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

Average Accuracy: 99.9% → 100.0%

Time: 4.6s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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(y * -0.5))
end

Wolfram

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

↓

code[x_, y_] := N[(x * 1.5 + N[(y * -0.5), $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 100.0%

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

   [Start] 99.9	\[ x + \frac{x - y}{2} \]
   +-commutative [=>] 99.9	\[ \color{blue}{\frac{x - y}{2} + x} \]
   div-sub [=>] 99.9	\[ \color{blue}{\left(\frac{x}{2} - \frac{y}{2}\right)} + x \]
   associate-+l- [=>] 99.8	\[ \color{blue}{\frac{x}{2} - \left(\frac{y}{2} - x\right)} \]
   sub-neg [=>] 99.8	\[ \frac{x}{2} - \color{blue}{\left(\frac{y}{2} + \left(-x\right)\right)} \]
   +-commutative [=>] 99.8	\[ \frac{x}{2} - \color{blue}{\left(\left(-x\right) + \frac{y}{2}\right)} \]
   associate--r+ [=>] 99.9	\[ \color{blue}{\left(\frac{x}{2} - \left(-x\right)\right) - \frac{y}{2}} \]
   sub-neg [=>] 99.9	\[ \color{blue}{\left(\frac{x}{2} + \left(-\left(-x\right)\right)\right)} - \frac{y}{2} \]
   remove-double-neg [=>] 99.9	\[ \left(\frac{x}{2} + \color{blue}{x}\right) - \frac{y}{2} \]
   +-commutative [<=] 99.9	\[ \color{blue}{\left(x + \frac{x}{2}\right)} - \frac{y}{2} \]
   *-lft-identity [<=] 99.9	\[ \left(\color{blue}{1 \cdot x} + \frac{x}{2}\right) - \frac{y}{2} \]
   metadata-eval [<=] 99.9	\[ \left(\color{blue}{\left(--1\right)} \cdot x + \frac{x}{2}\right) - \frac{y}{2} \]
   *-lft-identity [<=] 99.9	\[ \left(\left(--1\right) \cdot x + \frac{\color{blue}{1 \cdot x}}{2}\right) - \frac{y}{2} \]
   metadata-eval [<=] 99.9	\[ \left(\left(--1\right) \cdot x + \frac{\color{blue}{\left(--1\right)} \cdot x}{2}\right) - \frac{y}{2} \]
   associate-/l* [=>] 99.8	\[ \left(\left(--1\right) \cdot x + \color{blue}{\frac{--1}{\frac{2}{x}}}\right) - \frac{y}{2} \]
   associate-/r/ [=>] 99.9	\[ \left(\left(--1\right) \cdot x + \color{blue}{\frac{--1}{2} \cdot x}\right) - \frac{y}{2} \]
   distribute-rgt-out [=>] 99.9	\[ \color{blue}{x \cdot \left(\left(--1\right) + \frac{--1}{2}\right)} - \frac{y}{2} \]
   fma-neg [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(x, \left(--1\right) + \frac{--1}{2}, -\frac{y}{2}\right)} \]
   metadata-eval [=>] 100.0	\[ \mathsf{fma}\left(x, \color{blue}{1} + \frac{--1}{2}, -\frac{y}{2}\right) \]
   metadata-eval [=>] 100.0	\[ \mathsf{fma}\left(x, 1 + \frac{\color{blue}{1}}{2}, -\frac{y}{2}\right) \]
   metadata-eval [=>] 100.0	\[ \mathsf{fma}\left(x, 1 + \color{blue}{0.5}, -\frac{y}{2}\right) \]
   metadata-eval [=>] 100.0	\[ \mathsf{fma}\left(x, \color{blue}{1.5}, -\frac{y}{2}\right) \]
   distribute-neg-frac [=>] 100.0	\[ \mathsf{fma}\left(x, 1.5, \color{blue}{\frac{-y}{2}}\right) \]
   neg-mul-1 [=>] 100.0	\[ \mathsf{fma}\left(x, 1.5, \frac{\color{blue}{-1 \cdot y}}{2}\right) \]
   associate-/l* [=>] 99.9	\[ \mathsf{fma}\left(x, 1.5, \color{blue}{\frac{-1}{\frac{2}{y}}}\right) \]
   associate-/r/ [=>] 100.0	\[ \mathsf{fma}\left(x, 1.5, \color{blue}{\frac{-1}{2} \cdot y}\right) \]
   *-commutative [=>] 100.0	\[ \mathsf{fma}\left(x, 1.5, \color{blue}{y \cdot \frac{-1}{2}}\right) \]
   metadata-eval [=>] 100.0	\[ \mathsf{fma}\left(x, 1.5, y \cdot \color{blue}{-0.5}\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	74.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{+44}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-40}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	448

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

Alternative 3
Accuracy	99.9%
Cost	448

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

Alternative 4
Accuracy	50.5%
Cost	192

\[y \cdot -0.5 \]

Reproduce

herbie shell --seed 2023137 
(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)))