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

Average Accuracy: 79.8% → 100.0%

Time: 4.2s

Precision: binary64

Cost: 6784

Math

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

↓

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

FPCore

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

↓

(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 * y)) - (y * z);
}

↓

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 * y)) - Float64(y * z))
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 * y), $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]

↓

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

Target

Original	79.8%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 79.8%

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

2. Simplified 100.0%

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

   [Start] 79.8	\[ \left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z \]
   associate-+l- [=>] 87.5	\[ \color{blue}{\left(x \cdot y - \left(y \cdot y - y \cdot y\right)\right)} - y \cdot z \]
   +-inverses [=>] 100.0	\[ \left(x \cdot y - \color{blue}{0}\right) - y \cdot z \]
   associate--l- [=>] 100.0	\[ \color{blue}{x \cdot y - \left(0 + y \cdot z\right)} \]
   +-lft-identity [=>] 100.0	\[ x \cdot y - \color{blue}{y \cdot z} \]

3. Taylor expanded in x around 0 100.0%

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

4. Simplified 100.0%

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

   [Start] 100.0	\[ y \cdot x + -1 \cdot \left(y \cdot z\right) \]
   mul-1-neg [=>] 100.0	\[ y \cdot x + \color{blue}{\left(-y \cdot z\right)} \]
   distribute-rgt-neg-out [<=] 100.0	\[ y \cdot x + \color{blue}{y \cdot \left(-z\right)} \]
   fma-udef [<=] 100.0	\[ \color{blue}{\mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right)} \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	73.8%
Cost	786

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-115} \lor \neg \left(z \leq 6 \cdot 10^{-93}\right) \land \left(z \leq 5.6 \cdot 10^{-49} \lor \neg \left(z \leq 3.3 \cdot 10^{+29}\right)\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	320

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

Alternative 3
Accuracy	52.5%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, D"
  :precision binary64

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

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