Diagrams.Color.HSV:lerp from diagrams-contrib-1.3.0.5

Average Error: 0.03% → 0.01%

Time: 4.7s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	0.03%
Target	0.02%
Herbie	0.01%

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

Derivation

1. Initial program 0.03

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

2. Simplified 0.01

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

   [Start] 0.03	\[ \left(1 - x\right) \cdot y + x \cdot z \]
   *-commutative [=>] 0.03	\[ \color{blue}{y \cdot \left(1 - x\right)} + x \cdot z \]
   sub-neg [=>] 0.03	\[ y \cdot \color{blue}{\left(1 + \left(-x\right)\right)} + x \cdot z \]
   distribute-rgt-in [=>] 0.02	\[ \color{blue}{\left(1 \cdot y + \left(-x\right) \cdot y\right)} + x \cdot z \]
   associate-+l+ [=>] 0.02	\[ \color{blue}{1 \cdot y + \left(\left(-x\right) \cdot y + x \cdot z\right)} \]
   *-lft-identity [=>] 0.02	\[ \color{blue}{y} + \left(\left(-x\right) \cdot y + x \cdot z\right) \]
   +-commutative [=>] 0.02	\[ \color{blue}{\left(\left(-x\right) \cdot y + x \cdot z\right) + y} \]
   *-commutative [=>] 0.02	\[ \left(\color{blue}{y \cdot \left(-x\right)} + x \cdot z\right) + y \]
   neg-mul-1 [=>] 0.02	\[ \left(y \cdot \color{blue}{\left(-1 \cdot x\right)} + x \cdot z\right) + y \]
   associate-*r* [=>] 0.02	\[ \left(\color{blue}{\left(y \cdot -1\right) \cdot x} + x \cdot z\right) + y \]
   *-commutative [=>] 0.02	\[ \left(\left(y \cdot -1\right) \cdot x + \color{blue}{z \cdot x}\right) + y \]
   distribute-rgt-out [=>] 0.02	\[ \color{blue}{x \cdot \left(y \cdot -1 + z\right)} + y \]
   fma-def [=>] 0.01	\[ \color{blue}{\mathsf{fma}\left(x, y \cdot -1 + z, y\right)} \]
   +-commutative [=>] 0.01	\[ \mathsf{fma}\left(x, \color{blue}{z + y \cdot -1}, y\right) \]
   *-commutative [=>] 0.01	\[ \mathsf{fma}\left(x, z + \color{blue}{-1 \cdot y}, y\right) \]
   metadata-eval [<=] 0.01	\[ \mathsf{fma}\left(x, z + \color{blue}{\left(-1\right)} \cdot y, y\right) \]
   cancel-sign-sub-inv [<=] 0.01	\[ \mathsf{fma}\left(x, \color{blue}{z - 1 \cdot y}, y\right) \]
   *-lft-identity [=>] 0.01	\[ \mathsf{fma}\left(x, z - \color{blue}{y}, y\right) \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	26.44%
Cost	1115

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-127} \lor \neg \left(y \leq 2 \cdot 10^{-224} \lor \neg \left(y \leq 5 \cdot 10^{-193}\right) \land \left(y \leq 2.5 \cdot 10^{-144} \lor \neg \left(y \leq 3.5 \cdot 10^{-118}\right) \land y \leq 2 \cdot 10^{-38}\right)\right):\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 2
Error	36.68%
Cost	652

\[\begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-12}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+40}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.91%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-127} \lor \neg \left(y \leq 1.95 \cdot 10^{-38}\right):\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z - y\right)\\ \end{array} \]

Alternative 4
Error	1.4%
Cost	585

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

Alternative 5
Error	38.99%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-128}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-38}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6
Error	0.02%
Cost	448

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

Alternative 7
Error	53.7%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023089 
(FPCore (x y z)
  :name "Diagrams.Color.HSV:lerp  from diagrams-contrib-1.3.0.5"
  :precision binary64

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

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