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

Average Error: 0.0 → 0

Time: 1.3s

Precision: binary64

Cost: 6656

Math

\[x - y \cdot z \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, -y, x\right)} \]
   Proof
   (fma.f64 z (neg.f64 y) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (neg.f64 y)) x)): 3 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_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

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

Alternatives

Alternative 1
Error	17.5
Cost	784

\[\begin{array}{l} t_0 := z \cdot \left(-y\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-122}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-176}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	320

\[x - z \cdot y \]

Alternative 3
Error	27.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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)))