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

Average Error: 0.01% → 0%

Time: 3.8s

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(Float64(-y), 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	0.01%
Target	0.02%
Herbie	0%

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

Derivation

1. Initial program 0.01

   \[x - y \cdot z \]

2. Applied egg-rr 72.06

   \[\leadsto \color{blue}{\left(\sqrt{x} + \sqrt{y \cdot z}\right) \cdot \left(\sqrt{x} - \sqrt{y \cdot z}\right)} \]

3. Applied egg-rr 0.01

   \[\leadsto \color{blue}{x + \left(-y\right) \cdot z} \]

4. Simplified 0

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

   [Start] 0.01	\[ x + \left(-y\right) \cdot z \]
   +-commutative [=>] 0.01	\[ \color{blue}{\left(-y\right) \cdot z + x} \]
   fma-def [=>] 0	\[ \color{blue}{\mathsf{fma}\left(-y, z, x\right)} \]

5. Final simplification 0

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

Alternatives

Alternative 1
Error	27.88%
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-169}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.01%
Cost	320

\[x - y \cdot z \]

Alternative 3
Error	42.26%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023101 
(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)))