Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, C

Average Error: 0.0 → 0

Time: 2.7s

Precision: binary64

Cost: 6656

Math

\[x - y \cdot z \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	0.0
Target	0.0
Herbie	0

\[\frac{x + y \cdot z}{\frac{x + y \cdot z}{x - y \cdot z}} \]

Derivation

1. Initial program 0.0

   \[x - y \cdot z \]

2. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, -z, x\right)} \]
   Proof
   (fma.f64 y (neg.f64 z) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (neg.f64 z)) x)): 1 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y z))) x): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 y) z)) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 (neg.f64 y) z))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 x (*.f64 y z))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0

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

Alternatives

Alternative 1
Error	17.3
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -5.765247983000496 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.357817561832913 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	320

\[x - y \cdot z \]

Alternative 3
Error	27.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, C"
  :precision binary64

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

  (- x (* y z)))