Average Error: 6.2 → 1.5
Time: 4.2s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}} \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}} \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}
(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) a)))
↓
(FPCore (x y z t a)
 :precision binary64
 (+
  x
  (*
   (/
    (* (cbrt y) (cbrt y))
    (/ (* (cbrt a) (cbrt a)) (* (cbrt (- z t)) (cbrt (- z t)))))
   (/ (cbrt y) (/ (cbrt a) (cbrt (- z t)))))))
double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / a);
}
↓
double code(double x, double y, double z, double t, double a) {
	return x + (((cbrt(y) * cbrt(y)) / ((cbrt(a) * cbrt(a)) / (cbrt(z - t) * cbrt(z - t)))) * (cbrt(y) / (cbrt(a) / cbrt(z - t))));
}

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

Original6.2
Target0.7
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.2

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

  3. Applied associate-/l*_binary64_109355.8

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

  5. Applied add-cube-cbrt_binary64_110256.2

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

  6. Applied add-cube-cbrt_binary64_110256.3

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

  7. Applied times-frac_binary64_109966.3

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

  8. Applied add-cube-cbrt_binary64_110256.4

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

  9. Applied times-frac_binary64_109961.5

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

  10. Final simplification1.5

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

Reproduce

herbie shell --seed 2020356 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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