3frac (problem 3.3.3)

Average Error: 9.6 → 0.4

Time: 6.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -30.927103867772345:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \sqrt[3]{\frac{1}{\left(x - 1\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 0:\\ \;\;\;\;\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(1 + x\right) \cdot 2\right) + x \cdot \left(1 + x\right)}{\left(x - 1\right) \cdot \left(x \cdot \left(1 + x\right)\right)}\\ \end{array}\]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<=
      (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))
      -30.927103867772345)
   (+
    (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x))
    (cbrt (/ 1.0 (* (- x 1.0) (* (- x 1.0) (- x 1.0))))))
   (if (<= (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) 0.0)
     (+ (+ (/ (/ 2.0 x) (* x x)) (/ 2.0 (pow x 5.0))) (/ 2.0 (pow x 7.0)))
     (/
      (+ (* (- x 1.0) (- x (* (+ 1.0 x) 2.0))) (* x (+ 1.0 x)))
      (* (- x 1.0) (* x (+ 1.0 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 ((((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x - 1.0))) <= -30.927103867772345) {
		tmp = ((1.0 / (1.0 + x)) - (2.0 / x)) + cbrt(1.0 / ((x - 1.0) * ((x - 1.0) * (x - 1.0))));
	} else if ((((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x - 1.0))) <= 0.0) {
		tmp = (((2.0 / x) / (x * x)) + (2.0 / pow(x, 5.0))) + (2.0 / pow(x, 7.0));
	} else {
		tmp = (((x - 1.0) * (x - ((1.0 + x) * 2.0))) + (x * (1.0 + x))) / ((x - 1.0) * (x * (1.0 + x)));
	}
	return tmp;
}

Error

Bits error versus x

Target

Original	9.6
Target	0.2
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -30.927103867772345

   1. Initial program 0.0

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

   3. Applied add-cbrt-cube_binary64_2501 0.0

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

   4. Applied add-cbrt-cube_binary64_2501 0.0

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

   5. Applied cbrt-undiv_binary64_2499 0.0

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

   if -30.927103867772345 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 18.8

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

   2. Taylor expanded around inf 0.7

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}\right)}\]

   3. Simplified 0.7

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}}\]
   4. Using strategy rm

   5. Applied cube-mult_binary64_2495 0.7

      \[\leadsto \left(\frac{2}{\color{blue}{x \cdot \left(x \cdot x\right)}} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\]

   6. Applied associate-/r*_binary64_2409 0.3

      \[\leadsto \left(\color{blue}{\frac{\frac{2}{x}}{x \cdot x}} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\]

   if 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 1.3

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

   3. Applied frac-sub_binary64_2474 1.3

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

   4. Applied frac-add_binary64_2473 0.8

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -30.927103867772345:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \sqrt[3]{\frac{1}{\left(x - 1\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 0:\\ \;\;\;\;\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(1 + x\right) \cdot 2\right) + x \cdot \left(1 + x\right)}{\left(x - 1\right) \cdot \left(x \cdot \left(1 + x\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021032 
(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))))