Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 7.0 → 2.9

Time: 4.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -7.87198614559774051 \cdot 10^{-149} \lor \neg \left(x \le 5.9189247350309146 \cdot 10^{32}\right):\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{\frac{y - t}{2}}\\ \end{array}\]

C

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 VAR;
	if (((x <= -7.87198614559774e-149) || !(x <= 5.9189247350309146e+32))) {
		VAR = ((1.0 / z) * (x / ((y - t) / 2.0)));
	} else {
		VAR = ((x / z) * (1.0 / ((y - t) / 2.0)));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.0
Target	2.3
Herbie	2.9

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

Derivation

1. Split input into 2 regimes

2. if x < -7.87198614559774e-149 or 5.9189247350309146e+32 < x

   1. Initial program 9.6

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

   2. Simplified 8.7

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

   4. Applied *-un-lft-identity 8.7

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

   5. Applied times-frac 8.7

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

   6. Applied *-un-lft-identity 8.7

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

   7. Applied times-frac 3.2

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

   8. Simplified 3.2

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

   if -7.87198614559774e-149 < x < 5.9189247350309146e+32

   1. Initial program 3.7

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

   2. Simplified 2.2

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

   4. Applied *-un-lft-identity 2.2

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

   5. Applied times-frac 2.2

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

   6. Applied *-un-lft-identity 2.2

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

   7. Applied times-frac 8.0

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

   8. Simplified 8.0

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

   10. Applied div-inv 8.1

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

   11. Applied associate-*r* 2.4

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

   12. Simplified 2.4

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.87198614559774051 \cdot 10^{-149} \lor \neg \left(x \le 5.9189247350309146 \cdot 10^{32}\right):\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{\frac{y - t}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 +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))))