Average Error: 10.7 → 1.0
Time: 6.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\mathsf{fma}\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}{1}}, \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{z - a}}{y}}, x\right)\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\mathsf{fma}\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}{1}}, \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 r566820 = x;
        double r566821 = y;
        double r566822 = z;
        double r566823 = t;
        double r566824 = r566822 - r566823;
        double r566825 = r566821 * r566824;
        double r566826 = a;
        double r566827 = r566822 - r566826;
        double r566828 = r566825 / r566827;
        double r566829 = r566820 + r566828;
        return r566829;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r566830 = z;
        double r566831 = t;
        double r566832 = r566830 - r566831;
        double r566833 = cbrt(r566832);
        double r566834 = r566833 * r566833;
        double r566835 = a;
        double r566836 = r566830 - r566835;
        double r566837 = cbrt(r566836);
        double r566838 = r566837 * r566837;
        double r566839 = 1.0;
        double r566840 = r566838 / r566839;
        double r566841 = r566834 / r566840;
        double r566842 = y;
        double r566843 = r566837 / r566842;
        double r566844 = r566833 / r566843;
        double r566845 = x;
        double r566846 = fma(r566841, r566844, r566845);
        return r566846;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.7
Target1.4
Herbie1.0
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.7

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

  2. Simplified3.1

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

  4. Applied clear-num3.3

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

  6. Applied fma-udef3.3

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

  7. Simplified3.1

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

  9. Applied *-un-lft-identity3.1

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

  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-cbrt3.5

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

  13. Applied times-frac1.0

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

  14. Applied fma-def1.0

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

  15. Final simplification1.0

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

Reproduce

herbie shell --seed 2020083 +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))))