Average Error: 0.3 → 0.4
Time: 34.5s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(1 \cdot 1, 1, \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right) \cdot \left(\tan x \cdot \tan x\right)\right)} \cdot \left(\left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - \left(\tan x \cdot \tan x\right) \cdot 1\right) + 1 \cdot 1\right)\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(1 \cdot 1, 1, \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right) \cdot \left(\tan x \cdot \tan x\right)\right)} \cdot \left(\left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - \left(\tan x \cdot \tan x\right) \cdot 1\right) + 1 \cdot 1\right)
double f(double x) {
        double r827490 = 1.0;
        double r827491 = x;
        double r827492 = tan(r827491);
        double r827493 = r827492 * r827492;
        double r827494 = r827490 - r827493;
        double r827495 = r827490 + r827493;
        double r827496 = r827494 / r827495;
        return r827496;
}
↓
double f(double x) {
        double r827497 = 1.0;
        double r827498 = x;
        double r827499 = tan(r827498);
        double r827500 = r827499 * r827499;
        double r827501 = r827497 - r827500;
        double r827502 = r827497 * r827497;
        double r827503 = r827500 * r827500;
        double r827504 = r827503 * r827500;
        double r827505 = fma(r827502, r827497, r827504);
        double r827506 = r827501 / r827505;
        double r827507 = r827500 * r827497;
        double r827508 = r827503 - r827507;
        double r827509 = r827508 + r827502;
        double r827510 = r827506 * r827509;
        return r827510;
}

Error

Bits error versus x

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 flip3-+0.4

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

  4. Applied associate-/r/0.4

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

  5. Simplified0.4

    \[\leadsto \color{blue}{\frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(1 \cdot 1, 1, \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right) \cdot \left(\tan x \cdot \tan x\right)\right)}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - 1 \cdot \left(\tan x \cdot \tan x\right)\right)\right)\]

  6. Final simplification0.4

    \[\leadsto \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(1 \cdot 1, 1, \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right) \cdot \left(\tan x \cdot \tan x\right)\right)} \cdot \left(\left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - \left(\tan x \cdot \tan x\right) \cdot 1\right) + 1 \cdot 1\right)\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))