Average Error: 0.2 → 0.5
Time: 44.7s
Precision: 64
Internal Precision: 384
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}\right) \cdot \left(\sqrt[3]{b \cdot b + a \cdot a} \cdot \sqrt[3]{b \cdot b + a \cdot a}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.5

    \[\leadsto \left(\color{blue}{\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}\right) \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.5

    \[\leadsto \left(\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}\right) \cdot \sqrt[3]{{\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)}}^{2}} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  6. Applied unpow-prod-down0.5

    \[\leadsto \left(\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}\right) \cdot \sqrt[3]{\color{blue}{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2} \cdot {\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}}} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  7. Applied cbrt-prod0.5

    \[\leadsto \left(\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}\right) \cdot \color{blue}{\left(\sqrt[3]{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}}\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  8. Applied simplify0.5

    \[\leadsto \left(\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}\right) \cdot \left(\color{blue}{\sqrt[3]{b \cdot b + a \cdot a}} \cdot \sqrt[3]{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  9. Applied simplify0.5

    \[\leadsto \left(\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}\right) \cdot \left(\sqrt[3]{b \cdot b + a \cdot a} \cdot \color{blue}{\sqrt[3]{b \cdot b + a \cdot a}}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\]

Runtime

Time bar (total: 44.7s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))