Average Error: 60.0 → 0.0
Time: 1.1m
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 r7778208 = 1.0;
        double r7778209 = x;
        double r7778210 = r7778208 / r7778209;
        double r7778211 = tan(r7778209);
        double r7778212 = r7778208 / r7778211;
        double r7778213 = r7778210 - r7778212;
        return r7778213;
}
↓
double f(double x) {
        double r7778214 = x;
        double r7778215 = 0.022222222222222223;
        double r7778216 = r7778214 * r7778215;
        double r7778217 = r7778214 * r7778216;
        double r7778218 = -0.3333333333333333;
        double r7778219 = r7778217 + r7778218;
        double r7778220 = r7778217 * r7778219;
        double r7778221 = -0.1111111111111111;
        double r7778222 = r7778220 - r7778221;
        double r7778223 = r7778214 * r7778214;
        double r7778224 = r7778223 * r7778223;
        double r7778225 = 1.0973936899862826e-05;
        double r7778226 = r7778225 * r7778223;
        double r7778227 = r7778224 * r7778226;
        double r7778228 = 0.037037037037037035;
        double r7778229 = r7778227 + r7778228;
        double r7778230 = r7778222 / r7778229;
        double r7778231 = r7778214 / r7778230;
        double r7778232 = 5.0;
        double r7778233 = pow(r7778214, r7778232);
        double r7778234 = 0.0021164021164021165;
        double r7778235 = r7778233 * r7778234;
        double r7778236 = r7778231 + r7778235;
        return r7778236;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.0
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 60.0

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

  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 2019128 
(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))))