Average Error: 10.6 → 0.6
Time: 4.6s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[x + \left(y \cdot \left(\sqrt[3]{z - t} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right)\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\]
x + \frac{y \cdot \left(z - t\right)}{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

Original10.6
Target1.4
Herbie0.6
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Initial program 10.6

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

  2. Simplified1.5

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

  4. Applied add-cube-cbrt2.0

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

  5. Applied add-cube-cbrt1.8

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

  6. Applied times-frac1.8

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

  7. Applied associate-*r*0.6

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

  8. Simplified0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2020198 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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