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

Average Error: 0.0 → 0.0

Time: 4.0s

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	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, z - y, y\right)} \]
   Proof
   (fma.f64 x (-.f64 z y) y): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 z (neg.f64 y))) y): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 z (Rewrite=> neg-mul-1_binary64 (*.f64 -1 y))) y): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 z (Rewrite<= *-commutative_binary64 (*.f64 y -1))) y): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 y -1) z)) y): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 y -1) z)) y)): 3 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x (*.f64 y -1)) (*.f64 x z))) y): 0 points increase in error, 1 points decrease in error
   (+.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 y -1) x)) (*.f64 x z)) y): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 -1 x))) (*.f64 x z)) y): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 y (Rewrite<= neg-mul-1_binary64 (neg.f64 x))) (*.f64 x z)) y): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 y (+.f64 (*.f64 y (neg.f64 x)) (*.f64 x z)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 y (*.f64 y (neg.f64 x))) (*.f64 x z))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 y 1)) (*.f64 y (neg.f64 x))) (*.f64 x z)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> distribute-lft-out_binary64 (*.f64 y (+.f64 1 (neg.f64 x)))) (*.f64 x z)): 2 points increase in error, 2 points decrease in error
   (+.f64 (*.f64 y (Rewrite<= sub-neg_binary64 (-.f64 1 x))) (*.f64 x z)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1 x) y)) (*.f64 x z)): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	25.1
Cost	1180

\[\begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{+111}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{+16}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-121}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-43}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-24}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+21}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	19.2
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(1 - x\right)\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+150}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-16}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 3
Error	13.7
Cost	848

\[\begin{array}{l} t_0 := x \cdot \left(z - y\right)\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-121}:\\ \;\;\;\;y \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-43}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-23}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	25.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+16}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-121}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-43}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-24}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 5
Error	0.9
Cost	584

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

Alternative 6
Error	0.9
Cost	584

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

Alternative 7
Error	0.0
Cost	576

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

Alternative 8
Error	34.1
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022329 
(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)))