Average Error: 6.8 → 3.1
Time: 3.3s
Precision: binary64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.1834878622195064 \cdot 10^{103}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 2.727970289051773 \cdot 10^{121}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{elif}\;z \le 1.5000263552966557 \cdot 10^{248}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{\left(y - t\right) \cdot z}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.1834878622195064 \cdot 10^{103}:\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\

\mathbf{elif}\;z \le 2.727970289051773 \cdot 10^{121}:\\
\;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\

\mathbf{elif}\;z \le 1.5000263552966557 \cdot 10^{248}:\\
\;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{2}{\left(y - t\right) \cdot z}\\

\end{array}
double code(double x, double y, double z, double t) {
	return ((double) (((double) (x * 2.0)) / ((double) (((double) (y * z)) - ((double) (t * z))))));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if ((z <= -1.1834878622195064e+103)) {
		VAR = ((double) (((double) (x / z)) / ((double) (((double) (y - t)) / 2.0))));
	} else {
		double VAR_1;
		if ((z <= 2.7279702890517728e+121)) {
			VAR_1 = ((double) (x / ((double) (((double) (z * ((double) (y - t)))) / 2.0))));
		} else {
			double VAR_2;
			if ((z <= 1.5000263552966557e+248)) {
				VAR_2 = ((double) (((double) (x / ((double) (y - t)))) * ((double) (2.0 / z))));
			} else {
				VAR_2 = ((double) (x * ((double) (2.0 / ((double) (((double) (y - t)) * z))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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
Target2.1
Herbie3.1
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if z < -1.1834878622195064e103

    1. Initial program 13.4

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified10.1

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

    4. Applied *-un-lft-identity10.1

      \[\leadsto \frac{x}{\frac{z \cdot \left(y - t\right)}{\color{blue}{1 \cdot 2}}}\]

    5. Applied times-frac10.1

      \[\leadsto \frac{x}{\color{blue}{\frac{z}{1} \cdot \frac{y - t}{2}}}\]

    6. Applied associate-/r*2.1

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{z}{1}}}{\frac{y - t}{2}}}\]

    7. Simplified2.1

      \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{\frac{y - t}{2}}\]

    if -1.1834878622195064e103 < z < 2.727970289051773e121

    1. Initial program 3.2

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified3.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}}\]

    if 2.727970289051773e121 < z < 1.5000263552966557e248

    1. Initial program 13.4

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified11.2

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

    4. Applied div-inv11.2

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{z \cdot \left(y - t\right)}{2}}}\]

    5. Simplified10.3

      \[\leadsto x \cdot \color{blue}{\frac{\frac{2}{y - t}}{z}}\]
    6. Using strategy rm

    7. Applied div-inv10.3

      \[\leadsto x \cdot \frac{\color{blue}{2 \cdot \frac{1}{y - t}}}{z}\]

    8. Applied associate-/l*11.3

      \[\leadsto x \cdot \color{blue}{\frac{2}{\frac{z}{\frac{1}{y - t}}}}\]

    9. Simplified11.2

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

    11. Applied *-un-lft-identity11.2

      \[\leadsto x \cdot \frac{\color{blue}{1 \cdot 2}}{\left(y - t\right) \cdot z}\]

    12. Applied times-frac10.4

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{y - t} \cdot \frac{2}{z}\right)}\]

    13. Applied associate-*r*2.2

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{y - t}\right) \cdot \frac{2}{z}}\]

    14. Simplified2.2

      \[\leadsto \color{blue}{\frac{x}{y - t}} \cdot \frac{2}{z}\]

    if 1.5000263552966557e248 < z

    1. Initial program 14.3

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified8.4

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

    4. Applied div-inv8.4

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{z \cdot \left(y - t\right)}{2}}}\]

    5. Simplified7.8

      \[\leadsto x \cdot \color{blue}{\frac{\frac{2}{y - t}}{z}}\]
    6. Using strategy rm

    7. Applied div-inv7.8

      \[\leadsto x \cdot \frac{\color{blue}{2 \cdot \frac{1}{y - t}}}{z}\]

    8. Applied associate-/l*8.4

      \[\leadsto x \cdot \color{blue}{\frac{2}{\frac{z}{\frac{1}{y - t}}}}\]

    9. Simplified8.4

      \[\leadsto x \cdot \frac{2}{\color{blue}{\left(y - t\right) \cdot z}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.1834878622195064 \cdot 10^{103}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 2.727970289051773 \cdot 10^{121}:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{elif}\;z \le 1.5000263552966557 \cdot 10^{248}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{\left(y - t\right) \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.0450278273301259e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))