Average Error: 0.1 → 0.4
Time: 4.3s
Precision: 64
\[x \cdot \sin y + z \cdot \cos y\]
↓
\[\mathsf{fma}\left(x, \sin y, \left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos y}\right)\right)\right)\]
x \cdot \sin y + z \cdot \cos y
↓
\mathsf{fma}\left(x, \sin y, \left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos y}\right)\right)\right)
double code(double x, double y, double z) {
	return ((x * sin(y)) + (z * cos(y)));
}
↓
double code(double x, double y, double z) {
	return fma(x, sin(y), ((z * (cbrt(cos(y)) * cbrt(cos(y)))) * log1p(expm1(cbrt(cos(y))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[x \cdot \sin y + z \cdot \cos y\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.4

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

  5. Applied associate-*r*0.4

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

  7. Applied log1p-expm1-u0.4

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos y}\right)\right)}\right)\]

  8. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos y}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))