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

Average Accuracy: 99.8% → 99.8%

Time: 3.6s

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	99.8%
Target	99.8%
Herbie	99.8%

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

Derivation

1. Initial program 99.8%

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

2. Simplified 99.8%

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

   [Start] 99.8	\[ \left(x \cdot 3\right) \cdot y - z \]
   fma-neg [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(x \cdot 3, y, -z\right)} \]

3. Taylor expanded in x around 0 99.8%

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

4. Simplified 99.8%

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

   [Start] 99.8	\[ 3 \cdot \left(y \cdot x\right) + -1 \cdot z \]
   fma-def [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(3, y \cdot x, -1 \cdot z\right)} \]
   mul-1-neg [=>] 99.8	\[ \mathsf{fma}\left(3, y \cdot x, \color{blue}{-z}\right) \]

5. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	71.6%
Cost	849

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-29}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.12 \cdot 10^{-101} \lor \neg \left(z \leq 7.8 \cdot 10^{-50}\right) \land z \leq 8 \cdot 10^{+75}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	448

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

Alternative 3
Accuracy	57.2%
Cost	128

\[-z \]

Alternative 4
Accuracy	2.3%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023137 
(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))