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

Average Error: 2.4 → 1.1

Time: 4.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -2.7704548820906923 \cdot 10^{-177}:\\ \;\;\;\;\frac{t}{\frac{z - y}{x - y}}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 4.9868386023721 \cdot 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{z - y}{\left(x - y\right) \cdot t}}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 3.074373967977172 \cdot 10^{+201}:\\ \;\;\;\;\frac{t}{\frac{z - y}{x - y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{z - y}{t}}}{\frac{1}{x - y}}\\ \end{array}\]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ (- x y) (- z y)) t))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (- x y) (- z y)) -2.7704548820906923e-177)
   (/ t (/ (- z y) (- x y)))
   (if (<= (/ (- x y) (- z y)) 4.9868386023721e-316)
     (/ 1.0 (/ (- z y) (* (- x y) t)))
     (if (<= (/ (- x y) (- z y)) 3.074373967977172e+201)
       (/ t (/ (- z y) (- x y)))
       (/ (/ 1.0 (/ (- z y) t)) (/ 1.0 (- x y)))))))

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 tmp;
	if (((x - y) / (z - y)) <= -2.7704548820906923e-177) {
		tmp = t / ((z - y) / (x - y));
	} else if (((x - y) / (z - y)) <= 4.9868386023721e-316) {
		tmp = 1.0 / ((z - y) / ((x - y) * t));
	} else if (((x - y) / (z - y)) <= 3.074373967977172e+201) {
		tmp = t / ((z - y) / (x - y));
	} else {
		tmp = (1.0 / ((z - y) / t)) / (1.0 / (x - y));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.4
Target	2.3
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2.7704548820906923e-177 or 4.986838602e-316 < (/.f64 (-.f64 x y) (-.f64 z y)) < 3.07437396797717173e201

   1. Initial program 1.0

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

   3. Applied add-cube-cbrt_binary64 2.1

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

   4. Applied *-un-lft-identity_binary64 2.1

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

   5. Applied times-frac_binary64 2.1

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

   6. Applied associate-*l*_binary64 9.0

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

   7. Simplified 9.0

      \[\leadsto \frac{1}{\sqrt[3]{z - y} \cdot \sqrt[3]{z - y}} \cdot \color{blue}{\left(t \cdot \frac{x - y}{\sqrt[3]{z - y}}\right)}\]
   8. Using strategy rm

   9. Applied pow1_binary64 9.0

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

   10. Applied pow1_binary64 9.0

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

   11. Applied pow-prod-down_binary64 9.0

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

   12. Applied pow1_binary64 9.0

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

   13. Applied pow-prod-down_binary64 9.0

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

   14. Simplified 0.9

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

   if -2.7704548820906923e-177 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.986838602e-316

   1. Initial program 10.9

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

   3. Applied add-cube-cbrt_binary64 11.3

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

   4. Applied *-un-lft-identity_binary64 11.3

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

   5. Applied times-frac_binary64 11.3

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

   6. Applied associate-*l*_binary64 1.0

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

   7. Simplified 1.0

      \[\leadsto \frac{1}{\sqrt[3]{z - y} \cdot \sqrt[3]{z - y}} \cdot \color{blue}{\left(t \cdot \frac{x - y}{\sqrt[3]{z - y}}\right)}\]
   8. Using strategy rm

   9. Applied pow1_binary64 1.0

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

   10. Applied pow1_binary64 1.0

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

   11. Applied pow-prod-down_binary64 1.0

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

   12. Applied pow1_binary64 1.0

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

   13. Applied pow-prod-down_binary64 1.0

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

   14. Simplified 12.2

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

   16. Applied clear-num_binary64 12.5

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

   17. Simplified 2.4

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

   if 3.07437396797717173e201 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 18.8

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

   3. Applied add-cube-cbrt_binary64 19.6

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

   4. Applied *-un-lft-identity_binary64 19.6

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

   5. Applied times-frac_binary64 19.6

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

   6. Applied associate-*l*_binary64 6.8

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

   7. Simplified 6.8

      \[\leadsto \frac{1}{\sqrt[3]{z - y} \cdot \sqrt[3]{z - y}} \cdot \color{blue}{\left(t \cdot \frac{x - y}{\sqrt[3]{z - y}}\right)}\]
   8. Using strategy rm

   9. Applied pow1_binary64 6.8

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

   10. Applied pow1_binary64 6.8

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

   11. Applied pow-prod-down_binary64 6.8

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

   12. Applied pow1_binary64 6.8

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

   13. Applied pow-prod-down_binary64 6.8

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

   14. Simplified 16.2

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

   16. Applied div-inv_binary64 16.2

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

   17. Applied associate-/r*_binary64 2.1

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

   19. Applied clear-num_binary64 2.8

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -2.7704548820906923 \cdot 10^{-177}:\\ \;\;\;\;\frac{t}{\frac{z - y}{x - y}}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 4.9868386023721 \cdot 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{z - y}{\left(x - y\right) \cdot t}}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 3.074373967977172 \cdot 10^{+201}:\\ \;\;\;\;\frac{t}{\frac{z - y}{x - y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{z - y}{t}}}{\frac{1}{x - y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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))