Average Error: 10.6 → 1.0
Time: 5.3s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}{\sqrt[3]{z - t}}} \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{z - a}}{y}} + x\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}{\sqrt[3]{z - t}}} \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{z - a}}{y}} + x
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (((double) (y * ((double) (z - t)))) / ((double) (z - a))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (((double) (((double) cbrt(((double) (z - t)))) / ((double) (((double) (((double) cbrt(((double) (z - a)))) * ((double) cbrt(((double) (z - a)))))) / ((double) cbrt(((double) (z - t)))))))) * ((double) (((double) cbrt(((double) (z - t)))) / ((double) (((double) cbrt(((double) (z - a)))) / y)))))) + x));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

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Results

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Target

Original10.6
Target1.2
Herbie1.0
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.6

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

  2. Simplified2.8

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

  4. Applied clear-num3.0

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

  6. Applied fma-udef3.0

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

  7. Simplified2.8

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

  9. Applied *-un-lft-identity2.8

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

  10. Applied add-cube-cbrt3.3

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

  11. Applied times-frac3.3

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

  12. Applied add-cube-cbrt3.2

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

  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. Simplified1.0

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

  15. Final simplification1.0

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

Reproduce

herbie shell --seed 2020113 +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))))