sintan (problem 3.4.5)

Average Error: 31.3 → 0.1

Time: 11.5s

Precision: 64

Math

\[\frac{x - \sin x}{x - \tan x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.5623918316578669:\\ \;\;\;\;\sqrt[3]{{\left({\left(\frac{x - \sin x}{x - \tan x}\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}\right)}^{\left(\sqrt[3]{3}\right)}}\\ \mathbf{elif}\;x \le 0.024314219703607573:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\ \end{array}\]

C

double f(double x) {
        double r20684 = x;
        double r20685 = sin(r20684);
        double r20686 = r20684 - r20685;
        double r20687 = tan(r20684);
        double r20688 = r20684 - r20687;
        double r20689 = r20686 / r20688;
        return r20689;
}

↓

double f(double x) {
        double r20690 = x;
        double r20691 = -1.5623918316578669;
        bool r20692 = r20690 <= r20691;
        double r20693 = sin(r20690);
        double r20694 = r20690 - r20693;
        double r20695 = tan(r20690);
        double r20696 = r20690 - r20695;
        double r20697 = r20694 / r20696;
        double r20698 = 3.0;
        double r20699 = cbrt(r20698);
        double r20700 = r20699 * r20699;
        double r20701 = pow(r20697, r20700);
        double r20702 = pow(r20701, r20699);
        double r20703 = cbrt(r20702);
        double r20704 = 0.024314219703607573;
        bool r20705 = r20690 <= r20704;
        double r20706 = 0.225;
        double r20707 = 2.0;
        double r20708 = pow(r20690, r20707);
        double r20709 = 0.009642857142857142;
        double r20710 = 4.0;
        double r20711 = pow(r20690, r20710);
        double r20712 = 0.5;
        double r20713 = fma(r20709, r20711, r20712);
        double r20714 = -r20713;
        double r20715 = fma(r20706, r20708, r20714);
        double r20716 = pow(r20697, r20698);
        double r20717 = cbrt(r20716);
        double r20718 = r20705 ? r20715 : r20717;
        double r20719 = r20692 ? r20703 : r20718;
        return r20719;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -1.5623918316578669

   1. Initial program 0.0

      \[\frac{x - \sin x}{x - \tan x}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 41.9

      \[\leadsto \frac{x - \sin x}{\color{blue}{\sqrt[3]{\left(\left(x - \tan x\right) \cdot \left(x - \tan x\right)\right) \cdot \left(x - \tan x\right)}}}\]

   4. Applied add-cbrt-cube 43.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - \sin x\right) \cdot \left(x - \sin x\right)\right) \cdot \left(x - \sin x\right)}}}{\sqrt[3]{\left(\left(x - \tan x\right) \cdot \left(x - \tan x\right)\right) \cdot \left(x - \tan x\right)}}\]

   5. Applied cbrt-undiv 43.1

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x - \sin x\right) \cdot \left(x - \sin x\right)\right) \cdot \left(x - \sin x\right)}{\left(\left(x - \tan x\right) \cdot \left(x - \tan x\right)\right) \cdot \left(x - \tan x\right)}}}\]

   6. Simplified 0.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 0.0

      \[\leadsto \sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{\color{blue}{\left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}\right)}}}\]

   9. Applied pow-unpow 0.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(\frac{x - \sin x}{x - \tan x}\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}\right)}^{\left(\sqrt[3]{3}\right)}}}\]

   if -1.5623918316578669 < x < 0.024314219703607573

   1. Initial program 62.9

      \[\frac{x - \sin x}{x - \tan x}\]

   2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]

   3. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)}\]

   if 0.024314219703607573 < x

   1. Initial program 0.1

      \[\frac{x - \sin x}{x - \tan x}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 41.0

      \[\leadsto \frac{x - \sin x}{\color{blue}{\sqrt[3]{\left(\left(x - \tan x\right) \cdot \left(x - \tan x\right)\right) \cdot \left(x - \tan x\right)}}}\]

   4. Applied add-cbrt-cube 42.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - \sin x\right) \cdot \left(x - \sin x\right)\right) \cdot \left(x - \sin x\right)}}}{\sqrt[3]{\left(\left(x - \tan x\right) \cdot \left(x - \tan x\right)\right) \cdot \left(x - \tan x\right)}}\]

   5. Applied cbrt-undiv 42.1

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x - \sin x\right) \cdot \left(x - \sin x\right)\right) \cdot \left(x - \sin x\right)}{\left(\left(x - \tan x\right) \cdot \left(x - \tan x\right)\right) \cdot \left(x - \tan x\right)}}}\]

   6. Simplified 0.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5623918316578669:\\ \;\;\;\;\sqrt[3]{{\left({\left(\frac{x - \sin x}{x - \tan x}\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}\right)}^{\left(\sqrt[3]{3}\right)}}\\ \mathbf{elif}\;x \le 0.024314219703607573:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))