Average Error: 1.5 → 0.6
Time: 5.1s
Precision: binary64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[x + \left(y \cdot \left(\sqrt[3]{z - t} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right)\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\]
x + y \cdot \frac{z - t}{a - t}
↓
x + \left(y \cdot \left(\sqrt[3]{z - t} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right)\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}
(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- a t)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (+
  x
  (*
   (*
    y
    (* (cbrt (- z t)) (/ (cbrt (- z t)) (* (cbrt (- a t)) (cbrt (- a t))))))
   (/ (cbrt (- z t)) (cbrt (- a t))))))
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (y * (((double) (z - t)) / ((double) (a - t)))))));
}
↓
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) cbrt(((double) (z - t)))) / ((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t))))))))))) * (((double) cbrt(((double) (z - t)))) / ((double) cbrt(((double) (a - t)))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.5
Target0.4
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program Error: 1.5 bits

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

  3. Applied add-cube-cbrtError: 2.0 bits

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

  4. Applied add-cube-cbrtError: 1.9 bits

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

  5. Applied times-fracError: 1.9 bits

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

  6. Applied associate-*r*Error: 0.5 bits

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

  7. SimplifiedError: 0.6 bits

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

  8. Final simplificationError: 0.6 bits

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

Reproduce

herbie shell --seed 2020204 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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