Average Error: 6.8 → 1.9
Time: 23.6s
Precision: binary64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[x + \frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{y - x}}{\frac{\sqrt[3]{t}}{z}}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
x + \frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{y - x}}{\frac{\sqrt[3]{t}}{z}}
(FPCore (x y z t) :precision binary64 (+ x (/ (* (- y x) z) t)))
↓
(FPCore (x y z t)
 :precision binary64
 (+
  x
  (*
   (/ (* (cbrt (- y x)) (cbrt (- y x))) (* (cbrt t) (cbrt t)))
   (/ (cbrt (- y x)) (/ (cbrt t) z)))))
double code(double x, double y, double z, double t) {
	return x + (((y - x) * z) / t);
}
↓
double code(double x, double y, double z, double t) {
	return x + (((cbrt(y - x) * cbrt(y - x)) / (cbrt(t) * cbrt(t))) * (cbrt(y - x) / (cbrt(t) / z)));
}

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

Original6.8
Target2.0
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Initial program 6.8

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

  2. Simplified2.0

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

  4. Applied *-un-lft-identity_binary64_79212.0

    \[\leadsto x + \frac{y - x}{\frac{t}{\color{blue}{1 \cdot z}}}\]

  5. Applied add-cube-cbrt_binary64_79562.5

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

  6. Applied times-frac_binary64_79272.5

    \[\leadsto x + \frac{y - x}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{1} \cdot \frac{\sqrt[3]{t}}{z}}}\]

  7. Applied add-cube-cbrt_binary64_79562.6

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

  8. Applied times-frac_binary64_79271.9

    \[\leadsto x + \color{blue}{\frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{1}} \cdot \frac{\sqrt[3]{y - x}}{\frac{\sqrt[3]{t}}{z}}}\]

  9. Final simplification1.9

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

Reproduce

herbie shell --seed 2021176 
(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)))