Average Error: 6.2 → 1.9
Time: 5.6s
Precision: binary64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -1.0891107148063213 \cdot 10^{+162}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq -3.91463074575322 \cdot 10^{-215}:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot z - y \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \leq -1.0891107148063213 \cdot 10^{+162}:\\
\;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\

\mathbf{elif}\;y \cdot \left(z - t\right) \leq -3.91463074575322 \cdot 10^{-215}:\\
\;\;\;\;x - \frac{1}{\frac{a}{y \cdot z - y \cdot t}}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\left(z - t\right) \cdot \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}\\

\end{array}
(FPCore (x y z t a) :precision binary64 (- x (/ (* y (- z t)) a)))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* y (- z t)) -1.0891107148063213e+162)
   (- x (* (- z t) (/ y a)))
   (if (<= (* y (- z t)) -3.91463074575322e-215)
     (- x (/ 1.0 (/ a (- (* y z) (* y t)))))
     (- x (/ (* (- z t) (/ y (* (cbrt a) (cbrt a)))) (cbrt a))))))
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) {
	double tmp;
	if ((y * (z - t)) <= -1.0891107148063213e+162) {
		tmp = x - ((z - t) * (y / a));
	} else if ((y * (z - t)) <= -3.91463074575322e-215) {
		tmp = x - (1.0 / (a / ((y * z) - (y * t))));
	} else {
		tmp = x - (((z - t) * (y / (cbrt(a) * cbrt(a)))) / cbrt(a));
	}
	return tmp;
}

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
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 y (-.f64 z t)) < -1.0891107148063213e162

    1. Initial program 22.5

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

    3. Applied clear-num_binary6422.5

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

    4. Simplified22.5

      \[\leadsto x - \frac{1}{\color{blue}{\frac{a}{\left(z - t\right) \cdot y}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity_binary6422.5

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

    7. Applied times-frac_binary641.5

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

    8. Applied add-cube-cbrt_binary641.5

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

    9. Applied times-frac_binary641.6

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

    10. Simplified1.5

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

    11. Simplified1.4

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

    if -1.0891107148063213e162 < (*.f64 y (-.f64 z t)) < -3.91463074575321983e-215

    1. Initial program 0.1

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

    3. Applied clear-num_binary640.2

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

    4. Simplified0.2

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

    5. Taylor expanded around inf 0.2

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

    if -3.91463074575321983e-215 < (*.f64 y (-.f64 z t))

    1. Initial program 5.7

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

    3. Applied add-cube-cbrt_binary646.1

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

    4. Applied associate-/r*_binary646.1

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

    5. Simplified2.9

      \[\leadsto x - \frac{\color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(z - t\right)}}{\sqrt[3]{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -1.0891107148063213 \cdot 10^{+162}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq -3.91463074575322 \cdot 10^{-215}:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot z - y \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :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)))