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

Average Error: 17.2 → 0.0

Time: 7.0s

Precision: binary64

Cost: 6784

Math

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

↓

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

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(y * Float64(-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	17.2
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 17.2

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

2. Simplified 0.0

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

   [Start] 17.2	\[ \left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y \]
   associate--l- [=>] 17.2	\[ \color{blue}{\left(x \cdot y + y \cdot y\right) - \left(y \cdot z + y \cdot y\right)} \]
   +-commutative [=>] 17.2	\[ \left(x \cdot y + y \cdot y\right) - \color{blue}{\left(y \cdot y + y \cdot z\right)} \]
   associate--r+ [=>] 12.8	\[ \color{blue}{\left(\left(x \cdot y + y \cdot y\right) - y \cdot y\right) - y \cdot z} \]
   associate--l+ [=>] 8.0	\[ \color{blue}{\left(x \cdot y + \left(y \cdot y - y \cdot y\right)\right)} - y \cdot z \]
   +-inverses [=>] 0.0	\[ \left(x \cdot y + \color{blue}{0}\right) - y \cdot z \]
   +-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 30.8

   \[\leadsto \color{blue}{{\left(\sqrt{y \cdot \left(x - z\right)}\right)}^{2}} \]

4. Applied egg-rr 30.2

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

5. Taylor expanded in x around 0 0.0

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

6. Simplified 0.0

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

   [Start] 0.0	\[ y \cdot x + -1 \cdot \left(y \cdot z\right) \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(y, x, -1 \cdot \left(y \cdot z\right)\right)} \]
   associate-*r* [=>] 0.0	\[ \mathsf{fma}\left(y, x, \color{blue}{\left(-1 \cdot y\right) \cdot z}\right) \]
   neg-mul-1 [<=] 0.0	\[ \mathsf{fma}\left(y, x, \color{blue}{\left(-y\right)} \cdot z\right) \]

7. Final simplification 0.0

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

Alternatives

Alternative 1
Error	14.6
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-40} \lor \neg \left(z \leq 1.7 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	320

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

Alternative 3
Error	29.5
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2022354 
(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)))