Average Error: 11.0 → 1.1
Time: 5.6s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[x + \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{a - z}} \cdot \left(\frac{y - z}{\sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{a - z}}\right)\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
x + \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{a - z}} \cdot \left(\frac{y - z}{\sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{a - z}}\right)
(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) t) (- a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (+
  x
  (*
   (/ (* (cbrt t) (cbrt t)) (cbrt (- a z)))
   (* (/ (- y z) (cbrt (- a z))) (/ (cbrt t) (cbrt (- a z)))))))
double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * t) / (a - z));
}
↓
double code(double x, double y, double z, double t, double a) {
	return x + (((cbrt(t) * cbrt(t)) / cbrt(a - z)) * (((y - z) / cbrt(a - z)) * (cbrt(t) / cbrt(a - z))));
}

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

Original11.0
Target0.6
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program 11.0

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

  2. Taylor expanded around 0 11.0

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

  3. Simplified1.4

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

  5. Applied add-cube-cbrt_binary642.0

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

  6. Applied *-un-lft-identity_binary642.0

    \[\leadsto x + t \cdot \frac{\color{blue}{1 \cdot \left(y - z\right)}}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}\]

  7. Applied times-frac_binary642.0

    \[\leadsto x + t \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{y - z}{\sqrt[3]{a - z}}\right)}\]

  8. Applied associate-*r*_binary642.3

    \[\leadsto x + \color{blue}{\left(t \cdot \frac{1}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{y - z}{\sqrt[3]{a - z}}}\]

  9. Simplified2.3

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

  11. Applied add-cube-cbrt_binary642.4

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

  12. Applied times-frac_binary642.4

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

  13. Applied associate-*l*_binary641.1

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

  14. Simplified1.1

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

  15. Final simplification1.1

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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