Average Error: 7.3 → 2.8
Time: 3.6s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.2975885083579693 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 1.49678783062208739 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y - t}{2}}}{z}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.2975885083579693 \cdot 10^{-23}:\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\

\mathbf{elif}\;z \le 1.49678783062208739 \cdot 10^{-9}:\\
\;\;\;\;\frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{x}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{\frac{y - t}{2}}}{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 <= -5.297588508357969e-23)) {
		VAR = ((double) (((double) (x / z)) / ((double) (((double) (y - t)) / 2.0))));
	} else {
		double VAR_1;
		if ((z <= 1.4967878306220874e-09)) {
			VAR_1 = ((double) (1.0 / ((double) (((double) (((double) (z * ((double) (y - t)))) / 2.0)) / x))));
		} else {
			VAR_1 = ((double) (((double) (x / ((double) (((double) (y - t)) / 2.0)))) / z));
		}
		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

Original7.3
Target2.6
Herbie2.8
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -5.297588508357969e-23

    1. Initial program 10.2

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

    2. Simplified8.3

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

    4. Applied *-un-lft-identity8.3

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

    5. Applied times-frac8.3

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

    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 -5.297588508357969e-23 < z < 1.4967878306220874e-09

    1. Initial program 3.1

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

    2. Simplified3.1

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

    4. Applied clear-num3.5

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

    if 1.4967878306220874e-09 < z

    1. Initial program 10.9

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

    2. Simplified9.1

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

    4. Applied *-un-lft-identity9.1

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

    5. Applied times-frac9.0

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

    6. Applied *-un-lft-identity9.0

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

    7. Applied times-frac2.5

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

    8. Simplified2.5

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \frac{x}{\frac{y - t}{2}}\]
    9. Using strategy rm

    10. Applied associate-*l/2.4

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

    11. Simplified2.4

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y - t}{2}}}}{z}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.2975885083579693 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 1.49678783062208739 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y - t}{2}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 1.0450278273301259e-269) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2)))

  (/ (* x 2) (- (* y z) (* t z))))