Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3

Average Error: 0.0 → 0.0

Time: 3.7s

Precision: binary64

Cost: 6848

Math

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

↓

\[\mathsf{fma}\left(x, y, z \cdot \left(x + -1\right)\right) \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- x 1.0) z)))

↓

(FPCore (x y z) :precision binary64 (fma x y (* z (+ x -1.0))))

C

double code(double x, double y, double z) {
	return (x * y) + ((x - 1.0) * z);
}

↓

double code(double x, double y, double z) {
	return fma(x, y, (z * (x + -1.0)));
}

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(x - 1.0) * z))
end

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \left(x + -1\right) \cdot z\right)} \]
   Proof

   [Start] 0.0	\[ x \cdot y + \left(x - 1\right) \cdot z \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(x, y, \left(x - 1\right) \cdot z\right)} \]
   *-commutative [=>] 0.0	\[ \mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(x - 1\right)}\right) \]
   sub-neg [=>] 0.0	\[ \mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(x + \left(-1\right)\right)}\right) \]
   distribute-rgt-in [=>] 0.0	\[ \mathsf{fma}\left(x, y, \color{blue}{x \cdot z + \left(-1\right) \cdot z}\right) \]
   remove-double-neg [<=] 0.0	\[ \mathsf{fma}\left(x, y, \color{blue}{\left(-\left(-x\right)\right)} \cdot z + \left(-1\right) \cdot z\right) \]
   distribute-rgt-out [=>] 0.0	\[ \mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(\left(-\left(-x\right)\right) + \left(-1\right)\right)}\right) \]
   distribute-neg-in [<=] 0.0	\[ \mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(-\left(\left(-x\right) + 1\right)\right)}\right) \]
   *-commutative [<=] 0.0	\[ \mathsf{fma}\left(x, y, \color{blue}{\left(-\left(\left(-x\right) + 1\right)\right) \cdot z}\right) \]
   distribute-neg-in [=>] 0.0	\[ \mathsf{fma}\left(x, y, \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-1\right)\right)} \cdot z\right) \]
   remove-double-neg [=>] 0.0	\[ \mathsf{fma}\left(x, y, \left(\color{blue}{x} + \left(-1\right)\right) \cdot z\right) \]
   metadata-eval [=>] 0.0	\[ \mathsf{fma}\left(x, y, \left(x + \color{blue}{-1}\right) \cdot z\right) \]

3. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(x, y, z \cdot \left(x + -1\right)\right) \]

Alternatives

Alternative 1
Error	24.1
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+114}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{+68}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-13}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2
Error	12.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-82} \lor \neg \left(x \leq 4.2 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3
Error	13.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -160000 \lor \neg \left(z \leq 7 \cdot 10^{-118}\right):\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \end{array} \]

Alternative 4
Error	0.9
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - z\\ \end{array} \]

Alternative 5
Error	0.0
Cost	576

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

Alternative 6
Error	25.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -230000:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-117}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	0.0
Cost	448

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

Alternative 8
Error	35.0
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3"
  :precision binary64
  (+ (* x y) (* (- x 1.0) z)))