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

Average Error: 20.12% → 0.01%

Time: 6.1s

Precision: binary64

Cost: 6784

Math

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

↓

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

FPCore

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

↓

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

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

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

Target

Original	20.12%
Target	0.02%
Herbie	0.01%

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

Derivation

1. Initial program 20.12

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

2. Simplified 0.01

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

   [Start] 20.12	\[ \left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) - y \cdot z \]
   sub-neg [=>] 20.12	\[ \color{blue}{\left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right) + \left(-y \cdot z\right)} \]
   +-commutative [=>] 20.12	\[ \color{blue}{\left(-y \cdot z\right) + \left(\left(x \cdot y - y \cdot y\right) + y \cdot y\right)} \]
   associate-+l- [=>] 12.39	\[ \left(-y \cdot z\right) + \color{blue}{\left(x \cdot y - \left(y \cdot y - y \cdot y\right)\right)} \]
   associate-+r- [=>] 12.39	\[ \color{blue}{\left(\left(-y \cdot z\right) + x \cdot y\right) - \left(y \cdot y - y \cdot y\right)} \]
   +-inverses [=>] 0.02	\[ \left(\left(-y \cdot z\right) + x \cdot y\right) - \color{blue}{0} \]
   --rgt-identity [=>] 0.02	\[ \color{blue}{\left(-y \cdot z\right) + x \cdot y} \]
   distribute-rgt-neg-in [=>] 0.02	\[ \color{blue}{y \cdot \left(-z\right)} + x \cdot y \]
   fma-def [=>] 0.01	\[ \color{blue}{\mathsf{fma}\left(y, -z, x \cdot y\right)} \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	23.15%
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-66} \lor \neg \left(z \leq 3.6 \cdot 10^{-29}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	0.02%
Cost	320

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

Alternative 3
Error	45.62%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023115 
(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)))