Average Error: 0.3 → 0.4
Time: 5.0s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}
double f(double x) {
        double r11987 = 1.0;
        double r11988 = x;
        double r11989 = tan(r11988);
        double r11990 = r11989 * r11989;
        double r11991 = r11987 - r11990;
        double r11992 = r11987 + r11990;
        double r11993 = r11991 / r11992;
        return r11993;
}
↓
double f(double x) {
        double r11994 = 1.0;
        double r11995 = sqrt(r11994);
        double r11996 = x;
        double r11997 = tan(r11996);
        double r11998 = r11995 + r11997;
        double r11999 = exp(r11998);
        double r12000 = log(r11999);
        double r12001 = r11995 - r11997;
        double r12002 = r12000 * r12001;
        double r12003 = r11997 * r11997;
        double r12004 = r11994 + r12003;
        double r12005 = r12002 / r12004;
        return r12005;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 0.3

    \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.3

    \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}} - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]

  4. Applied difference-of-squares0.4

    \[\leadsto \frac{\color{blue}{\left(\sqrt{1} + \tan x\right) \cdot \left(\sqrt{1} - \tan x\right)}}{1 + \tan x \cdot \tan x}\]
  5. Using strategy rm

  6. Applied add-log-exp0.4

    \[\leadsto \frac{\left(\sqrt{1} + \color{blue}{\log \left(e^{\tan x}\right)}\right) \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}\]

  7. Applied add-log-exp0.4

    \[\leadsto \frac{\left(\color{blue}{\log \left(e^{\sqrt{1}}\right)} + \log \left(e^{\tan x}\right)\right) \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}\]

  8. Applied sum-log0.4

    \[\leadsto \frac{\color{blue}{\log \left(e^{\sqrt{1}} \cdot e^{\tan x}\right)} \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}\]

  9. Simplified0.4

    \[\leadsto \frac{\log \color{blue}{\left(e^{\sqrt{1} + \tan x}\right)} \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}\]

  10. Final simplification0.4

    \[\leadsto \frac{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))