Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3

Average Accuracy: 99.9% → 100.0%

Time: 4.1s

Precision: binary64

Cost: 6720

Math

\[x + \frac{x - y}{2} \]

↓

\[\mathsf{fma}\left(x, 1.5, -0.5 \cdot y\right) \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))

↓

(FPCore (x y) :precision binary64 (fma x 1.5 (* -0.5 y)))

C

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

↓

double code(double x, double y) {
	return fma(x, 1.5, (-0.5 * y));
}

Julia

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

↓

function code(x, y)
	return fma(x, 1.5, Float64(-0.5 * y))
end

Wolfram

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

↓

code[x_, y_] := N[(x * 1.5 + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision]

Target

Original	99.9%
Target	99.9%
Herbie	100.0%

\[1.5 \cdot x - 0.5 \cdot y \]

Derivation

1. Initial program 99.9%

   \[x + \frac{x - y}{2} \]

2. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5, x \cdot 1.5\right)} \]
   Proof

   [Start] 99.9	\[ x + \frac{x - y}{2} \]
   div-sub [=>] 99.9	\[ x + \color{blue}{\left(\frac{x}{2} - \frac{y}{2}\right)} \]
   associate-+r- [=>] 99.9	\[ \color{blue}{\left(x + \frac{x}{2}\right) - \frac{y}{2}} \]
   sub-neg [=>] 99.9	\[ \color{blue}{\left(x + \frac{x}{2}\right) + \left(-\frac{y}{2}\right)} \]
   +-commutative [=>] 99.9	\[ \color{blue}{\left(-\frac{y}{2}\right) + \left(x + \frac{x}{2}\right)} \]
   distribute-neg-frac [=>] 99.9	\[ \color{blue}{\frac{-y}{2}} + \left(x + \frac{x}{2}\right) \]
   neg-mul-1 [=>] 99.9	\[ \frac{\color{blue}{-1 \cdot y}}{2} + \left(x + \frac{x}{2}\right) \]
   associate-/l* [=>] 99.7	\[ \color{blue}{\frac{-1}{\frac{2}{y}}} + \left(x + \frac{x}{2}\right) \]
   associate-/r/ [=>] 99.9	\[ \color{blue}{\frac{-1}{2} \cdot y} + \left(x + \frac{x}{2}\right) \]
   *-commutative [=>] 99.9	\[ \color{blue}{y \cdot \frac{-1}{2}} + \left(x + \frac{x}{2}\right) \]
   fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(y, \frac{-1}{2}, x + \frac{x}{2}\right)} \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(y, \color{blue}{-0.5}, x + \frac{x}{2}\right) \]
   remove-double-neg [<=] 99.9	\[ \mathsf{fma}\left(y, -0.5, \color{blue}{\left(-\left(-x\right)\right)} + \frac{x}{2}\right) \]
   neg-mul-1 [=>] 99.9	\[ \mathsf{fma}\left(y, -0.5, \color{blue}{-1 \cdot \left(-x\right)} + \frac{x}{2}\right) \]
   remove-double-neg [<=] 99.9	\[ \mathsf{fma}\left(y, -0.5, -1 \cdot \left(-x\right) + \frac{\color{blue}{-\left(-x\right)}}{2}\right) \]
   neg-mul-1 [=>] 99.9	\[ \mathsf{fma}\left(y, -0.5, -1 \cdot \left(-x\right) + \frac{\color{blue}{-1 \cdot \left(-x\right)}}{2}\right) \]
   associate-/l* [=>] 99.9	\[ \mathsf{fma}\left(y, -0.5, -1 \cdot \left(-x\right) + \color{blue}{\frac{-1}{\frac{2}{-x}}}\right) \]
   associate-/r/ [=>] 99.9	\[ \mathsf{fma}\left(y, -0.5, -1 \cdot \left(-x\right) + \color{blue}{\frac{-1}{2} \cdot \left(-x\right)}\right) \]
   distribute-rgt-out [=>] 99.9	\[ \mathsf{fma}\left(y, -0.5, \color{blue}{\left(-x\right) \cdot \left(-1 + \frac{-1}{2}\right)}\right) \]
   distribute-lft-neg-in [<=] 99.9	\[ \mathsf{fma}\left(y, -0.5, \color{blue}{-x \cdot \left(-1 + \frac{-1}{2}\right)}\right) \]
   distribute-rgt-neg-in [=>] 99.9	\[ \mathsf{fma}\left(y, -0.5, \color{blue}{x \cdot \left(-\left(-1 + \frac{-1}{2}\right)\right)}\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(y, -0.5, x \cdot \left(-\left(-1 + \color{blue}{-0.5}\right)\right)\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(y, -0.5, x \cdot \left(-\color{blue}{-1.5}\right)\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(y, -0.5, x \cdot \color{blue}{1.5}\right) \]

3. Taylor expanded in y around 0 99.9%

   \[\leadsto \color{blue}{-0.5 \cdot y + 1.5 \cdot x} \]

4. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.5, -0.5 \cdot y\right)} \]
   Proof

   [Start] 99.9	\[ -0.5 \cdot y + 1.5 \cdot x \]
   +-commutative [=>] 99.9	\[ \color{blue}{1.5 \cdot x + -0.5 \cdot y} \]
   *-commutative [=>] 99.9	\[ \color{blue}{x \cdot 1.5} + -0.5 \cdot y \]
   *-commutative [=>] 99.9	\[ x \cdot 1.5 + \color{blue}{y \cdot -0.5} \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)} \]
   *-commutative [<=] 100.0	\[ \mathsf{fma}\left(x, 1.5, \color{blue}{-0.5 \cdot y}\right) \]

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, 1.5, -0.5 \cdot y\right) \]

Alternatives

Alternative 1
Accuracy	73.4%
Cost	721

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-50}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+20} \lor \neg \left(y \leq 5.2 \cdot 10^{+59}\right) \land y \leq 1.1 \cdot 10^{+92}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	448

\[x + \frac{x - y}{2} \]

Alternative 3
Accuracy	99.9%
Cost	448

\[-0.5 \cdot y + x \cdot 1.5 \]

Alternative 4
Accuracy	50.6%
Cost	192

\[-0.5 \cdot y \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* 1.5 x) (* 0.5 y))

  (+ x (/ (- x y) 2.0)))