Average Error: 6.8 → 2.5
Time: 3.6s
Precision: binary64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -5.4934368200861185 \cdot 10^{+45}:\\ \;\;\;\;\frac{x \cdot 2}{y - t} \cdot \frac{1}{z}\\ \mathbf{elif}\;z \leq 4.631462716974067 \cdot 10^{-61}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -5.4934368200861185 \cdot 10^{+45}:\\
\;\;\;\;\frac{x \cdot 2}{y - t} \cdot \frac{1}{z}\\

\mathbf{elif}\;z \leq 4.631462716974067 \cdot 10^{-61}:\\
\;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\

\end{array}
(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.4934368200861185e+45)
   (* (/ (* x 2.0) (- y t)) (/ 1.0 z))
   (if (<= z 4.631462716974067e-61)
     (/ (* x 2.0) (* z (- y t)))
     (/ (/ (* x 2.0) (- y t)) z))))
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 <= -5.4934368200861185e+45) {
		tmp = ((x * 2.0) / (y - t)) * (1.0 / z);
	} else if (z <= 4.631462716974067e-61) {
		tmp = (x * 2.0) / (z * (y - t));
	} else {
		tmp = ((x * 2.0) / (y - t)) / 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

Original6.8
Target2.3
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < 1.0450278273301259 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -5.4934368200861185e45

    1. Initial program 11.8

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

    2. Simplified8.8

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

    4. Applied associate-*r/_binary642.9

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

    6. Applied associate-*r/_binary642.8

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

    8. Applied div-inv_binary642.9

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

    if -5.4934368200861185e45 < z < 4.63146271697406693e-61

    1. Initial program 2.8

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

    2. Simplified2.9

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

    4. Applied associate-*r/_binary649.7

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

    6. Applied associate-*r/_binary649.7

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

    7. Applied associate-/l/_binary642.8

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

    8. Simplified2.8

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

    if 4.63146271697406693e-61 < z

    1. Initial program 9.0

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

    2. Simplified6.8

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

    4. Applied associate-*r/_binary641.9

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

    6. Applied associate-*r/_binary642.0

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4934368200861185 \cdot 10^{+45}:\\ \;\;\;\;\frac{x \cdot 2}{y - t} \cdot \frac{1}{z}\\ \mathbf{elif}\;z \leq 4.631462716974067 \cdot 10^{-61}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\ \end{array}\]

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))))