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

Average Error: 17.8 → 0.0

Time: 4.4s

Precision: binary64

Cost: 6784

Math

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

↓

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

FPCore

(FPCore (x y z)
 :precision binary64
 (- (- (+ (* x y) (* y y)) (* y z)) (* 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 * y)) - (y * z)) - (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 * y)) - Float64(y * z)) - Float64(y * y))
end

↓

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

Wolfram

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

↓

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

Target

Original	17.8
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 17.8

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

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-lft-out--_binary64 (-.f64 (*.f64 y x) (*.f64 y z))): 3 points increase in error, 1 points decrease in error
   (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (*.f64 y z)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 x y) 0)) (*.f64 y z)): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 x y) (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 y y) (*.f64 y y)))) (*.f64 y z)): 29 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 x y) (*.f64 y y)) (*.f64 y y))) (*.f64 y z)): 22 points increase in error, 0 points decrease in error
   (Rewrite<= associate--r+_binary64 (-.f64 (+.f64 (*.f64 x y) (*.f64 y y)) (+.f64 (*.f64 y y) (*.f64 y z)))): 33 points increase in error, 8 points decrease in error
   (-.f64 (+.f64 (*.f64 x y) (*.f64 y y)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 y z) (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l-_binary64 (-.f64 (-.f64 (+.f64 (*.f64 x y) (*.f64 y y)) (*.f64 y z)) (*.f64 y y))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 29.7

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

4. Taylor expanded in x around 0 0.0

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

5. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(-y\right) \cdot z\right)} \]
   Proof
   (fma.f64 y x (*.f64 (neg.f64 y) z)): 0 points increase in error, 0 points decrease in error
   (fma.f64 y x (*.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 y)) z)): 0 points increase in error, 0 points decrease in error
   (fma.f64 y x (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 y z)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y x) (*.f64 -1 (*.f64 y z)))): 3 points increase in error, 0 points decrease in error

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	17.2
Cost	1048

\[\begin{array}{l} t_0 := -y \cdot z\\ \mathbf{if}\;z \leq -4 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1020:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq -6.1 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-130}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9000000000:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.0
Cost	320

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

Alternative 3
Error	29.6
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2022338 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, C"
  :precision binary64

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

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