{\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + \left(-2 - 2\right)\right)} - -6} \]
   metadata-eval [=>] 0.0	\[ \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + \color{blue}{-4}\right) - -6} \]

4. Applied egg-rr 0.0

   \[\leadsto \frac{1 + \color{blue}{\left(\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -2\right) - 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) - -4\right)}}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -6} \]

5. Simplified 0.0

   \[\leadsto \frac{1 + \color{blue}{\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -4\right)}}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -6} \]
   Proof

   [Start] 0.0	\[ \frac{1 + \left(\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -2\right) - 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) - -4\right)}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -6} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ \frac{1 + \left(\left(\color{blue}{\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}} - 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) - -4\right)}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -6} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ \frac{1 + \left(\left(\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} - \color{blue}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot 2}\right) - -4\right)}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -6} \]
   rational_best_oopsla_all_46_json_45_simplify-102 [=>] 0.0	\[ \frac{1 + \left(\color{blue}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -2\right) - 2\right)} - -4\right)}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -6} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.0	\[ \frac{1 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\color{blue}{\left(-2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} - 2\right) - -4\right)}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -6} \]
   rational_best_oopsla_all_46_json_45_simplify-107 [=>] 0.0	\[ \frac{1 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \color{blue}{\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + \left(-2 - 2\right)\right)} - -4\right)}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -6} \]
   metadata-eval [=>] 0.0	\[ \frac{1 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + \color{blue}{-4}\right) - -4\right)}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -6} \]

6. Final simplification 0.0

   \[\leadsto \frac{1 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -4\right)}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + -4\right) - -6} \]

Alternatives

Alternative 1
Error	0.4
Cost	3396

\[\begin{array}{l} t_1 := \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := 2 - t_1\\ \mathbf{if}\;t_1 \leq 0.5:\\ \;\;\;\;\frac{1 + t_2 \cdot t_2}{t_1 \cdot \left(\frac{2}{t} + -4\right) - -6}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \left(4 - 2 \cdot t_1\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	3136

\[\begin{array}{l} t_1 := \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := 2 - t_1\\ \frac{1 + t_2 \cdot t_2}{t_1 \cdot \left(t_1 + -4\right) - -6} \end{array} \]

Alternative 3
Error	0.5
Cost	2244

\[\begin{array}{l} t_1 := \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;t_1 \leq 0.5:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t \cdot \left(4 - 2 \cdot t_1\right)}{2 + t \cdot \left(t \cdot 4\right)}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	1480

\[\begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222}{t}\\ t_2 := \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)\\ \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;\frac{1 + t_2}{2 + t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.6
Cost	584

\[\begin{array}{l} t_1 := 0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.485:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	1.0
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 7
Error	26.2
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023090 
(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))))))))