Average Error: 0.1 → 0.3
Time: 22.4s
Precision: 64
\[x \cdot \cos y + z \cdot \sin y\]
↓
\[\sin y \cdot z + \sqrt[3]{\cos y} \cdot \left(x \cdot {\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(y + y\right)\right)}^{\frac{1}{3}}\right)\]
x \cdot \cos y + z \cdot \sin y
↓
\sin y \cdot z + \sqrt[3]{\cos y} \cdot \left(x \cdot {\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(y + y\right)\right)}^{\frac{1}{3}}\right)
double f(double x, double y, double z) {
        double r10538418 = x;
        double r10538419 = y;
        double r10538420 = cos(r10538419);
        double r10538421 = r10538418 * r10538420;
        double r10538422 = z;
        double r10538423 = sin(r10538419);
        double r10538424 = r10538422 * r10538423;
        double r10538425 = r10538421 + r10538424;
        return r10538425;
}
↓
double f(double x, double y, double z) {
        double r10538426 = y;
        double r10538427 = sin(r10538426);
        double r10538428 = z;
        double r10538429 = r10538427 * r10538428;
        double r10538430 = cos(r10538426);
        double r10538431 = cbrt(r10538430);
        double r10538432 = x;
        double r10538433 = 0.5;
        double r10538434 = r10538426 + r10538426;
        double r10538435 = cos(r10538434);
        double r10538436 = r10538433 * r10538435;
        double r10538437 = r10538433 + r10538436;
        double r10538438 = 0.3333333333333333;
        double r10538439 = pow(r10538437, r10538438);
        double r10538440 = r10538432 * r10538439;
        double r10538441 = r10538431 * r10538440;
        double r10538442 = r10538429 + r10538441;
        return r10538442;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Derivation

  1. Initial program 0.1

    \[x \cdot \cos y + z \cdot \sin y\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.4

    \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right) \cdot \sqrt[3]{\cos y}\right)} + z \cdot \sin y\]

  4. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(x \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \sqrt[3]{\cos y}} + z \cdot \sin y\]
  5. Using strategy rm

  6. Applied pow1/316.3

    \[\leadsto \left(x \cdot \left(\sqrt[3]{\cos y} \cdot \color{blue}{{\left(\cos y\right)}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{\cos y} + z \cdot \sin y\]

  7. Applied pow1/316.2

    \[\leadsto \left(x \cdot \left(\color{blue}{{\left(\cos y\right)}^{\frac{1}{3}}} \cdot {\left(\cos y\right)}^{\frac{1}{3}}\right)\right) \cdot \sqrt[3]{\cos y} + z \cdot \sin y\]

  8. Applied pow-prod-down0.2

    \[\leadsto \left(x \cdot \color{blue}{{\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\cos y} + z \cdot \sin y\]
  9. Using strategy rm

  10. Applied sqr-cos0.3

    \[\leadsto \left(x \cdot {\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)}}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos y} + z \cdot \sin y\]

  11. Simplified0.3

    \[\leadsto \left(x \cdot {\left(\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \cos \left(y + y\right)}\right)}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos y} + z \cdot \sin y\]

  12. Final simplification0.3

    \[\leadsto \sin y \cdot z + \sqrt[3]{\cos y} \cdot \left(x \cdot {\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(y + y\right)\right)}^{\frac{1}{3}}\right)\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  (+ (* x (cos y)) (* z (sin y))))