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

Average Accuracy: 73.3% → 100.0%

Time: 9.1s

Precision: binary64

Cost: 6784

Math

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

↓

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

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (- (- (* x y) (* y z)) (* y y)) (* 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 * z)) - (y * y)) + (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 * z)) - Float64(y * y)) + 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 * z), $MachinePrecision]), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

↓

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

Target

Original	73.3%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 73.3%

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

2. Applied egg-rr 100.0%

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

   [Start] 73.3	\[ \left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
   associate-+l- [=>] 88.1	\[ \color{blue}{\left(x \cdot y - y \cdot z\right) - \left(y \cdot y - y \cdot y\right)} \]
   expm1-log1p-u [=>] 63.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot y - y \cdot z\right)\right)} - \left(y \cdot y - y \cdot y\right) \]
   expm1-udef [=>] 29.4	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot y - y \cdot z\right)} - 1\right)} - \left(y \cdot y - y \cdot y\right) \]
   associate--l- [=>] 29.4	\[ \color{blue}{e^{\mathsf{log1p}\left(x \cdot y - y \cdot z\right)} - \left(1 + \left(y \cdot y - y \cdot y\right)\right)} \]
   +-inverses [=>] 34.6	\[ e^{\mathsf{log1p}\left(x \cdot y - y \cdot z\right)} - \left(1 + \color{blue}{0}\right) \]
   metadata-eval [=>] 34.6	\[ e^{\mathsf{log1p}\left(x \cdot y - y \cdot z\right)} - \color{blue}{1} \]
   expm1-udef [<=] 69.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot y - y \cdot z\right)\right)} \]
   expm1-log1p-u [<=] 100.0	\[ \color{blue}{x \cdot y - y \cdot z} \]
   *-commutative [=>] 100.0	\[ \color{blue}{y \cdot x} - y \cdot z \]
   fma-neg [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(y, x, -y \cdot z\right)} \]
   distribute-rgt-neg-in [=>] 100.0	\[ \mathsf{fma}\left(y, x, \color{blue}{y \cdot \left(-z\right)}\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	75.4%
Cost	1050

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-8}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-55} \lor \neg \left(x \leq -6.9 \cdot 10^{-102}\right) \land \left(x \leq 3.3 \cdot 10^{-51} \lor \neg \left(x \leq 6.4 \cdot 10^{-18}\right) \land x \leq 1.18 \cdot 10^{+33}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	320

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

Alternative 3
Accuracy	52.9%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023137 
(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)))