Average Error: 45.7 → 45.6
Time: 3.2s
Precision: binary64
\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
↓
\[\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z} \cdot \left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z}\right)\]
\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)
↓
\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z} \cdot \left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z}\right)
(FPCore (x y z) :precision binary64 (- (fma x y z) (+ 1.0 (+ (* x y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (*
  (cbrt (- (- (+ (fma x y z) -1.0) (* x y)) z))
  (*
   (cbrt (- (- (+ (fma x y z) -1.0) (* x y)) z))
   (cbrt (- (- (+ (fma x y z) -1.0) (* x y)) z)))))
double code(double x, double y, double z) {
	return fma(x, y, z) - (1.0 + ((x * y) + z));
}
↓
double code(double x, double y, double z) {
	return cbrt(((fma(x, y, z) + -1.0) - (x * y)) - z) * (cbrt(((fma(x, y, z) + -1.0) - (x * y)) - z) * cbrt(((fma(x, y, z) + -1.0) - (x * y)) - z));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original45.7
Target0
Herbie45.6
\[-1\]

Derivation

  1. Initial program 45.7

    \[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
  2. Using strategy rm

  3. Applied associate--r+_binary64_103745.7

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - \left(x \cdot y + z\right)}\]

  4. Simplified45.7

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, y, z\right) + -1\right)} - \left(x \cdot y + z\right)\]
  5. Using strategy rm

  6. Applied associate--r+_binary64_103745.6

    \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt_binary64_113645.6

    \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z}\right) \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z}}\]

  9. Final simplification45.6

    \[\leadsto \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z} \cdot \left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) + -1\right) - x \cdot y\right) - z}\right)\]

Reproduce

herbie shell --seed 2020343 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

  (- (fma x y z) (+ 1.0 (+ (* x y) z))))