3frac (problem 3.3.3)

Average Error: 10.1 → 9.8

Time: 2.1min

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7391179435211844 \cdot 10^{+91} \lor \neg \left(x \leq 1.1103717260623167 \cdot 10^{+65}\right):\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}} \cdot \sqrt[3]{{\left(\sqrt[3]{\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\right)}^{6}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + x \cdot \left(x + 1\right)}{{x}^{3} - x}\\ \end{array}\]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

↓

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.7391179435211844e+91) (not (<= x 1.1103717260623167e+65)))
   (cbrt
    (pow
     (*
      (cbrt
       (/
        (-
         (* (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)))
         (/ (/ 1.0 (- x 1.0)) (- x 1.0)))
        (- (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))))
      (cbrt
       (pow
        (cbrt
         (/
          (-
           (* (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)))
           (/ (/ 1.0 (- x 1.0)) (- x 1.0)))
          (- (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))))
        6.0)))
     3.0))
   (/
    (+ (* (- x 1.0) (- x (* (+ x 1.0) 2.0))) (* x (+ x 1.0)))
    (- (pow x 3.0) x))))

C

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

↓

double code(double x) {
	double tmp;
	if ((x <= -1.7391179435211844e+91) || !(x <= 1.1103717260623167e+65)) {
		tmp = cbrt(pow((cbrt(((((1.0 / (x + 1.0)) - (2.0 / x)) * ((1.0 / (x + 1.0)) - (2.0 / x))) - ((1.0 / (x - 1.0)) / (x - 1.0))) / (((1.0 / (x + 1.0)) - (2.0 / x)) - (1.0 / (x - 1.0)))) * cbrt(pow(cbrt(((((1.0 / (x + 1.0)) - (2.0 / x)) * ((1.0 / (x + 1.0)) - (2.0 / x))) - ((1.0 / (x - 1.0)) / (x - 1.0))) / (((1.0 / (x + 1.0)) - (2.0 / x)) - (1.0 / (x - 1.0)))), 6.0))), 3.0));
	} else {
		tmp = (((x - 1.0) * (x - ((x + 1.0) * 2.0))) + (x * (x + 1.0))) / (pow(x, 3.0) - x);
	}
	return tmp;
}

Error

Bits error versus x

Target

Original	10.1
Target	0.2
Herbie	9.8

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

Derivation

1. Split input into 2 regimes

2. if x < -1.7391179435211844e91 or 1.1103717260623167e65 < x

   1. Initial program 7.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied flip-+_binary64 7.0

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

   4. Simplified 9.9

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

   5. Simplified 9.9

      \[\leadsto \frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\color{blue}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\]
   6. Using strategy rm

   7. Applied add-cbrt-cube_binary64 7.0

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}} \cdot \frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}\right) \cdot \frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}}}\]

   8. Simplified 7.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}\right)}^{3}}}\]
   9. Using strategy rm

   10. Applied add-cbrt-cube_binary64 7.0

       \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt[3]{\left(\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}} \cdot \frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}\right) \cdot \frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\right)}}^{3}}\]

   11. Simplified 7.0

       \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\color{blue}{{\left(\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}\right)}^{3}}}\right)}^{3}}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt_binary64 7.0

       \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{{\left(\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}\right)}^{3}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}\right)}^{3}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}\right)}^{3}}}\right)}}^{3}}\]

   14. Simplified 7.0

       \[\leadsto \sqrt[3]{{\left(\color{blue}{\sqrt[3]{{\left(\sqrt[3]{\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\right)}^{6}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}\right)}^{3}}}\right)}^{3}}\]

   15. Simplified 7.0

       \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\right)}^{6}} \cdot \color{blue}{\sqrt[3]{\frac{\left(\frac{1}{1 + x} - \frac{2}{x}\right) \cdot \left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}}}\right)}^{3}}\]

   if -1.7391179435211844e91 < x < 1.1103717260623167e65

   1. Initial program 12.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied frac-sub_binary64 12.0

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

   4. Applied frac-add_binary64 11.5

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

   5. Simplified 11.5

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

   6. Simplified 11.5

      \[\leadsto \frac{\left(x - 1\right) \cdot \left(x - \left(1 + x\right) \cdot 2\right) + x \cdot \left(1 + x\right)}{\color{blue}{{x}^{3} + \left(-x\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7391179435211844 \cdot 10^{+91} \lor \neg \left(x \leq 1.1103717260623167 \cdot 10^{+65}\right):\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}} \cdot \sqrt[3]{{\left(\sqrt[3]{\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\right)}^{6}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + x \cdot \left(x + 1\right)}{{x}^{3} - x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020285 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))