Average Error: 1.3 → 1.4
Time: 12.0s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(y, \frac{z}{a - t} - \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{a - t}}, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(y, \frac{z}{a - t} - \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{a - t}}, x\right)
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (y * ((double) (((double) (z - t)) / ((double) (a - t))))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((double) fma(y, ((double) (((double) (z / ((double) (a - t)))) - ((double) (((double) (((double) (((double) cbrt(t)) * ((double) cbrt(t)))) / ((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))) * ((double) (((double) cbrt(t)) / ((double) cbrt(((double) (a - t)))))))))), x));
}

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

Original1.3
Target0.4
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.3

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

  2. Simplified1.3

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

  4. Applied div-sub1.3

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

  6. Applied add-cube-cbrt1.6

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

  7. Applied add-cube-cbrt1.4

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

  8. Applied times-frac1.4

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

  9. Final simplification1.4

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(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 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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