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

Average Error: 0.1 → 0

Time: 4.4s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	0.1
Target	0.1
Herbie	0

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

Derivation

1. Initial program 0.1

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

2. Taylor expanded in x around 0 0.1

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

3. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)} \]
   Proof
   (fma.f64 3/2 x (*.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3/2 x) (*.f64 -1/2 y))): 18 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 1/2 1)) x) (*.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 1/2 x) x)) (*.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1/2 y) (+.f64 (*.f64 1/2 x) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 -1/2 y) (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 1/2 1) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 -1/2 y) (*.f64 (Rewrite=> metadata-eval 3/2) x)): 0 points increase in error, 0 points decrease in error

4. Final simplification 0

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

Alternatives

Alternative 1
Error	16.8
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -4.554913430096951 \cdot 10^{-13}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 4.648570085008161 \cdot 10^{-24}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;y \leq 14421863947182.963:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq 2.865447946667378 \cdot 10^{+83}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \]

Alternative 2
Error	0.1
Cost	448

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

Alternative 3
Error	31.8
Cost	192

\[1.5 \cdot x \]

Reproduce

herbie shell --seed 2022291 
(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)))