Diagrams.TwoD.Segment:bezierClip from diagrams-lib-1.3.0.3

Average Error: 0.0 → 0.0

Time: 7.4s

Precision: binary64

Cost: 6720

Math

\[x \cdot y + z \cdot \left(1 - y\right) \]

↓

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

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* z (- 1.0 y))))

↓

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

C

double code(double x, double y, double z) {
	return (x * y) + (z * (1.0 - y));
}

↓

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

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(z * Float64(1.0 - y)))
end

↓

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

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(z * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

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

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

   \[x \cdot y + z \cdot \left(1 - y\right) \]

2. Simplified 0.0

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

   [Start] 0.0	\[ x \cdot y + z \cdot \left(1 - y\right) \]
   +-commutative [=>] 0.0	\[ \color{blue}{z \cdot \left(1 - y\right) + x \cdot y} \]
   sub-neg [=>] 0.0	\[ z \cdot \color{blue}{\left(1 + \left(-y\right)\right)} + x \cdot y \]
   distribute-rgt-in [=>] 0.0	\[ \color{blue}{\left(1 \cdot z + \left(-y\right) \cdot z\right)} + x \cdot y \]
   associate-+l+ [=>] 0.0	\[ \color{blue}{1 \cdot z + \left(\left(-y\right) \cdot z + x \cdot y\right)} \]
   *-lft-identity [=>] 0.0	\[ \color{blue}{z} + \left(\left(-y\right) \cdot z + x \cdot y\right) \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(\left(-y\right) \cdot z + x \cdot y\right) + z} \]
   *-commutative [=>] 0.0	\[ \left(\color{blue}{z \cdot \left(-y\right)} + x \cdot y\right) + z \]
   neg-mul-1 [=>] 0.0	\[ \left(z \cdot \color{blue}{\left(-1 \cdot y\right)} + x \cdot y\right) + z \]
   associate-*r* [=>] 0.0	\[ \left(\color{blue}{\left(z \cdot -1\right) \cdot y} + x \cdot y\right) + z \]
   distribute-rgt-out [=>] 0.0	\[ \color{blue}{y \cdot \left(z \cdot -1 + x\right)} + z \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(y, z \cdot -1 + x, z\right)} \]
   +-commutative [=>] 0.0	\[ \mathsf{fma}\left(y, \color{blue}{x + z \cdot -1}, z\right) \]
   *-commutative [=>] 0.0	\[ \mathsf{fma}\left(y, x + \color{blue}{-1 \cdot z}, z\right) \]
   metadata-eval [<=] 0.0	\[ \mathsf{fma}\left(y, x + \color{blue}{\left(-1\right)} \cdot z, z\right) \]
   cancel-sign-sub-inv [<=] 0.0	\[ \mathsf{fma}\left(y, \color{blue}{x - 1 \cdot z}, z\right) \]
   *-lft-identity [=>] 0.0	\[ \mathsf{fma}\left(y, x - \color{blue}{z}, z\right) \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	24.9
Cost	916

\[\begin{array}{l} t_0 := z \cdot \left(-y\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-158}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-133}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+157}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	13.1
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(x - z\right)\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-158}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-133}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-11}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.2
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(x - z\right)\\ t_1 := z \cdot \left(1 - y\right)\\ \mathbf{if}\;y \leq -240000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-133}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	24.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-10}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-158}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-133}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-12}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Error	0.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.0076\right):\\ \;\;\;\;y \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]

Alternative 6
Error	0.0
Cost	576

\[z \cdot \left(1 - y\right) + y \cdot x \]

Alternative 7
Error	0.0
Cost	448

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

Alternative 8
Error	35.1
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023083 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment:bezierClip from diagrams-lib-1.3.0.3"
  :precision binary64

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

  (+ (* x y) (* z (- 1.0 y))))