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

Average Accuracy: 100.0% → 100.0%

Time: 4.7s

Precision: binary64

Cost: 6784

Math

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

↓

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

FPCore

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

↓

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

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), -z);
}

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, Float64(y + z), Float64(-z))
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 * N[(y + z), $MachinePrecision] + (-z)), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ x \cdot y + \left(x - 1\right) \cdot z \]
   *-commutative [=>] 100.0	\[ x \cdot y + \color{blue}{z \cdot \left(x - 1\right)} \]
   sub-neg [=>] 100.0	\[ x \cdot y + z \cdot \color{blue}{\left(x + \left(-1\right)\right)} \]
   distribute-rgt-in [=>] 100.0	\[ x \cdot y + \color{blue}{\left(x \cdot z + \left(-1\right) \cdot z\right)} \]
   associate-+r+ [=>] 100.0	\[ \color{blue}{\left(x \cdot y + x \cdot z\right) + \left(-1\right) \cdot z} \]
   distribute-lft-out [=>] 100.0	\[ \color{blue}{x \cdot \left(y + z\right)} + \left(-1\right) \cdot z \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(x, y + z, \left(-1\right) \cdot z\right)} \]
   metadata-eval [=>] 100.0	\[ \mathsf{fma}\left(x, y + z, \color{blue}{-1} \cdot z\right) \]
   mul-1-neg [=>] 100.0	\[ \mathsf{fma}\left(x, y + z, \color{blue}{-z}\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	81.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-14} \lor \neg \left(x \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Accuracy	98.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -58000000000000 \lor \neg \left(x \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - z\\ \end{array} \]

Alternative 3
Accuracy	62.6%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4
Accuracy	63.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 5
Accuracy	100.0%
Cost	448

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

Alternative 6
Accuracy	45.6%
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3"
  :precision binary64
  (+ (* x y) (* (- x 1.0) z)))