Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, B

Average Error: 0.2 → 0.1

Time: 2.2s

Precision: binary64

Cost: 6784

Math

\[\left(x \cdot 3\right) \cdot y - z \]

↓

\[\mathsf{fma}\left(3, y \cdot x, -z\right) \]

FPCore

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

↓

(FPCore (x y z) :precision binary64 (fma 3.0 (* y x) (- z)))

C

double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

↓

double code(double x, double y, double z) {
	return fma(3.0, (y * x), -z);
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * y) - z)
end

↓

function code(x, y, z)
	return fma(3.0, Float64(y * x), Float64(-z))
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(3.0 * N[(y * x), $MachinePrecision] + (-z)), $MachinePrecision]

Target

Original	0.2
Target	0.1
Herbie	0.1

\[x \cdot \left(3 \cdot y\right) - z \]

Derivation

1. Initial program 0.2

   \[\left(x \cdot 3\right) \cdot y - z \]

2. Simplified 0.1

   \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) - z} \]
   Proof
   (-.f64 (*.f64 x (*.f64 3 y)) z): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 3) y)) z): 19 points increase in error, 22 points decrease in error

3. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right) + -1 \cdot z} \]

4. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(3, y \cdot x, -z\right)} \]
   Proof
   (fma.f64 3 (*.f64 y x) (neg.f64 z)): 0 points increase in error, 0 points decrease in error
   (fma.f64 3 (*.f64 y x) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 z))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 3 (*.f64 y x)) (*.f64 -1 z))): 14 points increase in error, 10 points decrease in error

5. Final simplification 0.1

   \[\leadsto \mathsf{fma}\left(3, y \cdot x, -z\right) \]

Alternatives

Alternative 1
Error	16.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-116}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.14 \cdot 10^{-29}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	0.1
Cost	448

\[3 \cdot \left(y \cdot x\right) - z \]

Alternative 3
Error	27.1
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (* x (* 3.0 y)) z)

  (- (* (* x 3.0) y) z))