Average Error: 16.4 → 7.6
Time: 9.7s
Precision: binary64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[x + \left(y - \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\sqrt[3]{z - t} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\right)\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
x + \left(y - \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\sqrt[3]{z - t} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\right)
(FPCore (x y z t a) :precision binary64 (- (+ x y) (/ (* (- z t) y) (- a t))))
↓
(FPCore (x y z t a)
 :precision binary64
 (+
  x
  (-
   y
   (*
    (/ (cbrt (- z t)) (cbrt (- a t)))
    (*
     (cbrt (- z t))
     (* (/ (cbrt (- z t)) (cbrt (- a t))) (/ y (cbrt (- a t)))))))))
double code(double x, double y, double z, double t, double a) {
	return (x + y) - (((z - t) * y) / (a - t));
}
↓
double code(double x, double y, double z, double t, double a) {
	return x + (y - ((cbrt(z - t) / cbrt(a - t)) * (cbrt(z - t) * ((cbrt(z - t) / cbrt(a - t)) * (y / cbrt(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

Original16.4
Target8.7
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Initial program 16.4

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

  3. Applied add-cube-cbrt_binary64_1628516.6

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

  4. Applied times-frac_binary64_1625911.7

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

  6. Applied add-cube-cbrt_binary64_1628511.8

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

  7. Applied times-frac_binary64_1625911.8

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

  8. Applied associate-*l*_binary64_1619611.2

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

  9. Simplified11.2

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

  11. Applied associate-*r*_binary64_1619511.1

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

  12. Simplified11.2

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

  14. Applied associate--l+_binary64_161927.6

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

  15. Simplified7.6

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

  16. Final simplification7.6

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

Reproduce

herbie shell --seed 2020280 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

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