Linear.Quaternion:$c/ from linear-1.19.1.3, B

Average Error: 17.6 → 0.0

Time: 6.7s

Precision: binary64

Cost: 6784

Math

\[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]

↓

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

FPCore

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

↓

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

C

double code(double x, double y, double z) {
	return (((x * y) - (y * z)) - (y * y)) + (y * y);
}

↓

double code(double x, double y, double z) {
	return fma(y, x, (y * -z));
}

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	17.6
Target	0.0
Herbie	0.0

\[\left(x - z\right) \cdot y \]

Derivation

1. Initial program 17.6

   \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]

2. Simplified 0.0

   \[\leadsto \color{blue}{y \cdot \left(x - z\right)} \]
   Proof
   (*.f64 y (-.f64 x z)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 x y) (*.f64 z y))): 2 points increase in error, 3 points decrease in error
   (-.f64 (*.f64 x y) (Rewrite<= *-commutative_binary64 (*.f64 y z))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= --rgt-identity_binary64 (-.f64 (-.f64 (*.f64 x y) (*.f64 y z)) 0)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (*.f64 x y) (*.f64 y z)) (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 y y) (*.f64 y y)))): 30 points increase in error, 0 points decrease in error
   (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 (-.f64 (*.f64 x y) (*.f64 y z)) (*.f64 y y)) (*.f64 y y))): 52 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.0

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

4. Applied egg-rr 0.0

   \[\leadsto \color{blue}{y \cdot x - y \cdot z} \]

5. Applied egg-rr 0.0

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

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	15.5
Cost	784

\[\begin{array}{l} t_0 := y \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -4.3660531790651035 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.600760303331691 \cdot 10^{-74}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq 1.1354328350781584 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 12324.076275452126:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

\[y \cdot x - y \cdot z \]

Alternative 3
Error	0.0
Cost	320

\[y \cdot \left(x - z\right) \]

Alternative 4
Error	29.9
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2022291 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (* (- x z) y)

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