Average Error: 1.4 → 0.8
Time: 5.5min
Precision: binary64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[x + \left(y \cdot \frac{\sqrt[3]{z - t} \cdot \left(\sqrt[3]{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \sqrt[3]{\sqrt[3]{z - t}}\right)}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{z - a}}\]
x + y \cdot \frac{z - t}{z - a}
↓
x + \left(y \cdot \frac{\sqrt[3]{z - t} \cdot \left(\sqrt[3]{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}} \cdot \sqrt[3]{\sqrt[3]{z - t}}\right)}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{z - a}}
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (y * (((double) (z - t)) / ((double) (z - a)))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (((double) (y * (((double) (((double) cbrt(((double) (z - t)))) * ((double) (((double) cbrt(((double) (((double) cbrt(((double) (z - t)))) * ((double) cbrt(((double) (z - t)))))))) * ((double) cbrt(((double) cbrt(((double) (z - t)))))))))) / ((double) (((double) cbrt(((double) (z - a)))) * ((double) cbrt(((double) (z - a))))))))) * (((double) cbrt(((double) (z - t)))) / ((double) cbrt(((double) (z - a)))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

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Results

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Target

Original1.4
Target1.3
Herbie0.8
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 1.4

    \[x + y \cdot \frac{z - t}{z - a}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.9

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

  4. Applied add-cube-cbrt1.8

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

  5. Applied times-frac1.8

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

  6. Applied associate-*r*0.6

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

  8. Applied add-cube-cbrt0.7

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

  9. Applied cbrt-prod0.8

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

  10. Final simplification0.8

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

Reproduce

herbie shell --seed 2020181 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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