Average Error: 11.0 → 1.4
Time: 4.5s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r512693 = x;
        double r512694 = y;
        double r512695 = z;
        double r512696 = t;
        double r512697 = r512695 - r512696;
        double r512698 = r512694 * r512697;
        double r512699 = a;
        double r512700 = r512699 - r512696;
        double r512701 = r512698 / r512700;
        double r512702 = r512693 + r512701;
        return r512702;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r512703 = z;
        double r512704 = t;
        double r512705 = r512703 - r512704;
        double r512706 = a;
        double r512707 = r512706 - r512704;
        double r512708 = r512705 / r512707;
        double r512709 = y;
        double r512710 = x;
        double r512711 = fma(r512708, r512709, r512710);
        return r512711;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.0
Target1.3
Herbie1.4
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Initial program 11.0

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

  2. Simplified2.9

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

  4. Applied *-un-lft-identity2.9

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

  5. Applied add-cube-cbrt3.4

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

  6. Applied times-frac3.4

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

  7. Simplified3.4

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

  9. Applied fma-udef3.4

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

  10. Simplified3.0

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

  12. Applied *-un-lft-identity3.0

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

  13. Applied *-un-lft-identity3.0

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

  14. Applied distribute-lft-out3.0

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

  15. Simplified1.4

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

  16. Final simplification1.4

    \[\leadsto \mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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