2cbrt (problem 3.3.4)

Average Error: 29.7 → 0.4

Time: 7.3s

Precision: binary64

Math

\[\sqrt[3]{x + 1} - \sqrt[3]{x} \]

↓

\[\begin{array}{l} t_0 := \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 - \frac{0.1111111111111111}{x}\right) + 0.06172839506172839 \cdot \frac{\sqrt[3]{x}}{{x}^{3}}\\ \mathbf{if}\;x \leq -4527.3591884227435:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4285.791217009184:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt[3]{x + 1}\\ t_2 := \sqrt[3]{\sqrt[3]{x}}\\ t_3 := t_2 \cdot t_2\\ t_4 := t_2 \cdot t_3\\ \sqrt{e^{\log \left(t_1 - \sqrt[3]{x}\right)}} \cdot \sqrt{\mathsf{fma}\left(\sqrt[3]{t_1 \cdot t_1}, \sqrt[3]{t_1}, -t_4\right) + \mathsf{fma}\left(-t_2, t_3, t_4\right)} \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+
          (* (/ (cbrt x) x) (- 0.3333333333333333 (/ 0.1111111111111111 x)))
          (* 0.06172839506172839 (/ (cbrt x) (pow x 3.0))))))
   (if (<= x -4527.3591884227435)
     t_0
     (if (<= x 4285.791217009184)
       (let* ((t_1 (cbrt (+ x 1.0)))
              (t_2 (cbrt (cbrt x)))
              (t_3 (* t_2 t_2))
              (t_4 (* t_2 t_3)))
         (*
          (sqrt (exp (log (- t_1 (cbrt x)))))
          (sqrt
           (+
            (fma (cbrt (* t_1 t_1)) (cbrt t_1) (- t_4))
            (fma (- t_2) t_3 t_4)))))
       t_0))))

C

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

↓

double code(double x) {
	double t_0 = ((cbrt(x) / x) * (0.3333333333333333 - (0.1111111111111111 / x))) + (0.06172839506172839 * (cbrt(x) / pow(x, 3.0)));
	double tmp;
	if (x <= -4527.3591884227435) {
		tmp = t_0;
	} else if (x <= 4285.791217009184) {
		double t_1 = cbrt(x + 1.0);
		double t_2 = cbrt(cbrt(x));
		double t_3 = t_2 * t_2;
		double t_4 = t_2 * t_3;
		tmp = sqrt(exp(log(t_1 - cbrt(x)))) * sqrt(fma(cbrt(t_1 * t_1), cbrt(t_1), -t_4) + fma(-t_2, t_3, t_4));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -4527.3591884227435 or 4285.791217009184 < x

   1. Initial program 60.2

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]

   2. Applied add-cube-cbrt_binary64 60.3

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

   3. Applied cbrt-prod_binary64 60.4

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \]

   4. Taylor expanded in x around -inf 64.0

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{e^{0.3333333333333333 \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x} + \left(0.06172839506172839 \cdot \frac{e^{0.3333333333333333 \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{3}} + e^{0.3333333333333333 \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}\right)\right) - \left(0.1111111111111111 \cdot \frac{e^{0.3333333333333333 \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}} + {\left(-1 \cdot x\right)}^{0.3333333333333333} \cdot \sqrt[3]{-1}\right)} \]

   5. Simplified 0.6

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 - \frac{0.1111111111111111}{x}\right) + 0.06172839506172839 \cdot \frac{\sqrt[3]{x}}{{x}^{3}}} \]

   if -4527.3591884227435 < x < 4285.791217009184

   1. Initial program 0.1

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]

   2. Applied add-sqr-sqrt_binary64 0.2

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{x + 1} - \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x + 1} - \sqrt[3]{x}}} \]

   3. Applied add-cube-cbrt_binary64 0.2

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

   4. Applied add-cube-cbrt_binary64 0.2

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

   5. Applied cbrt-prod_binary64 0.2

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

   6. Applied prod-diff_binary64 0.2

      \[\leadsto \sqrt{\sqrt[3]{x + 1} - \sqrt[3]{x}} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}} \]

   7. Applied add-exp-log_binary64 0.2

      \[\leadsto \sqrt{\color{blue}{e^{\log \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}}} \cdot \sqrt{\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)} \]

   8. Simplified 0.2

      \[\leadsto \sqrt{e^{\color{blue}{\log \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)}}} \cdot \sqrt{\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4527.3591884227435:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 - \frac{0.1111111111111111}{x}\right) + 0.06172839506172839 \cdot \frac{\sqrt[3]{x}}{{x}^{3}}\\ \mathbf{elif}\;x \leq 4285.791217009184:\\ \;\;\;\;\sqrt{e^{\log \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}} \cdot \sqrt{\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}, \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 - \frac{0.1111111111111111}{x}\right) + 0.06172839506172839 \cdot \frac{\sqrt[3]{x}}{{x}^{3}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))