Average Error: 11.3 → 1.0
Time: 5.9s
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 r618122 = x;
        double r618123 = y;
        double r618124 = z;
        double r618125 = t;
        double r618126 = r618124 - r618125;
        double r618127 = r618123 * r618126;
        double r618128 = a;
        double r618129 = r618124 - r618128;
        double r618130 = r618127 / r618129;
        double r618131 = r618122 + r618130;
        return r618131;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r618132 = z;
        double r618133 = t;
        double r618134 = r618132 - r618133;
        double r618135 = cbrt(r618134);
        double r618136 = r618135 * r618135;
        double r618137 = a;
        double r618138 = r618132 - r618137;
        double r618139 = cbrt(r618138);
        double r618140 = r618139 * r618139;
        double r618141 = 1.0;
        double r618142 = r618140 / r618141;
        double r618143 = r618136 / r618142;
        double r618144 = y;
        double r618145 = r618139 / r618144;
        double r618146 = r618135 / r618145;
        double r618147 = x;
        double r618148 = fma(r618143, r618146, r618147);
        return r618148;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.3
Target1.2
Herbie1.0
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 11.3

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

  2. Simplified3.1

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

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

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

  5. Applied add-cube-cbrt3.5

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

  6. Applied times-frac3.5

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

  7. Simplified3.5

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

  9. Applied fma-udef3.5

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

  10. Simplified3.1

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

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

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

  13. Applied add-cube-cbrt3.6

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

  14. Applied times-frac3.6

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

  15. Applied add-cube-cbrt3.5

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

  16. Applied times-frac1.0

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

  17. Applied fma-def1.0

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

  18. 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 2020002 +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))))