Average Error: 0.3 → 0.4
Time: 35.9s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{\sin x}{\cos x} \cdot \tan x}{(\left(\log \left(e^{\frac{\sin x}{\cos x}}\right)\right) \cdot \left(\frac{\sin x}{\cos x}\right) + 1)_*}\]
double f(double x) {
        double r1147112 = 1.0;
        double r1147113 = x;
        double r1147114 = tan(r1147113);
        double r1147115 = r1147114 * r1147114;
        double r1147116 = r1147112 - r1147115;
        double r1147117 = r1147112 + r1147115;
        double r1147118 = r1147116 / r1147117;
        return r1147118;
}
↓
double f(double x) {
        double r1147119 = 1.0;
        double r1147120 = x;
        double r1147121 = sin(r1147120);
        double r1147122 = cos(r1147120);
        double r1147123 = r1147121 / r1147122;
        double r1147124 = tan(r1147120);
        double r1147125 = r1147123 * r1147124;
        double r1147126 = r1147119 - r1147125;
        double r1147127 = exp(r1147123);
        double r1147128 = log(r1147127);
        double r1147129 = fma(r1147128, r1147123, r1147119);
        double r1147130 = r1147126 / r1147129;
        return r1147130;
}


\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{\sin x}{\cos x} \cdot \tan x}{(\left(\log \left(e^{\frac{\sin x}{\cos x}}\right)\right) \cdot \left(\frac{\sin x}{\cos x}\right) + 1)_*}

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. Taylor expanded around -inf 0.4

    \[\leadsto \color{blue}{\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1}}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\frac{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x}\right) + 1)_*}}\]
  4. Using strategy rm

  5. Applied quot-tan0.4

    \[\leadsto \frac{1 - \frac{\sin x}{\cos x} \cdot \color{blue}{\tan x}}{(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x}\right) + 1)_*}\]
  6. Using strategy rm

  7. Applied add-log-exp0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))