Average Error: 6.8 → 2.4
Time: 7.2s
Precision: binary64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -7.066722543898828 \cdot 10^{-27} \lor \neg \left(z \leq 2.814061237892244 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{\frac{2 \cdot x}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{z \cdot \left(y - t\right)}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -7.066722543898828 \cdot 10^{-27} \lor \neg \left(z \leq 2.814061237892244 \cdot 10^{-40}\right):\\
\;\;\;\;\frac{\frac{2 \cdot x}{y - t}}{z}\\

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

\end{array}
(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7.066722543898828e-27) (not (<= z 2.814061237892244e-40)))
   (/ (/ (* 2.0 x) (- y t)) z)
   (/ (* 2.0 x) (* z (- y t)))))
double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.066722543898828e-27) || !(z <= 2.814061237892244e-40)) {
		tmp = ((2.0 * x) / (y - t)) / z;
	} else {
		tmp = (2.0 * x) / (z * (y - t));
	}
	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

Original6.8
Target2.3
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < 1.0450278273301259 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -7.06681e-27 or 2.81409e-40 < z

    1. Initial program 9.6

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

    2. Simplified7.2

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

    4. Applied associate-*r/_binary64_19192.2

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

    6. Applied associate-*r/_binary64_19192.1

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

    7. Simplified2.1

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

    if -7.06681e-27 < z < 2.81409e-40

    1. Initial program 2.8

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

    2. Simplified3.1

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

    4. Applied associate-*r/_binary64_191910.7

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

    6. Applied associate-*r/_binary64_191910.7

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

    7. Applied associate-/l/_binary64_19242.8

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.066722543898828 \cdot 10^{-27} \lor \neg \left(z \leq 2.814061237892244 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{\frac{2 \cdot x}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{z \cdot \left(y - t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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))))