Average Error: 17.1 → 0.0
Time: 5.9s
Precision: binary64
Cost: 6784
\[\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y \]
↓
\[\mathsf{fma}\left(y, x, y \cdot \left(-z\right)\right) \]
(FPCore (x y z)
 :precision binary64
 (- (- (+ (* x y) (* y y)) (* y z)) (* y y)))
↓
(FPCore (x y z) :precision binary64 (fma y x (* y (- z))))
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));
}

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

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]

\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)

Error

Target

Original17.1
Target0.0
Herbie0.0
\[\left(x - z\right) \cdot y \]

Derivation

  1. Initial program 17.1

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

  2. Simplified0.0

    \[\leadsto \color{blue}{y \cdot \left(x - z\right)} \]
    Proof
    (*.f64 y (-.f64 x z)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 y x) (*.f64 y z))): 2 points increase in error, 1 points decrease in error
    (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (*.f64 y z)): 0 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 x y) 0)) (*.f64 y z)): 0 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (*.f64 x y) (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 y y) (*.f64 y y)))) (*.f64 y z)): 35 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 x y) (*.f64 y y)) (*.f64 y y))) (*.f64 y z)): 24 points increase in error, 0 points decrease in error
    (Rewrite<= associate--r+_binary64 (-.f64 (+.f64 (*.f64 x y) (*.f64 y y)) (+.f64 (*.f64 y y) (*.f64 y z)))): 25 points increase in error, 10 points decrease in error
    (-.f64 (+.f64 (*.f64 x y) (*.f64 y y)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 y z) (*.f64 y y)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate--l-_binary64 (-.f64 (-.f64 (+.f64 (*.f64 x y) (*.f64 y y)) (*.f64 y z)) (*.f64 y y))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.0

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

  4. Applied egg-rr0.0

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

  5. Applied egg-rr0.0

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

  6. Final simplification0.0

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

Alternatives

Alternative 1
Error15.4
Cost784
\[\begin{array}{l} t_0 := y \cdot \left(-z\right)\\ \mathbf{if}\;x \leq -5.7 \cdot 10^{+69}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -8747248413094373000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -0.597696618661753:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.2156616957824783 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
Alternative 2
Error0.0
Cost320
\[y \cdot \left(x - z\right) \]
Alternative 3
Error30.1
Cost192
\[y \cdot x \]

Error

Reproduce

herbie shell --seed 2022294 
(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)))