tanhf (example 3.4)

Average Error: 30.1 → 0.6

Time: 21.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.02088855374907699902209401443542446941137:\\ \;\;\;\;\frac{\left(\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)\right) \cdot \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}{\left(\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)\right) \cdot \left(\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)\right)}\\ \mathbf{elif}\;x \le 0.01743449574326636983268024039261945290491:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)} - \frac{\sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \end{array}\]

C

double f(double x) {
        double r27012 = 1.0;
        double r27013 = x;
        double r27014 = cos(r27013);
        double r27015 = r27012 - r27014;
        double r27016 = sin(r27013);
        double r27017 = r27015 / r27016;
        return r27017;
}

↓

double f(double x) {
        double r27018 = x;
        double r27019 = -0.020888553749077;
        bool r27020 = r27018 <= r27019;
        double r27021 = sin(r27018);
        double r27022 = cos(r27018);
        double r27023 = 1.0;
        double r27024 = r27023 + r27022;
        double r27025 = r27022 * r27024;
        double r27026 = r27023 * r27023;
        double r27027 = r27025 + r27026;
        double r27028 = r27021 * r27027;
        double r27029 = 3.0;
        double r27030 = pow(r27023, r27029);
        double r27031 = pow(r27022, r27029);
        double r27032 = r27030 - r27031;
        double r27033 = r27028 * r27032;
        double r27034 = r27028 * r27028;
        double r27035 = r27033 / r27034;
        double r27036 = 0.01743449574326637;
        bool r27037 = r27018 <= r27036;
        double r27038 = 0.041666666666666664;
        double r27039 = pow(r27018, r27029);
        double r27040 = r27038 * r27039;
        double r27041 = 0.004166666666666667;
        double r27042 = 5.0;
        double r27043 = pow(r27018, r27042);
        double r27044 = r27041 * r27043;
        double r27045 = 0.5;
        double r27046 = r27045 * r27018;
        double r27047 = r27044 + r27046;
        double r27048 = r27040 + r27047;
        double r27049 = r27030 / r27028;
        double r27050 = pow(r27031, r27029);
        double r27051 = cbrt(r27050);
        double r27052 = r27051 / r27028;
        double r27053 = r27049 - r27052;
        double r27054 = r27037 ? r27048 : r27053;
        double r27055 = r27020 ? r27035 : r27054;
        return r27055;
}

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 < -0.020888553749077

   1. Initial program 0.9

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

   3. Applied add-log-exp 1.1

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

   5. Applied flip3-- 1.2

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

   6. Applied associate-/l/ 1.2

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

   7. Simplified 1.2

      \[\leadsto \log \left(e^{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}}}\right)\]
   8. Using strategy rm

   9. Applied div-sub 1.2

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

   10. Applied exp-diff 1.3

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

   11. Applied log-div 1.3

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

   12. Simplified 1.2

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

   13. Simplified 1.0

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

   15. Applied frac-sub 1.1

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

   16. Simplified 1.1

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

   if -0.020888553749077 < x < 0.01743449574326637

   1. Initial program 59.9

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

   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 0.01743449574326637 < x

   1. Initial program 1.0

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

   3. Applied add-log-exp 1.2

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

   5. Applied flip3-- 1.3

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

   6. Applied associate-/l/ 1.3

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

   7. Simplified 1.3

      \[\leadsto \log \left(e^{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}}}\right)\]
   8. Using strategy rm

   9. Applied div-sub 1.3

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

   10. Applied exp-diff 1.3

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

   11. Applied log-div 1.2

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

   12. Simplified 1.2

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

   13. Simplified 1.1

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

   15. Applied add-cbrt-cube 1.2

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

   16. Simplified 1.2

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02088855374907699902209401443542446941137:\\ \;\;\;\;\frac{\left(\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)\right) \cdot \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}{\left(\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)\right) \cdot \left(\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)\right)}\\ \mathbf{elif}\;x \le 0.01743449574326636983268024039261945290491:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)} - \frac{\sqrt[3]{{\left({\left(\cos x\right)}^{3}\right)}^{3}}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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