simple fma test

Average Error: 45.3 → 24.5

Time: 8.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(z + x \cdot y\right)\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\mathsf{fma}\left(x, y, z\right) + z \cdot \sqrt[3]{-1}\right)\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (- (fma x y z) (+ 1.0 (+ (* x y) z))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (- (fma x y z) (+ 1.0 (+ z (* x y)))) -1.0)
   (- (fma x y z) (+ 1.0 (+ z (* x y))))
   (+ -1.0 (+ (fma x y z) (* z (cbrt -1.0))))))

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) {
	double tmp;
	if ((fma(x, y, z) - (1.0 + (z + (x * y)))) <= -1.0) {
		tmp = fma(x, y, z) - (1.0 + (z + (x * y)));
	} else {
		tmp = -1.0 + (fma(x, y, z) + (z * cbrt(-1.0)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	45.3
Target	0
Herbie	24.5

\[-1\]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (fma.f64 x y z) (+.f64 1 (+.f64 (*.f64 x y) z))) < -1

   1. Initial program 0.3

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

   if -1 < (-.f64 (fma.f64 x y z) (+.f64 1 (+.f64 (*.f64 x y) z)))

   1. Initial program 62.1

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

   3. Applied add-cube-cbrt_binary64_2500 62.1

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

   4. Simplified 62.1

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

   5. Simplified 62.1

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

   6. Taylor expanded around inf 63.3

      \[\leadsto \mathsf{fma}\left(x, y, z\right) - \left(1 + \left(\sqrt[3]{z + x \cdot y} \cdot \sqrt[3]{z + x \cdot y}\right) \cdot \color{blue}{{z}^{0.3333333333333333}}\right)\]

   7. Simplified 62.2

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

   8. Taylor expanded around -inf 33.5

      \[\leadsto \color{blue}{\left(z \cdot \sqrt[3]{-1} + \mathsf{fma}\left(x, y, z\right)\right) - 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 24.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(z + x \cdot y\right)\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\mathsf{fma}\left(x, y, z\right) + z \cdot \sqrt[3]{-1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

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