Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1

Average Error: 34.4 → 2.0

Time: 5.6s

Precision: binary64

Math

\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -8.615083971814368 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{\frac{z}{t}}{t}\\ \mathbf{elif}\;x \leq -7.167240071025731 \cdot 10^{-222} \lor \neg \left(x \leq 1.8116009526472228 \cdot 10^{-272}\right) \land x \leq 1.1255167977815244 \cdot 10^{+199}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot \frac{x}{y}\right) + \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{1}{t} \cdot \left(z \cdot \frac{z}{t}\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((double) ((((double) (x * x)) / ((double) (y * y))) + (((double) (z * z)) / ((double) (t * t)))));
}

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((x <= -8.615083971814368e+45)) {
		VAR = ((double) (((double) ((x / y) * (x / y))) + ((double) (z * ((z / t) / t)))));
	} else {
		double VAR_1;
		if (((x <= -7.167240071025731e-222) || (!(x <= 1.8116009526472228e-272) && (x <= 1.1255167977815244e+199)))) {
			VAR_1 = ((double) (((double) ((1.0 / y) * ((double) (x * (x / y))))) + ((double) ((z / t) * (z / t)))));
		} else {
			VAR_1 = ((double) (((double) ((x / y) * (x / y))) + ((double) ((1.0 / t) * ((double) (z * (z / t)))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	34.4
Target	0.4
Herbie	2.0

\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

1. Split input into 3 regimes

2. if x < -8.615083971814368e45

   1. Initial program Error: 44.6 bits

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]

   2. Simplified Error: 29.3 bits

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

   4. Applied add-sqr-sqrt Error: 64.0 bits

      \[\leadsto x \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y \cdot y} + z \cdot \frac{z}{t \cdot t}\]

   5. Applied times-frac Error: 64.0 bits

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

   6. Applied add-sqr-sqrt Error: 64.0 bits

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

   7. Applied unswap-sqr Error: 64.0 bits

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

   8. Simplified Error: 64.0 bits

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

   9. Simplified Error: 11.2 bits

      \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} + z \cdot \frac{z}{t \cdot t}\]
   10. Using strategy rm

   11. Applied associate-/r* Error: 3.7 bits

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

   if -8.615083971814368e45 < x < -7.167240071025731e-222 or 1.8116009526472228e-272 < x < 1.1255167977815244e199

   1. Initial program Error: 30.8 bits

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]

   2. Simplified Error: 23.7 bits

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

   4. Applied add-sqr-sqrt Error: 39.4 bits

      \[\leadsto x \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y \cdot y} + z \cdot \frac{z}{t \cdot t}\]

   5. Applied times-frac Error: 35.9 bits

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

   6. Applied add-sqr-sqrt Error: 36.0 bits

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

   7. Applied unswap-sqr Error: 33.3 bits

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

   8. Simplified Error: 33.2 bits

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

   9. Simplified Error: 14.2 bits

      \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} + z \cdot \frac{z}{t \cdot t}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt Error: 39.2 bits

       \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{t \cdot t}\]

   12. Applied times-frac Error: 34.5 bits

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

   13. Applied add-sqr-sqrt Error: 34.6 bits

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

   14. Applied unswap-sqr Error: 32.3 bits

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

   15. Simplified Error: 32.3 bits

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

   16. Simplified Error: 0.4 bits

       \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}}\]
   17. Using strategy rm

   18. Applied add-sqr-sqrt Error: 31.8 bits

       \[\leadsto \frac{x}{y} \cdot \frac{x}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} + \frac{z}{t} \cdot \frac{z}{t}\]

   19. Applied *-un-lft-identity Error: 31.8 bits

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

   20. Applied times-frac Error: 31.8 bits

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

   21. Applied add-sqr-sqrt Error: 31.9 bits

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

   22. Applied *-un-lft-identity Error: 31.9 bits

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

   23. Applied times-frac Error: 31.9 bits

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

   24. Applied swap-sqr Error: 32.3 bits

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

   25. Simplified Error: 32.2 bits

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

   26. Simplified Error: 1.2 bits

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

   if -7.167240071025731e-222 < x < 1.8116009526472228e-272 or 1.1255167977815244e199 < x

   1. Initial program Error: 40.2 bits

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]

   2. Simplified Error: 30.4 bits

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

   4. Applied add-sqr-sqrt Error: 50.1 bits

      \[\leadsto x \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y \cdot y} + z \cdot \frac{z}{t \cdot t}\]

   5. Applied times-frac Error: 40.4 bits

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

   6. Applied add-sqr-sqrt Error: 40.5 bits

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

   7. Applied unswap-sqr Error: 39.4 bits

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

   8. Simplified Error: 39.4 bits

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

   9. Simplified Error: 13.4 bits

      \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} + z \cdot \frac{z}{t \cdot t}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt Error: 38.4 bits

       \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{t \cdot t}\]

   12. Applied times-frac Error: 34.7 bits

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

   13. Applied add-sqr-sqrt Error: 34.7 bits

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

   14. Applied unswap-sqr Error: 32.3 bits

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

   15. Simplified Error: 32.2 bits

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

   16. Simplified Error: 0.4 bits

       \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}}\]
   17. Using strategy rm

   18. Applied add-sqr-sqrt Error: 32.1 bits

       \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \frac{z}{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}\]

   19. Applied *-un-lft-identity Error: 32.1 bits

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

   20. Applied times-frac Error: 32.0 bits

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

   21. Applied add-sqr-sqrt Error: 32.1 bits

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

   22. Applied *-un-lft-identity Error: 32.1 bits

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

   23. Applied times-frac Error: 32.1 bits

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

   24. Applied swap-sqr Error: 34.1 bits

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

   25. Simplified Error: 34.0 bits

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

   26. Simplified Error: 3.8 bits

       \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{1}{t} \cdot \color{blue}{\left(z \cdot \frac{z}{t}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification Error: 2.0 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.615083971814368 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{\frac{z}{t}}{t}\\ \mathbf{elif}\;x \leq -7.167240071025731 \cdot 10^{-222} \lor \neg \left(x \leq 1.8116009526472228 \cdot 10^{-272}\right) \land x \leq 1.1255167977815244 \cdot 10^{+199}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot \frac{x}{y}\right) + \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{1}{t} \cdot \left(z \cdot \frac{z}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :herbie-target
  (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))