Average Error: 10.9 → 0.9
Time: 17.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\mathsf{fma}\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}, \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{z - a}}{y}}, x\right)\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\mathsf{fma}\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}, \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{z - a}}{y}}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r321529 = x;
        double r321530 = y;
        double r321531 = z;
        double r321532 = t;
        double r321533 = r321531 - r321532;
        double r321534 = r321530 * r321533;
        double r321535 = a;
        double r321536 = r321531 - r321535;
        double r321537 = r321534 / r321536;
        double r321538 = r321529 + r321537;
        return r321538;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r321539 = z;
        double r321540 = t;
        double r321541 = r321539 - r321540;
        double r321542 = cbrt(r321541);
        double r321543 = r321542 * r321542;
        double r321544 = a;
        double r321545 = r321539 - r321544;
        double r321546 = cbrt(r321545);
        double r321547 = r321546 * r321546;
        double r321548 = r321543 / r321547;
        double r321549 = y;
        double r321550 = r321546 / r321549;
        double r321551 = r321542 / r321550;
        double r321552 = x;
        double r321553 = fma(r321548, r321551, r321552);
        return r321553;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.9
Target1.1
Herbie0.9
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.9

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

  2. Simplified3.1

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

  4. Applied clear-num3.4

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

  6. Applied fma-udef3.4

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

  7. Simplified3.2

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

  9. Applied *-un-lft-identity3.2

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

  10. Applied add-cube-cbrt3.6

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

  11. Applied times-frac3.6

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

  12. Applied add-cube-cbrt3.6

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

  13. Applied times-frac0.9

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

  14. Applied fma-def0.9

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

  15. Final simplification0.9

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

Reproduce

herbie shell --seed 2019347 +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))))