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

Average Accuracy: 73.1% → 100.0%

Time: 5.8s

Precision: binary64

Cost: 6784

Math

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

↓

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

FPCore

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

↓

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

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 fma(-z, y, (y * x));
}

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

Target

Original	73.1%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 73.1%

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

2. Taylor expanded in x around 0 100.0%

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

3. Simplified 100.0%

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

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

4. Applied egg-rr 100.0%

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

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

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	76.1%
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-49} \lor \neg \left(z \leq 3.9 \cdot 10^{+20}\right):\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	320

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

Alternative 3
Accuracy	52.4%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023151 
(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)))