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

Average Error: 0.03% → 0.02%

Time: 4.2s

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 0.03

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

2. Simplified 0.02

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

   [Start] 0.03	\[ x \cdot y + \left(x - 1\right) \cdot z \]
   fma-def [=>] 0.02	\[ \color{blue}{\mathsf{fma}\left(x, y, \left(x - 1\right) \cdot z\right)} \]
   *-commutative [=>] 0.02	\[ \mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(x - 1\right)}\right) \]
   sub-neg [=>] 0.02	\[ \mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(x + \left(-1\right)\right)}\right) \]
   distribute-rgt-in [=>] 0.01	\[ \mathsf{fma}\left(x, y, \color{blue}{x \cdot z + \left(-1\right) \cdot z}\right) \]
   remove-double-neg [<=] 0.01	\[ \mathsf{fma}\left(x, y, \color{blue}{\left(-\left(-x\right)\right)} \cdot z + \left(-1\right) \cdot z\right) \]
   distribute-rgt-out [=>] 0.02	\[ \mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(\left(-\left(-x\right)\right) + \left(-1\right)\right)}\right) \]
   distribute-neg-in [<=] 0.02	\[ \mathsf{fma}\left(x, y, z \cdot \color{blue}{\left(-\left(\left(-x\right) + 1\right)\right)}\right) \]
   *-commutative [<=] 0.02	\[ \mathsf{fma}\left(x, y, \color{blue}{\left(-\left(\left(-x\right) + 1\right)\right) \cdot z}\right) \]
   distribute-neg-in [=>] 0.02	\[ \mathsf{fma}\left(x, y, \color{blue}{\left(\left(-\left(-x\right)\right) + \left(-1\right)\right)} \cdot z\right) \]
   remove-double-neg [=>] 0.02	\[ \mathsf{fma}\left(x, y, \left(\color{blue}{x} + \left(-1\right)\right) \cdot z\right) \]
   metadata-eval [=>] 0.02	\[ \mathsf{fma}\left(x, y, \left(x + \color{blue}{-1}\right) \cdot z\right) \]

3. Final simplification 0.02

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

Alternatives

Alternative 1
Error	37.06%
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00075:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-13}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{+99}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+206}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2
Error	1.32%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 0.0003:\\ \;\;\;\;x \cdot y - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + x \cdot z\\ \end{array} \]

Alternative 3
Error	19.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00075 \lor \neg \left(x \leq 7.5 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4
Error	21.32%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-140} \lor \neg \left(z \leq 7.2 \cdot 10^{-90}\right):\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \end{array} \]

Alternative 5
Error	1.32%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.0003\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - z\\ \end{array} \]

Alternative 6
Error	21.43%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-143}:\\ \;\;\;\;x \cdot z - z\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 7
Error	0.03%
Cost	576

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

Alternative 8
Error	37.46%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00075:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-12}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9
Error	0.02%
Cost	448

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

Alternative 10
Error	54.82%
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3"
  :precision binary64
  (+ (* x y) (* (- x 1.0) z)))