tanhf (example 3.4)

Average Error: 30.1 → 0.6

Time: 7.3s

Precision: 64

Math

\[\frac{1 - \cos x}{\sin x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0199905813077168142:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;x \le 0.021452837467613652:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{\sin x} \cdot \sqrt{1 - \cos x}\\ \end{array}\]

C

double f(double x) {
        double r72038 = 1.0;
        double r72039 = x;
        double r72040 = cos(r72039);
        double r72041 = r72038 - r72040;
        double r72042 = sin(r72039);
        double r72043 = r72041 / r72042;
        return r72043;
}

↓

double f(double x) {
        double r72044 = x;
        double r72045 = -0.019990581307716814;
        bool r72046 = r72044 <= r72045;
        double r72047 = 1.0;
        double r72048 = cos(r72044);
        double r72049 = r72047 - r72048;
        double r72050 = sin(r72044);
        double r72051 = r72049 / r72050;
        double r72052 = 3.0;
        double r72053 = pow(r72051, r72052);
        double r72054 = cbrt(r72053);
        double r72055 = pow(r72054, r72052);
        double r72056 = cbrt(r72055);
        double r72057 = 0.02145283746761365;
        bool r72058 = r72044 <= r72057;
        double r72059 = 0.041666666666666664;
        double r72060 = pow(r72044, r72052);
        double r72061 = r72059 * r72060;
        double r72062 = 0.004166666666666667;
        double r72063 = 5.0;
        double r72064 = pow(r72044, r72063);
        double r72065 = r72062 * r72064;
        double r72066 = 0.5;
        double r72067 = r72066 * r72044;
        double r72068 = r72065 + r72067;
        double r72069 = r72061 + r72068;
        double r72070 = sqrt(r72049);
        double r72071 = r72070 / r72050;
        double r72072 = r72071 * r72070;
        double r72073 = r72058 ? r72069 : r72072;
        double r72074 = r72046 ? r72056 : r72073;
        return r72074;
}

Error

Bits error versus x

Target

Original	30.1
Target	0
Herbie	0.6

\[\tan \left(\frac{x}{2}\right)\]

Derivation

1. Split input into 3 regimes

2. if x < -0.019990581307716814

   1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 1.1

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

   4. Applied add-cbrt-cube 1.3

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

   5. Applied cbrt-undiv 1.1

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

   6. Simplified 1.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}}\]
   7. Using strategy rm

   8. Applied add-cbrt-cube 1.3

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

   9. Applied add-cbrt-cube 1.4

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

   10. Applied cbrt-undiv 1.3

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

   11. Simplified 1.2

       \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\color{blue}{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}}\right)}^{3}}\]

   if -0.019990581307716814 < x < 0.02145283746761365

   1. Initial program 59.8

      \[\frac{1 - \cos x}{\sin x}\]

   2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]

   if 0.02145283746761365 < x

   1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x}\]
   2. Using strategy rm

   3. Applied add-log-exp 1.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 1.0

      \[\leadsto \log \left(e^{\frac{1 - \cos x}{\color{blue}{1 \cdot \sin x}}}\right)\]

   6. Applied add-sqr-sqrt 1.2

      \[\leadsto \log \left(e^{\frac{\color{blue}{\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}}}{1 \cdot \sin x}}\right)\]

   7. Applied times-frac 1.2

      \[\leadsto \log \left(e^{\color{blue}{\frac{\sqrt{1 - \cos x}}{1} \cdot \frac{\sqrt{1 - \cos x}}{\sin x}}}\right)\]

   8. Applied exp-prod 1.2

      \[\leadsto \log \color{blue}{\left({\left(e^{\frac{\sqrt{1 - \cos x}}{1}}\right)}^{\left(\frac{\sqrt{1 - \cos x}}{\sin x}\right)}\right)}\]

   9. Applied log-pow 1.1

      \[\leadsto \color{blue}{\frac{\sqrt{1 - \cos x}}{\sin x} \cdot \log \left(e^{\frac{\sqrt{1 - \cos x}}{1}}\right)}\]

   10. Simplified 1.1

       \[\leadsto \frac{\sqrt{1 - \cos x}}{\sin x} \cdot \color{blue}{\sqrt{1 - \cos x}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0199905813077168142:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{1 - \cos x}{\sin x}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;x \le 0.021452837467613652:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{\sin x} \cdot \sqrt{1 - \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))