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

Average Error: 17.5 → 0.0

Time: 6.0s

Precision: binary64

Cost: 7296

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (- (+ (* y x) (fma (- y) z (* y z))) (* 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 ((y * x) + fma(-y, z, (y * z))) - (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 Float64(Float64(Float64(y * x) + fma(Float64(-y), z, Float64(y * z))) - Float64(y * 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[(N[(N[(y * x), $MachinePrecision] + N[((-y) * z + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]

Target

Original	17.5
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 17.5

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

2. Simplified 0.0

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

   [Start] 17.5	\[ \left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
   associate-+l- [=>] 8.0	\[ \color{blue}{\left(x \cdot y - y \cdot z\right) - \left(y \cdot y - y \cdot y\right)} \]
   +-inverses [=>] 0.0	\[ \left(x \cdot y - y \cdot z\right) - \color{blue}{0} \]
   --rgt-identity [=>] 0.0	\[ \color{blue}{x \cdot y - y \cdot z} \]
   *-commutative [=>] 0.0	\[ x \cdot y - \color{blue}{z \cdot y} \]
   distribute-rgt-out-- [=>] 0.0	\[ \color{blue}{y \cdot \left(x - z\right)} \]

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

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

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	576

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

Alternative 2
Error	15.0
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-8} \lor \neg \left(z \leq 4.1 \cdot 10^{-44}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	0.0
Cost	320

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

Alternative 4
Error	29.7
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023041 
(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)))