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

Average Accuracy: 72.2% → 100.0%

Time: 9.5s

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	72.2%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 72.2%

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

2. Simplified 100.0%

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

   [Start] 72.2	\[ \left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y \]
   associate--l- [=>] 72.2	\[ \color{blue}{\left(x \cdot y + y \cdot y\right) - \left(y \cdot z + y \cdot y\right)} \]
   +-commutative [=>] 72.2	\[ \left(x \cdot y + y \cdot y\right) - \color{blue}{\left(y \cdot y + y \cdot z\right)} \]
   associate--r+ [=>] 79.6	\[ \color{blue}{\left(\left(x \cdot y + y \cdot y\right) - y \cdot y\right) - y \cdot z} \]
   associate--l+ [=>] 87.1	\[ \color{blue}{\left(x \cdot y + \left(y \cdot y - y \cdot y\right)\right)} - y \cdot z \]
   +-inverses [=>] 100.0	\[ \left(x \cdot y + \color{blue}{0}\right) - y \cdot z \]
   +-rgt-identity [=>] 100.0	\[ \color{blue}{x \cdot y} - y \cdot z \]
   *-commutative [=>] 100.0	\[ x \cdot y - \color{blue}{z \cdot y} \]
   distribute-rgt-out-- [=>] 100.0	\[ \color{blue}{y \cdot \left(x - z\right)} \]

3. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ y \cdot \left(x - z\right) \]
   sub-neg [=>] 100.0	\[ y \cdot \color{blue}{\left(x + \left(-z\right)\right)} \]
   distribute-lft-in [=>] 100.0	\[ \color{blue}{y \cdot x + y \cdot \left(-z\right)} \]

4. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ y \cdot x + y \cdot \left(-z\right) \]
   distribute-rgt-neg-out [=>] 100.0	\[ y \cdot x + \color{blue}{\left(-y \cdot z\right)} \]
   unsub-neg [=>] 100.0	\[ \color{blue}{y \cdot x - y \cdot z} \]

5. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ y \cdot x - y \cdot z \]
   fma-neg [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(y, x, -y \cdot z\right)} \]
   distribute-rgt-neg-in [=>] 100.0	\[ \mathsf{fma}\left(y, x, \color{blue}{y \cdot \left(-z\right)}\right) \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	74.7%
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-90} \lor \neg \left(z \leq 9.2 \cdot 10^{+70}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	448

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

Alternative 3
Accuracy	100.0%
Cost	320

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

Alternative 4
Accuracy	53.4%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023129 
(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)))