Average Error: 0.2 → 0.1
Time: 1.6m
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\sqrt{\frac{1}{\sqrt{1}}}}{\left|\sqrt[3]{9}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{9} \cdot \sqrt[3]{9}}}} \cdot \frac{\sqrt{\frac{1}{\sqrt{1}}} \cdot rand}{\sqrt{\sqrt[3]{\sqrt[3]{9}} \cdot \left(a - \frac{1}{3}\right)}}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\sqrt{\frac{1}{\sqrt{1}}}}{\left|\sqrt[3]{9}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{9} \cdot \sqrt[3]{9}}}} \cdot \frac{\sqrt{\frac{1}{\sqrt{1}}} \cdot rand}{\sqrt{\sqrt[3]{\sqrt[3]{9}} \cdot \left(a - \frac{1}{3}\right)}}\right)
double code(double a, double rand) {
	return ((double) (((double) (a - ((double) (1.0 / 3.0)))) * ((double) (1.0 + ((double) (((double) (1.0 / ((double) sqrt(((double) (9.0 * ((double) (a - ((double) (1.0 / 3.0)))))))))) * rand))))));
}
↓
double code(double a, double rand) {
	return ((double) (((double) (a - ((double) (1.0 / 3.0)))) * ((double) (1.0 + ((double) (((double) (((double) sqrt(((double) (1.0 / ((double) sqrt(1.0)))))) / ((double) (((double) fabs(((double) cbrt(9.0)))) * ((double) sqrt(((double) cbrt(((double) (((double) cbrt(9.0)) * ((double) cbrt(9.0)))))))))))) * ((double) (((double) (((double) sqrt(((double) (1.0 / ((double) sqrt(1.0)))))) * rand)) / ((double) sqrt(((double) (((double) cbrt(((double) cbrt(9.0)))) * ((double) (a - ((double) (1.0 / 3.0))))))))))))))));
}

Error

Bits error versus a

Bits error versus rand

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.2

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1 \cdot \left(9 \cdot \left(a - \frac{1}{3}\right)\right)}}} \cdot rand\right)\]

  4. Applied sqrt-prod0.2

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\color{blue}{\sqrt{1} \cdot \sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}} \cdot rand\right)\]

  5. Applied associate-/r*0.2

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \color{blue}{\frac{\frac{1}{\sqrt{1}}}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}} \cdot rand\right)\]

  6. Applied associate-*l/0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \color{blue}{\frac{\frac{1}{\sqrt{1}} \cdot rand}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right)\]
  7. Using strategy rm

  8. Applied add-cube-cbrt0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{1}} \cdot rand}{\sqrt{\color{blue}{\left(\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \sqrt[3]{9}\right)} \cdot \left(a - \frac{1}{3}\right)}}\right)\]

  9. Applied associate-*l*0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{1}} \cdot rand}{\sqrt{\color{blue}{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(a - \frac{1}{3}\right)\right)}}}\right)\]

  10. Applied sqrt-prod0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{1}} \cdot rand}{\color{blue}{\sqrt{\sqrt[3]{9} \cdot \sqrt[3]{9}} \cdot \sqrt{\sqrt[3]{9} \cdot \left(a - \frac{1}{3}\right)}}}\right)\]

  11. Simplified0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{1}} \cdot rand}{\color{blue}{\left|\sqrt[3]{9}\right|} \cdot \sqrt{\sqrt[3]{9} \cdot \left(a - \frac{1}{3}\right)}}\right)\]
  12. Using strategy rm

  13. Applied add-cube-cbrt0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{1}} \cdot rand}{\left|\sqrt[3]{9}\right| \cdot \sqrt{\sqrt[3]{\color{blue}{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \sqrt[3]{9}}} \cdot \left(a - \frac{1}{3}\right)}}\right)\]

  14. Applied cbrt-prod0.2

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{1}} \cdot rand}{\left|\sqrt[3]{9}\right| \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\sqrt[3]{9} \cdot \sqrt[3]{9}} \cdot \sqrt[3]{\sqrt[3]{9}}\right)} \cdot \left(a - \frac{1}{3}\right)}}\right)\]

  15. Applied associate-*l*0.2

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{1}} \cdot rand}{\left|\sqrt[3]{9}\right| \cdot \sqrt{\color{blue}{\sqrt[3]{\sqrt[3]{9} \cdot \sqrt[3]{9}} \cdot \left(\sqrt[3]{\sqrt[3]{9}} \cdot \left(a - \frac{1}{3}\right)\right)}}}\right)\]

  16. Applied sqrt-prod0.2

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{1}} \cdot rand}{\left|\sqrt[3]{9}\right| \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\sqrt[3]{9} \cdot \sqrt[3]{9}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{9}} \cdot \left(a - \frac{1}{3}\right)}\right)}}\right)\]

  17. Applied associate-*r*0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{1}{\sqrt{1}} \cdot rand}{\color{blue}{\left(\left|\sqrt[3]{9}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{9} \cdot \sqrt[3]{9}}}\right) \cdot \sqrt{\sqrt[3]{\sqrt[3]{9}} \cdot \left(a - \frac{1}{3}\right)}}}\right)\]

  18. Applied add-sqr-sqrt0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\color{blue}{\left(\sqrt{\frac{1}{\sqrt{1}}} \cdot \sqrt{\frac{1}{\sqrt{1}}}\right)} \cdot rand}{\left(\left|\sqrt[3]{9}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{9} \cdot \sqrt[3]{9}}}\right) \cdot \sqrt{\sqrt[3]{\sqrt[3]{9}} \cdot \left(a - \frac{1}{3}\right)}}\right)\]

  19. Applied associate-*l*0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\color{blue}{\sqrt{\frac{1}{\sqrt{1}}} \cdot \left(\sqrt{\frac{1}{\sqrt{1}}} \cdot rand\right)}}{\left(\left|\sqrt[3]{9}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{9} \cdot \sqrt[3]{9}}}\right) \cdot \sqrt{\sqrt[3]{\sqrt[3]{9}} \cdot \left(a - \frac{1}{3}\right)}}\right)\]

  20. Applied times-frac0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \color{blue}{\frac{\sqrt{\frac{1}{\sqrt{1}}}}{\left|\sqrt[3]{9}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{9} \cdot \sqrt[3]{9}}}} \cdot \frac{\sqrt{\frac{1}{\sqrt{1}}} \cdot rand}{\sqrt{\sqrt[3]{\sqrt[3]{9}} \cdot \left(a - \frac{1}{3}\right)}}}\right)\]

  21. Final simplification0.1

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\sqrt{\frac{1}{\sqrt{1}}}}{\left|\sqrt[3]{9}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{9} \cdot \sqrt[3]{9}}}} \cdot \frac{\sqrt{\frac{1}{\sqrt{1}}} \cdot rand}{\sqrt{\sqrt[3]{\sqrt[3]{9}} \cdot \left(a - \frac{1}{3}\right)}}\right)\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))