Average Error: 14.9 → 0.7
Time: 3.8s
Precision: binary64
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2.6141934126185 \cdot 10^{+231}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -5.6230639977695645 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 9.8813129168249 \cdot 10^{-324}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 9.28092931853481 \cdot 10^{+259}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]
x \cdot \frac{\frac{y}{z} \cdot t}{t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} \leq -2.6141934126185 \cdot 10^{+231}:\\
\;\;\;\;y \cdot \frac{x}{z}\\

\mathbf{elif}\;\frac{y}{z} \leq -5.6230639977695645 \cdot 10^{-102}:\\
\;\;\;\;\frac{y}{z} \cdot x\\

\mathbf{elif}\;\frac{y}{z} \leq 9.8813129168249 \cdot 10^{-324}:\\
\;\;\;\;y \cdot \frac{x}{z}\\

\mathbf{elif}\;\frac{y}{z} \leq 9.28092931853481 \cdot 10^{+259}:\\
\;\;\;\;\frac{y}{z} \cdot x\\

\mathbf{else}:\\
\;\;\;\;\frac{y \cdot x}{z}\\

\end{array}
(FPCore (x y z t) :precision binary64 (* x (/ (* (/ y z) t) t)))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= (/ y z) -2.6141934126185e+231)
   (* y (/ x z))
   (if (<= (/ y z) -5.6230639977695645e-102)
     (* (/ y z) x)
     (if (<= (/ y z) 9.8813129168249e-324)
       (* y (/ x z))
       (if (<= (/ y z) 9.28092931853481e+259) (* (/ y z) x) (/ (* y x) z))))))
double code(double x, double y, double z, double t) {
	return x * (((y / z) * t) / t);
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y / z) <= -2.6141934126185e+231) {
		tmp = y * (x / z);
	} else if ((y / z) <= -5.6230639977695645e-102) {
		tmp = (y / z) * x;
	} else if ((y / z) <= 9.8813129168249e-324) {
		tmp = y * (x / z);
	} else if ((y / z) <= 9.28092931853481e+259) {
		tmp = (y / z) * x;
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

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

Original14.9
Target1.6
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} < -1.20672205123045 \cdot 10^{+245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < -5.907522236933906 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 5.658954423153415 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} < 2.0087180502407133 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 y z) < -2.6141934126185e231 or -5.62306399776956453e-102 < (/.f64 y z) < 9.88131e-324

    1. Initial program 21.3

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

    2. Simplified13.5

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

    4. Applied add-cube-cbrt_binary6413.9

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

    5. Applied *-un-lft-identity_binary6413.9

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

    6. Applied times-frac_binary6413.9

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

    7. Applied associate-*r*_binary643.6

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

    8. Simplified3.6

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\sqrt[3]{z}}\]

    9. Taylor expanded around 0 1.4

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]

    10. Simplified1.4

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}}\]

    if -2.6141934126185e231 < (/.f64 y z) < -5.62306399776956453e-102 or 9.88131e-324 < (/.f64 y z) < 9.28092931853481e259

    1. Initial program 9.5

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

    2. Simplified0.4

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]

    if 9.28092931853481e259 < (/.f64 y z)

    1. Initial program 50.5

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

    2. Simplified42.3

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

    4. Applied associate-*r/_binary640.5

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq -2.6141934126185 \cdot 10^{+231}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq -5.6230639977695645 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} \leq 9.8813129168249 \cdot 10^{-324}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} \leq 9.28092931853481 \cdot 10^{+259}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

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