Average Error: 1.4 → 0.5
Time: 7.4s
Precision: binary64
\[x + y \cdot \frac{z - t}{a - t} \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{z - t}\\ t_2 := \sqrt[3]{a - t}\\ x + \left(y \cdot \frac{t_1 \cdot t_1}{t_2 \cdot t_2}\right) \cdot \frac{t_1}{t_2} \end{array} \]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
t_1 := \sqrt[3]{z - t}\\
t_2 := \sqrt[3]{a - t}\\
x + \left(y \cdot \frac{t_1 \cdot t_1}{t_2 \cdot t_2}\right) \cdot \frac{t_1}{t_2}
\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- a t)))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (cbrt (- z t))) (t_2 (cbrt (- a t))))
   (+ x (* (* y (/ (* t_1 t_1) (* t_2 t_2))) (/ t_1 t_2)))))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (a - t)));
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = cbrt(z - t);
	double t_2 = cbrt(a - t);
	return x + ((y * ((t_1 * t_1) / (t_2 * t_2))) * (t_1 / t_2));
}

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

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

Derivation

  1. Initial program 1.4

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

  2. Applied add-cube-cbrt_binary641.9

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

  3. Applied add-cube-cbrt_binary641.8

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

  4. Applied times-frac_binary641.8

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

  5. Applied associate-*r*_binary640.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2022076 
(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)))))