simple fma test

Average Error: 45.2 → 45.0

Time: 6.2s

Precision: binary64

Math

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

↓

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

FPCore

(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
   (*
    (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)))))))

C

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(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

Original	45.2
Target	0
Herbie	45.0

\[-1\]

Derivation

1. Initial program 45.2

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

3. Applied associate--r+_binary64_1026 45.2

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

5. Applied associate--r+_binary64_1026 45.1

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

7. Applied add-cube-cbrt_binary64_1120 45.0

   \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - x \cdot y\right) - z} \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - x \cdot y\right) - z}\right) \cdot \sqrt[3]{\left(\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - x \cdot y\right) - z}}\]
8. Using strategy rm

9. Applied add-cube-cbrt_binary64_1120 45.0

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

10. Final simplification 45.0

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

Reproduce

herbie shell --seed 2020303 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

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