Average Error: 6.2 → 0.9
Time: 4.1s
Precision: binary64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[x + \left(\left(y - x\right) \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
x + \left(\left(y - x\right) \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}
double code(double x, double y, double z, double t) {
	return ((double) (x + (((double) (((double) (y - x)) * z)) / t)));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (x + ((double) (((double) (((double) (y - x)) * ((double) (((double) cbrt(z)) * (((double) cbrt(z)) / ((double) (((double) cbrt(t)) * ((double) cbrt(t))))))))) * (((double) cbrt(z)) / ((double) cbrt(t)))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.2
Target2.1
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Initial program Error: 6.2 bits

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

  2. SimplifiedError: 1.9 bits

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

  4. Applied add-cube-cbrtError: 2.4 bits

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

  5. Applied add-cube-cbrtError: 2.5 bits

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

  6. Applied times-fracError: 2.5 bits

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

  7. Applied associate-*r*Error: 0.9 bits

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

  8. SimplifiedError: 0.9 bits

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

  9. Final simplificationError: 0.9 bits

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

Reproduce

herbie shell --seed 2020200 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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