Error

Target

Derivation

1. Initial program 46.8

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]

2. Taylor expanded around 0 35.6

   \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \color{blue}{1}\]

3. Applied simplify35.4

   \[\leadsto \color{blue}{x \cdot \cos \left(\frac{2.0 \cdot y + 1.0}{\frac{16.0}{t \cdot z}}\right)}\]
4. Using strategy rm

5. Applied add-cube-cbrt35.4

   \[\leadsto x \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\frac{2.0 \cdot y + 1.0}{\frac{16.0}{t \cdot z}}} \cdot \sqrt[3]{\frac{2.0 \cdot y + 1.0}{\frac{16.0}{t \cdot z}}}\right) \cdot \sqrt[3]{\frac{2.0 \cdot y + 1.0}{\frac{16.0}{t \cdot z}}}\right)}\]
6. Using strategy rm

7. Applied cbrt-div35.5

   \[\leadsto x \cdot \cos \left(\left(\sqrt[3]{\frac{2.0 \cdot y + 1.0}{\frac{16.0}{t \cdot z}}} \cdot \sqrt[3]{\frac{2.0 \cdot y + 1.0}{\frac{16.0}{t \cdot z}}}\right) \cdot \color{blue}{\frac{\sqrt[3]{2.0 \cdot y + 1.0}}{\sqrt[3]{\frac{16.0}{t \cdot z}}}}\right)\]
8. Using strategy rm

9. Applied add-cube-cbrt35.4

   \[\leadsto x \cdot \cos \left(\left(\sqrt[3]{\frac{2.0 \cdot y + 1.0}{\frac{16.0}{t \cdot z}}} \cdot \sqrt[3]{\frac{2.0 \cdot y + 1.0}{\color{blue}{\left(\sqrt[3]{\frac{16.0}{t \cdot z}} \cdot \sqrt[3]{\frac{16.0}{t \cdot z}}\right) \cdot \sqrt[3]{\frac{16.0}{t \cdot z}}}}}\right) \cdot \frac{\sqrt[3]{2.0 \cdot y + 1.0}}{\sqrt[3]{\frac{16.0}{t \cdot z}}}\right)\]

10. Applied add-cube-cbrt35.4

    \[\leadsto x \cdot \cos \left(\left(\sqrt[3]{\frac{2.0 \cdot y + 1.0}{\frac{16.0}{t \cdot z}}} \cdot \sqrt[3]{\frac{\color{blue}{\left(\sqrt[3]{2.0 \cdot y + 1.0} \cdot \sqrt[3]{2.0 \cdot y + 1.0}\right) \cdot \sqrt[3]{2.0 \cdot y + 1.0}}}{\left(\sqrt[3]{\frac{16.0}{t \cdot z}} \cdot \sqrt[3]{\frac{16.0}{t \cdot z}}\right) \cdot \sqrt[3]{\frac{16.0}{t \cdot z}}}}\right) \cdot \frac{\sqrt[3]{2.0 \cdot y + 1.0}}{\sqrt[3]{\frac{16.0}{t \cdot z}}}\right)\]

11. Applied times-frac35.4

    \[\leadsto x \cdot \cos \left(\left(\sqrt[3]{\frac{2.0 \cdot y + 1.0}{\frac{16.0}{t \cdot z}}} \cdot \sqrt[3]{\color{blue}{\frac{\sqrt[3]{2.0 \cdot y + 1.0} \cdot \sqrt[3]{2.0 \cdot y + 1.0}}{\sqrt[3]{\frac{16.0}{t \cdot z}} \cdot \sqrt[3]{\frac{16.0}{t \cdot z}}} \cdot \frac{\sqrt[3]{2.0 \cdot y + 1.0}}{\sqrt[3]{\frac{16.0}{t \cdot z}}}}}\right) \cdot \frac{\sqrt[3]{2.0 \cdot y + 1.0}}{\sqrt[3]{\frac{16.0}{t \cdot z}}}\right)\]

12. Applied cbrt-prod35.5

    \[\leadsto x \cdot \cos \left(\left(\sqrt[3]{\frac{2.0 \cdot y + 1.0}{\frac{16.0}{t \cdot z}}} \cdot \color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{2.0 \cdot y + 1.0} \cdot \sqrt[3]{2.0 \cdot y + 1.0}}{\sqrt[3]{\frac{16.0}{t \cdot z}} \cdot \sqrt[3]{\frac{16.0}{t \cdot z}}}} \cdot \sqrt[3]{\frac{\sqrt[3]{2.0 \cdot y + 1.0}}{\sqrt[3]{\frac{16.0}{t \cdot z}}}}\right)}\right) \cdot \frac{\sqrt[3]{2.0 \cdot y + 1.0}}{\sqrt[3]{\frac{16.0}{t \cdot z}}}\right)\]
13. Removed slow pow expressions.

Runtime

Time bar (total: 2.5m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))