\[\left(\frac{x}{2} + y \cdot x\right) + z \]

↓

\[\mathsf{fma}\left(0.5 + y, x, z\right) \]

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

↓

(FPCore (x y z) :precision binary64 (fma (+ 0.5 y) x z))

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

↓

double code(double x, double y, double z) {
	return fma((0.5 + y), x, z);
}

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

↓

function code(x, y, z)
	return fma(Float64(0.5 + y), x, z)
end

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(0.5 + y), $MachinePrecision] * x + z), $MachinePrecision]

\left(\frac{x}{2} + y \cdot x\right) + z

↓

\mathsf{fma}\left(0.5 + y, x, z\right)

Derivation

Initial program 0.0
\[\left(\frac{x}{2} + y \cdot x\right) + z \]
Simplified0.0
\[\leadsto \color{blue}{\frac{x}{2} + \mathsf{fma}\left(x, y, z\right)} \]
Proof
(+.f64 (/.f64 x 2) (fma.f64 x y z)): 0 points increase in error, 0 points decrease in error
(+.f64 (/.f64 x 2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) z))): 0 points increase in error, 0 points decrease in error
(+.f64 (/.f64 x 2) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 y x)) z)): 0 points increase in error, 0 points decrease in error
(Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (/.f64 x 2) (*.f64 y x)) z)): 0 points increase in error, 0 points decrease in error
Taylor expanded in x around 0 0.0
\[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + y, x, z\right)} \]
Proof
(fma.f64 (+.f64 1/2 y) x z): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 1/2 y) x) z)): 0 points increase in error, 0 points decrease in error
(Rewrite<= +-commutative_binary64 (+.f64 z (*.f64 (+.f64 1/2 y) x))): 0 points increase in error, 0 points decrease in error
Final simplification0.0
\[\leadsto \mathsf{fma}\left(0.5 + y, x, z\right) \]

Alternatives

Alternative 1
Error	30.5
Cost	1116

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+45}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-188}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-221}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-263}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+29}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+191}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	16.6
Cost	849

\[\begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{+91}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+53} \lor \neg \left(z \leq -1.95 \cdot 10^{-31}\right) \land z \leq 2.8 \cdot 10^{-41}:\\ \;\;\;\;\left(0.5 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3
Error	11.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+147} \lor \neg \left(x \leq 2000000000000\right):\\ \;\;\;\;\left(0.5 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]

Alternative 4
Error	1.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+14} \lor \neg \left(y \leq 2.45 \cdot 10^{-14}\right):\\ \;\;\;\;z + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + \frac{x}{2}\\ \end{array} \]

Alternative 5
Error	28.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+145}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+18}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 6
Error	0.0
Cost	448

\[z + \left(0.5 + y\right) \cdot x \]

Alternative 7
Error	35.1
Cost	64

\[z \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2.0) (* y x)) z))

Error

Derivation

Alternatives

Error

Reproduce