Average Error: 30.5 → 0.4
Time: 1.1m
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\frac{\sin x}{x}}{x} \cdot \sin x}{\cos x + 1}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\frac{\sin x}{x}}{x} \cdot \sin x}{\cos x + 1}
double f(double x) {
        double r5373118 = 1.0;
        double r5373119 = x;
        double r5373120 = cos(r5373119);
        double r5373121 = r5373118 - r5373120;
        double r5373122 = r5373119 * r5373119;
        double r5373123 = r5373121 / r5373122;
        return r5373123;
}
↓
double f(double x) {
        double r5373124 = x;
        double r5373125 = sin(r5373124);
        double r5373126 = r5373125 / r5373124;
        double r5373127 = r5373126 / r5373124;
        double r5373128 = r5373127 * r5373125;
        double r5373129 = cos(r5373124);
        double r5373130 = 1.0;
        double r5373131 = r5373129 + r5373130;
        double r5373132 = r5373128 / r5373131;
        return r5373132;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.5

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

  3. Applied flip--30.6

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

  4. Applied associate-/l/30.6

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

  5. Simplified14.7

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

  7. Applied clear-num14.7

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

  8. Taylor expanded around -inf 14.7

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

  9. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x + 1}}\]
  10. Using strategy rm

  11. Applied *-un-lft-identity0.3

    \[\leadsto \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{\color{blue}{1 \cdot x}}}{\cos x + 1}\]

  12. Applied add-sqr-sqrt32.0

    \[\leadsto \frac{\frac{\sin x}{x} \cdot \frac{\color{blue}{\sqrt{\sin x} \cdot \sqrt{\sin x}}}{1 \cdot x}}{\cos x + 1}\]

  13. Applied times-frac32.0

    \[\leadsto \frac{\frac{\sin x}{x} \cdot \color{blue}{\left(\frac{\sqrt{\sin x}}{1} \cdot \frac{\sqrt{\sin x}}{x}\right)}}{\cos x + 1}\]

  14. Applied *-un-lft-identity32.0

    \[\leadsto \frac{\frac{\sin x}{\color{blue}{1 \cdot x}} \cdot \left(\frac{\sqrt{\sin x}}{1} \cdot \frac{\sqrt{\sin x}}{x}\right)}{\cos x + 1}\]

  15. Applied add-sqr-sqrt32.1

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{\sin x} \cdot \sqrt{\sin x}}}{1 \cdot x} \cdot \left(\frac{\sqrt{\sin x}}{1} \cdot \frac{\sqrt{\sin x}}{x}\right)}{\cos x + 1}\]

  16. Applied times-frac32.1

    \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{\sin x}}{1} \cdot \frac{\sqrt{\sin x}}{x}\right)} \cdot \left(\frac{\sqrt{\sin x}}{1} \cdot \frac{\sqrt{\sin x}}{x}\right)}{\cos x + 1}\]

  17. Applied swap-sqr32.2

    \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{\sin x}}{1} \cdot \frac{\sqrt{\sin x}}{1}\right) \cdot \left(\frac{\sqrt{\sin x}}{x} \cdot \frac{\sqrt{\sin x}}{x}\right)}}{\cos x + 1}\]

  18. Simplified32.1

    \[\leadsto \frac{\color{blue}{\sin x} \cdot \left(\frac{\sqrt{\sin x}}{x} \cdot \frac{\sqrt{\sin x}}{x}\right)}{\cos x + 1}\]

  19. Simplified0.4

    \[\leadsto \frac{\sin x \cdot \color{blue}{\frac{\frac{\sin x}{x}}{x}}}{\cos x + 1}\]

  20. Final simplification0.4

    \[\leadsto \frac{\frac{\frac{\sin x}{x}}{x} \cdot \sin x}{\cos x + 1}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))