Average Error: 10.7 → 1.4
Time: 3.3s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[y \cdot \frac{z - t}{z - a} + x\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
y \cdot \frac{z - t}{z - a} + x
double f(double x, double y, double z, double t, double a) {
        double r562681 = x;
        double r562682 = y;
        double r562683 = z;
        double r562684 = t;
        double r562685 = r562683 - r562684;
        double r562686 = r562682 * r562685;
        double r562687 = a;
        double r562688 = r562683 - r562687;
        double r562689 = r562686 / r562688;
        double r562690 = r562681 + r562689;
        return r562690;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r562691 = y;
        double r562692 = z;
        double r562693 = t;
        double r562694 = r562692 - r562693;
        double r562695 = a;
        double r562696 = r562692 - r562695;
        double r562697 = r562694 / r562696;
        double r562698 = r562691 * r562697;
        double r562699 = x;
        double r562700 = r562698 + r562699;
        return r562700;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.7
Target1.3
Herbie1.4
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.7

    \[x + \frac{y \cdot \left(z - t\right)}{z - a}\]

  2. Simplified2.8

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)}\]
  3. Using strategy rm

  4. Applied div-inv2.9

    \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \frac{1}{z - a}}, z - t, x\right)\]
  5. Using strategy rm

  6. Applied add-cube-cbrt3.3

    \[\leadsto \mathsf{fma}\left(y \cdot \frac{1}{\color{blue}{\left(\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}\right) \cdot \sqrt[3]{z - a}}}, z - t, x\right)\]

  7. Applied associate-/r*3.3

    \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\frac{\frac{1}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}}{\sqrt[3]{z - a}}}, z - t, x\right)\]
  8. Using strategy rm

  9. Applied fma-udef3.3

    \[\leadsto \color{blue}{\left(y \cdot \frac{\frac{1}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}}{\sqrt[3]{z - a}}\right) \cdot \left(z - t\right) + x}\]

  10. Simplified10.7

    \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{z - a}} + x\]
  11. Using strategy rm

  12. Applied *-un-lft-identity10.7

    \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{1 \cdot \left(z - a\right)}} + x\]

  13. Applied times-frac1.4

    \[\leadsto \color{blue}{\frac{y}{1} \cdot \frac{z - t}{z - a}} + x\]

  14. Simplified1.4

    \[\leadsto \color{blue}{y} \cdot \frac{z - t}{z - a} + x\]

  15. Final simplification1.4

    \[\leadsto y \cdot \frac{z - t}{z - a} + x\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

  (+ x (/ (* y (- z t)) (- z a))))