Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1

Average Error: 2.3 → 2.4

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -9.3910125961314414 \cdot 10^{-161} \lor \neg \left(y \le 6.0401320354982989 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{x - y}{z - y} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1}}{1} \cdot \left(\left(x - y\right) \cdot \frac{t}{z - y}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double temp;
	if (((y <= -9.391012596131441e-161) || !(y <= 6.040132035498299e-75))) {
		temp = (((x - y) / (z - y)) * t);
	} else {
		temp = ((sqrt(1.0) / 1.0) * ((x - y) * (t / (z - y))));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.3
Target	2.3
Herbie	2.4

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

Derivation

1. Split input into 2 regimes

2. if y < -9.391012596131441e-161 or 6.040132035498299e-75 < y

   1. Initial program 1.0

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

   if -9.391012596131441e-161 < y < 6.040132035498299e-75

   1. Initial program 5.7

      \[\frac{x - y}{z - y} \cdot t\]
   2. Using strategy rm

   3. Applied clear-num 6.1

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

   5. Applied *-un-lft-identity 6.1

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

   6. Applied *-un-lft-identity 6.1

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

   7. Applied times-frac 6.1

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

   8. Applied add-sqr-sqrt 6.1

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

   9. Applied times-frac 6.1

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

   10. Applied associate-*l* 6.1

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

   11. Simplified 5.7

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

   13. Applied div-sub 5.7

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

   15. Applied div-inv 5.7

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

   16. Applied div-inv 5.7

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

   17. Applied distribute-rgt-out-- 5.7

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

   18. Applied *-un-lft-identity 5.7

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

   19. Applied times-frac 6.3

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

   20. Simplified 6.2

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -9.3910125961314414 \cdot 10^{-161} \lor \neg \left(y \le 6.0401320354982989 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{x - y}{z - y} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1}}{1} \cdot \left(\left(x - y\right) \cdot \frac{t}{z - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (/ t (/ (- z y) (- x y)))

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