Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 7.1 → 2.7

Time: 5.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.8263896362557053 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{1}}{1} \cdot \frac{\frac{x}{\frac{y - t}{2}}}{z}\\ \mathbf{elif}\;x \le 3.4056700725501648 \cdot 10^{171}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\frac{y - t}{2}} \cdot \sqrt[3]{\frac{y - t}{2}}}}{z} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\frac{y - t}{2}}}\\ \end{array}\]

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 ((x <= -8.826389636255705e-53)) {
		temp = ((sqrt(1.0) / 1.0) * ((x / ((y - t) / 2.0)) / z));
	} else {
		double temp_1;
		if ((x <= 3.405670072550165e+171)) {
			temp_1 = ((x / ((y - t) * z)) * 2.0);
		} else {
			temp_1 = ((((cbrt(x) * cbrt(x)) / (cbrt(((y - t) / 2.0)) * cbrt(((y - t) / 2.0)))) / z) * (cbrt(x) / cbrt(((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	7.1
Target	2.1
Herbie	2.7

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

Derivation

1. Split input into 3 regimes

2. if x < -8.826389636255705e-53

   1. Initial program 9.9

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

   2. Simplified 8.9

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

   4. Applied *-un-lft-identity 8.9

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

   5. Applied times-frac 8.9

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

   6. Applied *-un-lft-identity 8.9

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

   7. Applied times-frac 2.4

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

   8. Simplified 2.4

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

   10. Applied *-un-lft-identity 2.4

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

   11. Applied add-sqr-sqrt 2.4

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

   12. Applied times-frac 2.4

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

   13. Applied associate-*l* 2.4

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

   14. Simplified 2.3

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

   if -8.826389636255705e-53 < x < 3.405670072550165e+171

   1. Initial program 4.3

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

   2. Simplified 2.9

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

   4. Applied *-un-lft-identity 2.9

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

   5. Applied times-frac 2.9

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

   6. Applied *-un-lft-identity 2.9

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

   7. Applied times-frac 6.9

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

   8. Simplified 6.9

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

   10. Applied associate-/r/ 6.9

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

   11. Applied associate-*r* 6.9

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

   12. Simplified 2.9

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

   if 3.405670072550165e+171 < x

   1. Initial program 17.8

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

   2. Simplified 17.5

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

   4. Applied *-un-lft-identity 17.5

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

   5. Applied times-frac 17.5

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

   6. Applied *-un-lft-identity 17.5

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

   7. Applied times-frac 4.6

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

   8. Simplified 4.6

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

   10. Applied add-cube-cbrt 5.5

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

   11. Applied add-cube-cbrt 5.6

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

   12. Applied times-frac 5.6

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

   13. Applied associate-*r* 1.8

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

   14. Simplified 1.7

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.8263896362557053 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{1}}{1} \cdot \frac{\frac{x}{\frac{y - t}{2}}}{z}\\ \mathbf{elif}\;x \le 3.4056700725501648 \cdot 10^{171}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\frac{y - t}{2}} \cdot \sqrt[3]{\frac{y - t}{2}}}}{z} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\frac{y - t}{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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))))