Data.Colour.RGB:hslsv from colour-2.3.3, B

Average Error: 0.4 → 0.1

Time: 7.0s

Precision: binary64

Math

\[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]

↓

\[\mathsf{fma}\left(a, 120, \frac{60}{z - t} \cdot \left(x - y\right)\right) \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (/ (* 60.0 (- x y)) (- z t)) (* a 120.0)))

↓

(FPCore (x y z t a)
 :precision binary64
 (fma a 120.0 (* (/ 60.0 (- z t)) (- x y))))

C

double code(double x, double y, double z, double t, double a) {
	return ((60.0 * (x - y)) / (z - t)) + (a * 120.0);
}

↓

double code(double x, double y, double z, double t, double a) {
	return fma(a, 120.0, ((60.0 / (z - t)) * (x - y)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.4
Target	0.2
Herbie	0.1

\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]

Derivation

1. Initial program 0.4

   \[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]

2. Simplified 0.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, 120, \frac{60 \cdot \left(x - y\right)}{z - t}\right)} \]

3. Applied associate-/l*_binary64 0.1

   \[\leadsto \mathsf{fma}\left(a, 120, \color{blue}{\frac{60}{\frac{z - t}{x - y}}}\right) \]

4. Applied div-inv_binary64 0.2

   \[\leadsto \mathsf{fma}\left(a, 120, \frac{60}{\color{blue}{\left(z - t\right) \cdot \frac{1}{x - y}}}\right) \]

5. Applied associate-/r*_binary64 0.2

   \[\leadsto \mathsf{fma}\left(a, 120, \color{blue}{\frac{\frac{60}{z - t}}{\frac{1}{x - y}}}\right) \]

6. Applied *-un-lft-identity_binary64 0.2

   \[\leadsto \mathsf{fma}\left(a, 120, \frac{\frac{60}{z - t}}{\frac{1}{\color{blue}{1 \cdot \left(x - y\right)}}}\right) \]

7. Applied add-cube-cbrt_binary64 0.2

   \[\leadsto \mathsf{fma}\left(a, 120, \frac{\frac{60}{z - t}}{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \left(x - y\right)}}\right) \]

8. Applied times-frac_binary64 0.2

   \[\leadsto \mathsf{fma}\left(a, 120, \frac{\frac{60}{z - t}}{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{x - y}}}\right) \]

9. Applied *-un-lft-identity_binary64 0.2

   \[\leadsto \mathsf{fma}\left(a, 120, \frac{\frac{60}{\color{blue}{1 \cdot \left(z - t\right)}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{x - y}}\right) \]

10. Applied *-un-lft-identity_binary64 0.2

    \[\leadsto \mathsf{fma}\left(a, 120, \frac{\frac{\color{blue}{1 \cdot 60}}{1 \cdot \left(z - t\right)}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{x - y}}\right) \]

11. Applied times-frac_binary64 0.2

    \[\leadsto \mathsf{fma}\left(a, 120, \frac{\color{blue}{\frac{1}{1} \cdot \frac{60}{z - t}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{x - y}}\right) \]

12. Applied times-frac_binary64 0.2

    \[\leadsto \mathsf{fma}\left(a, 120, \color{blue}{\frac{\frac{1}{1}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}} \cdot \frac{\frac{60}{z - t}}{\frac{\sqrt[3]{1}}{x - y}}}\right) \]

13. Simplified 0.2

    \[\leadsto \mathsf{fma}\left(a, 120, \color{blue}{1} \cdot \frac{\frac{60}{z - t}}{\frac{\sqrt[3]{1}}{x - y}}\right) \]

14. Simplified 0.1

    \[\leadsto \mathsf{fma}\left(a, 120, 1 \cdot \color{blue}{\left(\frac{60}{z - t} \cdot \left(x - y\right)\right)}\right) \]

15. Final simplification 0.1

    \[\leadsto \mathsf{fma}\left(a, 120, \frac{60}{z - t} \cdot \left(x - y\right)\right) \]

Reproduce

herbie shell --seed 2022077 
(FPCore (x y z t a)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (+ (/ 60.0 (/ (- z t) (- x y))) (* a 120.0))

  (+ (/ (* 60.0 (- x y)) (- z t)) (* a 120.0)))