Average Error: 0.2 → 0.3
Time: 8.1s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 4} \cdot \left(\left|\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 4}\right| \cdot \sqrt{\sqrt[3]{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 4}} \cdot \sqrt[3]{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 4}}}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 4} \cdot \left(\left|\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 4}\right| \cdot \sqrt{\sqrt[3]{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 4}} \cdot \sqrt[3]{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 4}}}\right) - 1
(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))
↓
(FPCore (a b)
 :precision binary64
 (-
  (*
   (sqrt (+ (pow (+ (* a a) (* b b)) 2.0) (* (* b b) 4.0)))
   (*
    (fabs (cbrt (+ (pow (+ (* a a) (* b b)) 2.0) (* (* b b) 4.0))))
    (sqrt
     (*
      (cbrt (sqrt (+ (pow (+ (* a a) (* b b)) 2.0) (* (* b b) 4.0))))
      (cbrt (sqrt (+ (pow (+ (* a a) (* b b)) 2.0) (* (* b b) 4.0))))))))
  1.0))
double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}
↓
double code(double a, double b) {
	return (sqrt(pow(((a * a) + (b * b)), 2.0) + ((b * b) * 4.0)) * (fabs(cbrt(pow(((a * a) + (b * b)), 2.0) + ((b * b) * 4.0))) * sqrt(cbrt(sqrt(pow(((a * a) + (b * b)), 2.0) + ((b * b) * 4.0))) * cbrt(sqrt(pow(((a * a) + (b * b)), 2.0) + ((b * b) * 4.0)))))) - 1.0;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-sqr-sqrt_binary640.2

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied add-cube-cbrt_binary640.3

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

  8. Applied sqrt-prod_binary640.3

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

  9. Simplified0.3

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

  11. Applied add-sqr-sqrt_binary640.3

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

  12. Applied cbrt-prod_binary640.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020268 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))