simple fma test

Average Error: 45.0 → 0

Time: 5.2s

Precision: binary64

Cost: 64

Math

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

↓

\[-1 \]

FPCore

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

↓

(FPCore (x y z) :precision binary64 -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) {
	return -1.0;
}

Julia

function code(x, y, z)
	return Float64(fma(x, y, z) - Float64(1.0 + Float64(Float64(x * y) + z)))
end

↓

function code(x, y, z)
	return -1.0
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y + z), $MachinePrecision] - N[(1.0 + N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := -1.0

Target

Original	45.0
Target	0
Herbie	0

\[-1 \]

Derivation

1. Initial program 45.0

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

2. Simplified 7.9

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

   [Start] 45.0	\[ \mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right) \]
   rational.json-simplify-46 [=>] 45.0	\[ \color{blue}{\left(\mathsf{fma}\left(x, y, z\right) - 1\right) - \left(x \cdot y + z\right)} \]
   rational.json-simplify-42 [=>] 7.9	\[ \color{blue}{\left(\mathsf{fma}\left(x, y, z\right) - \left(x \cdot y + z\right)\right) - 1} \]
   rational.json-simplify-16 [=>] 7.9	\[ \color{blue}{\left(\mathsf{fma}\left(x, y, z\right) - \left(x \cdot y + z\right)\right) + -1} \]
   rational.json-simplify-1 [=>] 7.9	\[ \left(\mathsf{fma}\left(x, y, z\right) - \color{blue}{\left(z + x \cdot y\right)}\right) + -1 \]

3. Taylor expanded in x around 0 0

   \[\leadsto \color{blue}{0} + -1 \]

4. Final simplification 0

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x y z)
  :name "simple fma test"
  :precision binary64

  :herbie-target
  -1.0

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