Average Error: 59.8 → 0.0
Time: 1.0m
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{x}{\frac{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right) + \frac{-1}{3}\right) - \frac{-1}{9}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right) + \frac{1}{27}}} + {x}^{5} \cdot \frac{2}{945}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{x}{\frac{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right) + \frac{-1}{3}\right) - \frac{-1}{9}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right) + \frac{1}{27}}} + {x}^{5} \cdot \frac{2}{945}
double f(double x) {
        double r7931132 = 1.0;
        double r7931133 = x;
        double r7931134 = r7931132 / r7931133;
        double r7931135 = tan(r7931133);
        double r7931136 = r7931132 / r7931135;
        double r7931137 = r7931134 - r7931136;
        return r7931137;
}
↓
double f(double x) {
        double r7931138 = x;
        double r7931139 = 0.022222222222222223;
        double r7931140 = r7931138 * r7931139;
        double r7931141 = r7931138 * r7931140;
        double r7931142 = -0.3333333333333333;
        double r7931143 = r7931141 + r7931142;
        double r7931144 = r7931141 * r7931143;
        double r7931145 = -0.1111111111111111;
        double r7931146 = r7931144 - r7931145;
        double r7931147 = r7931138 * r7931138;
        double r7931148 = r7931147 * r7931147;
        double r7931149 = 1.0973936899862826e-05;
        double r7931150 = r7931149 * r7931147;
        double r7931151 = r7931148 * r7931150;
        double r7931152 = 0.037037037037037035;
        double r7931153 = r7931151 + r7931152;
        double r7931154 = r7931146 / r7931153;
        double r7931155 = r7931138 / r7931154;
        double r7931156 = 5.0;
        double r7931157 = pow(r7931138, r7931156);
        double r7931158 = 0.0021164021164021165;
        double r7931159 = r7931157 * r7931158;
        double r7931160 = r7931155 + r7931159;
        return r7931160;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.8
Target0.1
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.8

    \[\frac{1}{x} - \frac{1}{\tan x}\]

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\frac{1}{3} \cdot x + \left(\frac{1}{45} \cdot {x}^{3} + \frac{2}{945} \cdot {x}^{5}\right)}\]

  3. Simplified0.3

    \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + \frac{1}{45} \cdot \left(x \cdot x\right)\right) + {x}^{5} \cdot \frac{2}{945}}\]
  4. Using strategy rm

  5. Applied flip3-+1.2

    \[\leadsto x \cdot \color{blue}{\frac{{\frac{1}{3}}^{3} + {\left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}^{3}}{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) - \frac{1}{3} \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)\right)}} + {x}^{5} \cdot \frac{2}{945}\]

  6. Applied associate-*r/1.1

    \[\leadsto \color{blue}{\frac{x \cdot \left({\frac{1}{3}}^{3} + {\left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)}^{3}\right)}{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) - \frac{1}{3} \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)\right)}} + {x}^{5} \cdot \frac{2}{945}\]

  7. Simplified0.3

    \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{27} + \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}}{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) - \frac{1}{3} \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)\right)} + {x}^{5} \cdot \frac{2}{945}\]
  8. Using strategy rm

  9. Applied associate-/l*0.0

    \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{3} \cdot \frac{1}{3} + \left(\left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right) - \frac{1}{3} \cdot \left(\frac{1}{45} \cdot \left(x \cdot x\right)\right)\right)}{\frac{1}{27} + \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}} + {x}^{5} \cdot \frac{2}{945}\]

  10. Simplified0.0

    \[\leadsto \frac{x}{\color{blue}{\frac{\left(x \cdot \left(\frac{1}{45} \cdot x\right)\right) \cdot \left(\frac{-1}{3} + x \cdot \left(\frac{1}{45} \cdot x\right)\right) - \frac{-1}{9}}{\frac{1}{27} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right)}}} + {x}^{5} \cdot \frac{2}{945}\]

  11. Final simplification0.0

    \[\leadsto \frac{x}{\frac{\left(x \cdot \left(x \cdot \frac{1}{45}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{45}\right) + \frac{-1}{3}\right) - \frac{-1}{9}}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{91125} \cdot \left(x \cdot x\right)\right) + \frac{1}{27}}} + {x}^{5} \cdot \frac{2}{945}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (x)
  :name "invcot (example 3.9)"
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

  (- (/ 1 x) (/ 1 (tan x))))