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

Average Error: 0.0 → 0.0

Time: 3.1s

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 (-.f64 z (Rewrite<= *-lft-identity_binary64 (*.f64 1 y))) y): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 z (*.f64 (neg.f64 1) y))) y): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (+.f64 z (*.f64 (Rewrite=> metadata-eval -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)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 y -1) x) (*.f64 z x))) y): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 -1 x))) (*.f64 z x)) y): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 y (Rewrite<= neg-mul-1_binary64 (neg.f64 x))) (*.f64 z x)) y): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 x) y)) (*.f64 z x)) y): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 (neg.f64 x) y) (Rewrite<= *-commutative_binary64 (*.f64 x z))) y): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 y (+.f64 (*.f64 (neg.f64 x) y) (*.f64 x z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 y)) (+.f64 (*.f64 (neg.f64 x) y) (*.f64 x z))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 1 y) (*.f64 (neg.f64 x) y)) (*.f64 x z))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 y (+.f64 1 (neg.f64 x)))) (*.f64 x z)): 0 points increase in error, 0 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	26.0
Cost	784

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+37}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-105}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-23}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+33}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Error	11.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-29} \lor \neg \left(x \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;x \cdot \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 3
Error	1.0
Cost	585

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

Alternative 4
Error	0.0
Cost	576

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

Alternative 5
Error	23.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-29}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 6
Error	34.9
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022343 
(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)))