Average Error: 59.8 → 0.3
Time: 37.3s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[0.3333333333333333148296162562473909929395 \cdot x + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + \left(\left(x \cdot x\right) \cdot 0.02222222222222222307030925492199457949027\right) \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
0.3333333333333333148296162562473909929395 \cdot x + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + \left(\left(x \cdot x\right) \cdot 0.02222222222222222307030925492199457949027\right) \cdot x\right)
double f(double x) {
        double r3830382 = 1.0;
        double r3830383 = x;
        double r3830384 = r3830382 / r3830383;
        double r3830385 = tan(r3830383);
        double r3830386 = r3830382 / r3830385;
        double r3830387 = r3830384 - r3830386;
        return r3830387;
}
↓
double f(double x) {
        double r3830388 = 0.3333333333333333;
        double r3830389 = x;
        double r3830390 = r3830388 * r3830389;
        double r3830391 = 0.0021164021164021165;
        double r3830392 = 5.0;
        double r3830393 = pow(r3830389, r3830392);
        double r3830394 = r3830391 * r3830393;
        double r3830395 = r3830389 * r3830389;
        double r3830396 = 0.022222222222222223;
        double r3830397 = r3830395 * r3830396;
        double r3830398 = r3830397 * r3830389;
        double r3830399 = r3830394 + r3830398;
        double r3830400 = r3830390 + r3830399;
        return r3830400;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.8
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988065102485279567190446:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.8

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{0.3333333333333333148296162562473909929395 \cdot x + \left(0.02222222222222222307030925492199457949027 \cdot {x}^{3} + 0.002116402116402116544841005563171165704262 \cdot {x}^{5}\right)}\]

  3. Simplified0.3

    \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333148296162562473909929395 + \left(x \cdot x\right) \cdot 0.02222222222222222307030925492199457949027\right) + {x}^{5} \cdot 0.002116402116402116544841005563171165704262}\]
  4. Using strategy rm

  5. Applied distribute-rgt-in0.3

    \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot x + \left(\left(x \cdot x\right) \cdot 0.02222222222222222307030925492199457949027\right) \cdot x\right)} + {x}^{5} \cdot 0.002116402116402116544841005563171165704262\]

  6. Applied associate-+l+0.3

    \[\leadsto \color{blue}{0.3333333333333333148296162562473909929395 \cdot x + \left(\left(\left(x \cdot x\right) \cdot 0.02222222222222222307030925492199457949027\right) \cdot x + {x}^{5} \cdot 0.002116402116402116544841005563171165704262\right)}\]

  7. Final simplification0.3

    \[\leadsto 0.3333333333333333148296162562473909929395 \cdot x + \left(0.002116402116402116544841005563171165704262 \cdot {x}^{5} + \left(\left(x \cdot x\right) \cdot 0.02222222222222222307030925492199457949027\right) \cdot x\right)\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "invcot (example 3.9)"
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3.0) (+ 1.0 (/ (* x x) 15.0))) (- (/ 1.0 x) (/ 1.0 (tan x))))

  (- (/ 1.0 x) (/ 1.0 (tan x))))