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

Average Error: 0.1 → 0

Time: 4.3s

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	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. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)} \]
   Proof
   (fma.f64 x 3/2 (*.f64 y -1/2)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (Rewrite<= metadata-eval (+.f64 1 1/2)) (*.f64 y -1/2)): 0 points increase in error, 19 points decrease in error
   (fma.f64 x (+.f64 (Rewrite<= metadata-eval (neg.f64 -1)) 1/2) (*.f64 y -1/2)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 (neg.f64 -1) (Rewrite<= metadata-eval (/.f64 1 2))) (*.f64 y -1/2)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 (neg.f64 -1) (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) 2)) (*.f64 y -1/2)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 (neg.f64 -1) (/.f64 (neg.f64 -1) 2)) (*.f64 y (Rewrite<= metadata-eval (/.f64 -1 2)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 (neg.f64 -1) (/.f64 (neg.f64 -1) 2)) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 -1 2) y))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 (neg.f64 -1) (/.f64 (neg.f64 -1) 2)) (Rewrite<= associate-/r/_binary64 (/.f64 -1 (/.f64 2 y)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 (neg.f64 -1) (/.f64 (neg.f64 -1) 2)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 y) 2))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 (neg.f64 -1) (/.f64 (neg.f64 -1) 2)) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 y)) 2)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 (neg.f64 -1) (/.f64 (neg.f64 -1) 2)) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 y 2)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x (+.f64 (neg.f64 -1) (/.f64 (neg.f64 -1) 2))) (/.f64 y 2))): 18 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (neg.f64 -1) x) (*.f64 (/.f64 (neg.f64 -1) 2) x))) (/.f64 y 2)): 0 points increase in error, 18 points decrease in error
   (-.f64 (+.f64 (*.f64 (Rewrite=> metadata-eval 1) x) (*.f64 (/.f64 (neg.f64 -1) 2) x)) (/.f64 y 2)): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (Rewrite=> *-lft-identity_binary64 x) (*.f64 (/.f64 (neg.f64 -1) 2) x)) (/.f64 y 2)): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 x (Rewrite<= associate-/r/_binary64 (/.f64 (neg.f64 -1) (/.f64 2 x)))) (/.f64 y 2)): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (neg.f64 -1) x) 2))) (/.f64 y 2)): 19 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 x (/.f64 (*.f64 (Rewrite=> metadata-eval 1) x) 2)) (/.f64 y 2)): 0 points increase in error, 19 points decrease in error
   (-.f64 (+.f64 x (/.f64 (Rewrite=> *-lft-identity_binary64 x) 2)) (/.f64 y 2)): 19 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (+.f64 x (/.f64 x 2)) 0)) (/.f64 y 2)): 0 points increase in error, 19 points decrease in error
   (-.f64 (Rewrite=> associate--l+_binary64 (+.f64 x (-.f64 (/.f64 x 2) 0))) (/.f64 y 2)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 (/.f64 x 2) 0) x)) (/.f64 y 2)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate--r-_binary64 (-.f64 (/.f64 x 2) (-.f64 0 x))) (/.f64 y 2)): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (/.f64 x 2) (Rewrite<= neg-sub0_binary64 (neg.f64 x))) (/.f64 y 2)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--r+_binary64 (-.f64 (/.f64 x 2) (+.f64 (neg.f64 x) (/.f64 y 2)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 x 2) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 y 2) (neg.f64 x)))): 19 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 x 2) (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 y 2) x))): 1 points increase in error, 19 points decrease in error
   (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (/.f64 x 2) (/.f64 y 2)) x)): 0 points increase in error, 1 points decrease in error
   (+.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 x y) 2)) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (-.f64 x y) 2))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0

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

Alternatives

Alternative 1
Error	16.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+45}:\\ \;\;\;\;x \cdot 1.5\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-35}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 2
Error	0.1
Cost	448

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

Alternative 3
Error	31.3
Cost	192

\[y \cdot -0.5 \]

Reproduce

herbie shell --seed 2022343 
(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)))