Average Error: 59.8 → 0.0
Time: 37.8s
Precision: 64
\[-0.026 \lt x \land x \lt 0.026\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{x}{\frac{\frac{1}{9} + \left(\frac{1}{2025} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - \left(x \cdot x\right) \cdot \frac{1}{135}\right)}{\frac{1}{27} + \left(x \cdot x\right) \cdot \left(\frac{1}{91125} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}} + {x}^{5} \cdot \frac{2}{945}\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{x}{\frac{\frac{1}{9} + \left(\frac{1}{2025} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - \left(x \cdot x\right) \cdot \frac{1}{135}\right)}{\frac{1}{27} + \left(x \cdot x\right) \cdot \left(\frac{1}{91125} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}} + {x}^{5} \cdot \frac{2}{945}
double f(double x) {
        double r2892936 = 1.0;
        double r2892937 = x;
        double r2892938 = r2892936 / r2892937;
        double r2892939 = tan(r2892937);
        double r2892940 = r2892936 / r2892939;
        double r2892941 = r2892938 - r2892940;
        return r2892941;
}
↓
double f(double x) {
        double r2892942 = x;
        double r2892943 = 0.1111111111111111;
        double r2892944 = 0.0004938271604938272;
        double r2892945 = r2892942 * r2892942;
        double r2892946 = r2892945 * r2892945;
        double r2892947 = r2892944 * r2892946;
        double r2892948 = 0.007407407407407408;
        double r2892949 = r2892945 * r2892948;
        double r2892950 = r2892947 - r2892949;
        double r2892951 = r2892943 + r2892950;
        double r2892952 = 0.037037037037037035;
        double r2892953 = 1.0973936899862826e-05;
        double r2892954 = r2892953 * r2892946;
        double r2892955 = r2892945 * r2892954;
        double r2892956 = r2892952 + r2892955;
        double r2892957 = r2892951 / r2892956;
        double r2892958 = r2892942 / r2892957;
        double r2892959 = 5.0;
        double r2892960 = pow(r2892942, r2892959);
        double r2892961 = 0.0021164021164021165;
        double r2892962 = r2892960 * r2892961;
        double r2892963 = r2892958 + r2892962;
        return r2892963;
}

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

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

  5. Applied flip3-+1.2

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

  6. Applied associate-*l/1.1

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

  7. Simplified0.3

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

  9. Applied associate-/l*0.0

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

  10. Simplified0.0

    \[\leadsto \frac{2}{945} \cdot {x}^{5} + \frac{x}{\color{blue}{\frac{\frac{1}{9} + \left(\frac{1}{2025} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - \left(x \cdot x\right) \cdot \frac{1}{135}\right)}{\frac{1}{27} + \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{91125}\right)}}}\]

  11. Final simplification0.0

    \[\leadsto \frac{x}{\frac{\frac{1}{9} + \left(\frac{1}{2025} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - \left(x \cdot x\right) \cdot \frac{1}{135}\right)}{\frac{1}{27} + \left(x \cdot x\right) \cdot \left(\frac{1}{91125} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}} + {x}^{5} \cdot \frac{2}{945}\]

Reproduce

herbie shell --seed 2019138 
(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))))