Average Error: 6.8 → 3.5
Time: 4.8s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot 2 \le -2.07896827281715923 \cdot 10^{-181}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{\frac{z}{2}}\\ \mathbf{elif}\;x \cdot 2 \le 2.88293000746779607 \cdot 10^{154}:\\ \;\;\;\;\frac{1}{z \cdot \left(y - t\right)} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot 2 \le -2.07896827281715923 \cdot 10^{-181}:\\
\;\;\;\;\frac{\frac{x}{y - t}}{\frac{z}{2}}\\

\mathbf{elif}\;x \cdot 2 \le 2.88293000746779607 \cdot 10^{154}:\\
\;\;\;\;\frac{1}{z \cdot \left(y - t\right)} \cdot \left(x \cdot 2\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\

\end{array}
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 temp;
	if (((x * 2.0) <= -2.0789682728171592e-181)) {
		temp = ((x / (y - t)) / (z / 2.0));
	} else {
		double temp_1;
		if (((x * 2.0) <= 2.882930007467796e+154)) {
			temp_1 = ((1.0 / (z * (y - t))) * (x * 2.0));
		} else {
			temp_1 = ((1.0 / z) * (x / ((y - t) / 2.0)));
		}
		temp = temp_1;
	}
	return temp;
}

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
Target2.2
Herbie3.5
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x 2.0) < -2.0789682728171592e-181

    1. Initial program 8.1

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

    2. Simplified6.8

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

    4. Applied *-un-lft-identity6.8

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

    5. Applied *-commutative6.8

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

    6. Applied times-frac6.8

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

    7. Applied associate-/r*3.9

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

    8. Simplified3.9

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

    if -2.0789682728171592e-181 < (* x 2.0) < 2.882930007467796e+154

    1. Initial program 4.0

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

    2. Simplified2.8

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

    4. Applied div-inv2.8

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

    5. Applied *-un-lft-identity2.8

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

    6. Applied times-frac3.0

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

    7. Simplified3.0

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

    if 2.882930007467796e+154 < (* x 2.0)

    1. Initial program 16.0

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

    2. Simplified15.8

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

    4. Applied *-un-lft-identity15.8

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

    5. Applied times-frac15.7

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

    6. Applied *-un-lft-identity15.7

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

    7. Applied times-frac4.4

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

    8. Simplified4.4

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \frac{x}{\frac{y - t}{2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \le -2.07896827281715923 \cdot 10^{-181}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{\frac{z}{2}}\\ \mathbf{elif}\;x \cdot 2 \le 2.88293000746779607 \cdot 10^{154}:\\ \;\;\;\;\frac{1}{z \cdot \left(y - t\right)} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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))))