Average Error: 59.9 → 0.3
Time: 29.7s
Precision: 64
\[-0.0259999999999999988065102485279567190446 \lt x \land x \lt 0.0259999999999999988065102485279567190446\]
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\frac{6405119470038039}{288230376151711744} \cdot {x}^{3} + \left(\frac{4880091024790887}{2305843009213693952} \cdot {x}^{5} + \frac{6004799503160661}{18014398509481984} \cdot x\right)\]
\frac{1}{x} - \frac{1}{\tan x}
↓
\frac{6405119470038039}{288230376151711744} \cdot {x}^{3} + \left(\frac{4880091024790887}{2305843009213693952} \cdot {x}^{5} + \frac{6004799503160661}{18014398509481984} \cdot x\right)
double f(double x) {
        double r75010 = 1.0;
        double r75011 = x;
        double r75012 = r75010 / r75011;
        double r75013 = tan(r75011);
        double r75014 = r75010 / r75013;
        double r75015 = r75012 - r75014;
        return r75015;
}
↓
double f(double x) {
        double r75016 = 6405119470038039.0;
        double r75017 = 2.8823037615171174e+17;
        double r75018 = r75016 / r75017;
        double r75019 = x;
        double r75020 = 3.0;
        double r75021 = pow(r75019, r75020);
        double r75022 = r75018 * r75021;
        double r75023 = 4880091024790887.0;
        double r75024 = 2.305843009213694e+18;
        double r75025 = r75023 / r75024;
        double r75026 = 5.0;
        double r75027 = pow(r75019, r75026);
        double r75028 = r75025 * r75027;
        double r75029 = 6004799503160661.0;
        double r75030 = 18014398509481984.0;
        double r75031 = r75029 / r75030;
        double r75032 = r75031 * r75019;
        double r75033 = r75028 + r75032;
        double r75034 = r75022 + r75033;
        return r75034;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 59.9

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

    \[\leadsto \color{blue}{\frac{6405119470038039}{288230376151711744} \cdot {x}^{3} + \left(\frac{4880091024790887}{2305843009213693952} \cdot {x}^{5} + \frac{6004799503160661}{18014398509481984} \cdot x\right)}\]

  4. Final simplification0.3

    \[\leadsto \frac{6405119470038039}{288230376151711744} \cdot {x}^{3} + \left(\frac{4880091024790887}{2305843009213693952} \cdot {x}^{5} + \frac{6004799503160661}{18014398509481984} \cdot x\right)\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.0259999999999999988 x) (< x 0.0259999999999999988))

  :herbie-target
  (if (< (fabs x) 0.0259999999999999988) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

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