Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.8 → 2.3

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.4988469650336908 \cdot 10^{67}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{\sqrt{2}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\frac{y - t}{\sqrt{2}}}{\sqrt[3]{x}}}\\ \mathbf{elif}\;z \le 2.51302619308367974 \cdot 10^{-42}:\\ \;\;\;\;1 \cdot \frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \end{array}\]

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 temp;
	if ((z <= -1.4988469650336908e+67)) {
		temp = (1.0 / (((z / sqrt(2.0)) / (cbrt(x) * cbrt(x))) * (((y - t) / sqrt(2.0)) / cbrt(x))));
	} else {
		double temp_1;
		if ((z <= 2.5130261930836797e-42)) {
			temp_1 = (1.0 * (x / ((z * (y - t)) / 2.0)));
		} else {
			temp_1 = ((x / z) / ((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

Target

Original	6.8
Target	2.1
Herbie	2.3

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

Derivation

1. Split input into 3 regimes

2. if z < -1.4988469650336908e+67

   1. Initial program 11.9

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

   2. Simplified 9.5

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

   4. Applied clear-num 9.8

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

   6. Applied add-cube-cbrt 10.2

      \[\leadsto \frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}\]

   7. Applied add-sqr-sqrt 10.2

      \[\leadsto \frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\]

   8. Applied times-frac 10.2

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{z}{\sqrt{2}} \cdot \frac{y - t}{\sqrt{2}}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\]

   9. Applied times-frac 2.2

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z}{\sqrt{2}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\frac{y - t}{\sqrt{2}}}{\sqrt[3]{x}}}}\]

   if -1.4988469650336908e+67 < z < 2.5130261930836797e-42

   1. Initial program 2.5

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

   2. Simplified 2.5

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

   4. Applied clear-num 2.8

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

   6. Applied *-un-lft-identity 2.8

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

   7. Applied *-un-lft-identity 2.8

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

   8. Applied times-frac 2.8

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

   9. Applied add-sqr-sqrt 2.8

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

   10. Applied times-frac 2.8

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

   11. Simplified 2.8

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

   12. Simplified 2.5

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

   if 2.5130261930836797e-42 < z

   1. Initial program 9.8

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

   2. Simplified 8.0

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

   4. Applied *-un-lft-identity 8.0

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

   5. Applied times-frac 8.0

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

   6. Applied associate-/r* 2.1

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

   7. Simplified 2.1

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.4988469650336908 \cdot 10^{67}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{\sqrt{2}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\frac{y - t}{\sqrt{2}}}{\sqrt[3]{x}}}\\ \mathbf{elif}\;z \le 2.51302619308367974 \cdot 10^{-42}:\\ \;\;\;\;1 \cdot \frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +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))))