Average Error: 5.6 → 0.1
Time: 7.0s
Precision: binary64
Cost: 6720
\[x \cdot \left(1 + y \cdot y\right) \]
↓
\[\mathsf{fma}\left(y, y \cdot x, x\right) \]
(FPCore (x y) :precision binary64 (* x (+ 1.0 (* y y))))
↓
(FPCore (x y) :precision binary64 (fma y (* y x) x))
double code(double x, double y) {
	return x * (1.0 + (y * y));
}
↓
double code(double x, double y) {
	return fma(y, (y * x), x);
}

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(y * y)))
end
↓
function code(x, y)
	return fma(y, Float64(y * x), x)
end

code[x_, y_] := N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(y * N[(y * x), $MachinePrecision] + x), $MachinePrecision]

x \cdot \left(1 + y \cdot y\right)
↓
\mathsf{fma}\left(y, y \cdot x, x\right)

Error

Target

Original5.6
Target0.1
Herbie0.1
\[x + \left(x \cdot y\right) \cdot y \]

Derivation

  1. Initial program 5.6

    \[x \cdot \left(1 + y \cdot y\right) \]

  2. Taylor expanded in x around 0 5.6

    \[\leadsto \color{blue}{\left(1 + {y}^{2}\right) \cdot x} \]

  3. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot x, x\right)} \]
    Proof
    (fma.f64 y (*.f64 y x) x): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (*.f64 y x)) x)): 1 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y y) x)) x): 40 points increase in error, 7 points decrease in error
    (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) x) x): 0 points increase in error, 0 points decrease in error
    (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 (pow.f64 y 2) 1) x)): 2 points increase in error, 1 points decrease in error
    (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 y 2))) x): 0 points increase in error, 0 points decrease in error

  4. Final simplification0.1

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

Alternatives

Alternative 1
Error1.1
Cost584
\[\begin{array}{l} t_0 := y \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -287.87126243190335:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.07914621420229247:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error6.6
Cost580
\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.006264123222555162:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]
Alternative 3
Error0.1
Cost448
\[x + y \cdot \left(y \cdot x\right) \]
Alternative 4
Error20.8
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022294 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64

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

  (* x (+ 1.0 (* y y))))