Average Error: 1.4 → 1.0
Time: 5.9s
Precision: binary64
\[x + y \cdot \frac{z - t}{a - t} \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{a - t}\\ x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{t_1 \cdot t_1} \cdot \left(\frac{\sqrt[3]{y}}{t_1} \cdot \left(z - t\right)\right) \end{array} \]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
t_1 := \sqrt[3]{a - t}\\
x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{t_1 \cdot t_1} \cdot \left(\frac{\sqrt[3]{y}}{t_1} \cdot \left(z - t\right)\right)
\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 (- a t))))
   (+
    x
    (* (/ (* (cbrt y) (cbrt y)) (* t_1 t_1)) (* (/ (cbrt y) t_1) (- z t))))))
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(a - t);
	return x + (((cbrt(y) * cbrt(y)) / (t_1 * t_1)) * ((cbrt(y) / t_1) * (z - 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

Original1.4
Target0.5
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

  1. Initial program 1.4

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

  2. Simplified1.4

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

  3. Taylor expanded in y around 0 10.5

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

  4. Simplified3.1

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

  5. Applied add-cube-cbrt_binary643.5

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

  6. Applied add-cube-cbrt_binary643.7

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

  7. Applied times-frac_binary643.7

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

  8. Applied associate-*l*_binary641.0

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

  9. Final simplification1.0

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

Reproduce

herbie shell --seed 2021215 
(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)))))