Average Error: 0.0 → 0.0
Time: 8.7s
Precision: binary64
Cost: 6720
\[x + \left(y - x\right) \cdot z \]
↓
\[\mathsf{fma}\left(y - x, z, x\right) \]
(FPCore (x y z) :precision binary64 (+ x (* (- y x) z)))
↓
(FPCore (x y z) :precision binary64 (fma (- y x) z x))
double code(double x, double y, double z) {
	return x + ((y - x) * z);
}
↓
double code(double x, double y, double z) {
	return fma((y - x), z, x);
}

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

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

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

Error

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, z, x\right)} \]
    Proof
    (fma.f64 (-.f64 y x) z x): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 y x) z) x)): 3 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 (-.f64 y x) z))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.0

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

Alternatives

Alternative 1
Error26.9
Cost1312
\[\begin{array}{l} \mathbf{if}\;z \leq -2.335692013546964 \cdot 10^{-24}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -4.228206953008226 \cdot 10^{-73}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.3911802600254683 \cdot 10^{-114}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.0665849218522088 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -6.867932340465391 \cdot 10^{-199}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2.0208099600057898 \cdot 10^{-70}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 717315.7439188524:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+79}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
Alternative 2
Error15.3
Cost1112
\[\begin{array}{l} t_0 := \left(y - x\right) \cdot z\\ \mathbf{if}\;z \leq -2.335692013546964 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.228206953008226 \cdot 10^{-73}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.3911802600254683 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.0665849218522088 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -6.867932340465391 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.0208099600057898 \cdot 10^{-70}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error26.7
Cost984
\[\begin{array}{l} \mathbf{if}\;z \leq -2.335692013546964 \cdot 10^{-24}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -4.228206953008226 \cdot 10^{-73}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.3911802600254683 \cdot 10^{-114}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.0665849218522088 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -6.867932340465391 \cdot 10^{-199}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2.0208099600057898 \cdot 10^{-70}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
Alternative 4
Error1.2
Cost712
\[\begin{array}{l} \mathbf{if}\;z \leq -4339070671785.9253:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{elif}\;z \leq 2.8755868229424205 \cdot 10^{-8}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot z - x \cdot z\\ \end{array} \]
Alternative 5
Error1.2
Cost584
\[\begin{array}{l} t_0 := \left(y - x\right) \cdot z\\ \mathbf{if}\;z \leq -4339070671785.9253:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.8755868229424205 \cdot 10^{-8}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error0.0
Cost448
\[x + \left(y - x\right) \cdot z \]
Alternative 7
Error35.2
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022300 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))