Average Error: 0.1 → 0.3
Time: 17.1s
Precision: 64
\[x \cdot \cos y + z \cdot \sin y\]
↓
\[z \cdot \sin y + x \cdot \sqrt[3]{\left(\cos \left(y + y\right) \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot \cos y}\]
x \cdot \cos y + z \cdot \sin y
↓
z \cdot \sin y + x \cdot \sqrt[3]{\left(\cos \left(y + y\right) \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot \cos y}
double f(double x, double y, double z) {
        double r10211561 = x;
        double r10211562 = y;
        double r10211563 = cos(r10211562);
        double r10211564 = r10211561 * r10211563;
        double r10211565 = z;
        double r10211566 = sin(r10211562);
        double r10211567 = r10211565 * r10211566;
        double r10211568 = r10211564 + r10211567;
        return r10211568;
}
↓
double f(double x, double y, double z) {
        double r10211569 = z;
        double r10211570 = y;
        double r10211571 = sin(r10211570);
        double r10211572 = r10211569 * r10211571;
        double r10211573 = x;
        double r10211574 = r10211570 + r10211570;
        double r10211575 = cos(r10211574);
        double r10211576 = 0.5;
        double r10211577 = r10211575 * r10211576;
        double r10211578 = r10211577 + r10211576;
        double r10211579 = cos(r10211570);
        double r10211580 = r10211578 * r10211579;
        double r10211581 = cbrt(r10211580);
        double r10211582 = r10211573 * r10211581;
        double r10211583 = r10211572 + r10211582;
        return r10211583;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

Enter valid numbers for all inputs

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. Taylor expanded around inf 16.2

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

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))))