Average Error: 59.9 → 0.4
Time: 59.7s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\log \left({\left(e^{{x}^{5}}\right)}^{0.002116402116402116544841005563171165704262}\right) + \left(0.3333333333333333148296162562473909929395 \cdot x + {x}^{3} \cdot 0.02222222222222222307030925492199457949027\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\log \left({\left(e^{{x}^{5}}\right)}^{0.002116402116402116544841005563171165704262}\right) + \left(0.3333333333333333148296162562473909929395 \cdot x + {x}^{3} \cdot 0.02222222222222222307030925492199457949027\right)
double f(double x) {
        double r82138 = 1.0;
        double r82139 = x;
        double r82140 = r82138 / r82139;
        double r82141 = tan(r82139);
        double r82142 = r82138 / r82141;
        double r82143 = r82140 - r82142;
        return r82143;
}
↓
double f(double x) {
        double r82144 = x;
        double r82145 = 5.0;
        double r82146 = pow(r82144, r82145);
        double r82147 = exp(r82146);
        double r82148 = 0.0021164021164021165;
        double r82149 = pow(r82147, r82148);
        double r82150 = log(r82149);
        double r82151 = 0.3333333333333333;
        double r82152 = r82151 * r82144;
        double r82153 = 3.0;
        double r82154 = pow(r82144, r82153);
        double r82155 = 0.022222222222222223;
        double r82156 = r82154 * r82155;
        double r82157 = r82152 + r82156;
        double r82158 = r82150 + r82157;
        return r82158;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.9
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.0259999999999999988065102485279567190446:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.9

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

  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

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

  5. Applied add-log-exp0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019196 
(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))))