Average Error: 10.8 → 1.9
Time: 5.2s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t}{\sqrt[3]{a - z}}\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t}{\sqrt[3]{a - z}}
(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) t) (- a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (+ x (* (/ (- y z) (* (cbrt (- a z)) (cbrt (- a z)))) (/ t (cbrt (- a z))))))
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + (((double) (((double) (y - z)) * t)) / ((double) (a - z)))));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) ((((double) (y - z)) / ((double) (((double) cbrt(((double) (a - z)))) * ((double) cbrt(((double) (a - z))))))) * (t / ((double) cbrt(((double) (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

Original10.8
Target0.6
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program Error: 10.8 bits

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

  2. SimplifiedError: 2.9 bits

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

  4. Applied add-cube-cbrtError: 3.3 bits

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

  5. Applied *-un-lft-identityError: 3.3 bits

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

  6. Applied times-fracError: 3.3 bits

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

  7. Applied associate-*r*Error: 1.9 bits

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

  8. SimplifiedError: 1.9 bits

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

  9. Final simplificationError: 1.9 bits

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

Reproduce

herbie shell --seed 2020204 
(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))))