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

Percentage Accurate: 95.8% → 98.0%

Time: 7.8s

Precision: binary64

Cost: 7044

• Unsound rule application detected in e-graph. Results may not be sound.

\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+266}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 1e+266) (/ x (fma (- z) t y)) (/ (/ (- x) z) t)))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 1e+266) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = (-x / z) / t;
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 1e+266)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(Float64(-x) / z) / t);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], 1e+266], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]]

Target

Original	95.8%
Target	96.6%
Herbie	98.0%

\[\begin{array}{l} \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 1e266

   1. Initial program 98.2%

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

   2. Applied egg-rr 98.2%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(-z, t, y\right)}} \]
      Step-by-step derivation

      [Start] 98.2%	\[ \frac{x}{y - z \cdot t} \]
      sub-neg [=>] 98.2%	\[ \frac{x}{\color{blue}{y + \left(-z \cdot t\right)}} \]
      +-commutative [=>] 98.2%	\[ \frac{x}{\color{blue}{\left(-z \cdot t\right) + y}} \]
      distribute-lft-neg-in [=>] 98.2%	\[ \frac{x}{\color{blue}{\left(-z\right) \cdot t} + y} \]
      fma-def [=>] 98.2%	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(-z, t, y\right)}} \]

   if 1e266 < (*.f64 z t)

   1. Initial program 69.8%

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

   2. Applied egg-rr 69.7%

      \[\leadsto \color{blue}{{\left(\frac{y - z \cdot t}{x}\right)}^{-1}} \]
      Step-by-step derivation

      [Start] 69.8%	\[ \frac{x}{y - z \cdot t} \]
      clear-num [=>] 69.7%	\[ \color{blue}{\frac{1}{\frac{y - z \cdot t}{x}}} \]
      inv-pow [=>] 69.7%	\[ \color{blue}{{\left(\frac{y - z \cdot t}{x}\right)}^{-1}} \]

   3. Taylor expanded in y around 0 69.7%

      \[\leadsto {\color{blue}{\left(-1 \cdot \frac{t \cdot z}{x}\right)}}^{-1} \]

   4. Simplified 96.8%

      \[\leadsto {\color{blue}{\left(\frac{-t}{\frac{x}{z}}\right)}}^{-1} \]
      Step-by-step derivation

      [Start] 69.7%	\[ {\left(-1 \cdot \frac{t \cdot z}{x}\right)}^{-1} \]
      mul-1-neg [=>] 69.7%	\[ {\color{blue}{\left(-\frac{t \cdot z}{x}\right)}}^{-1} \]
      associate-/l* [=>] 96.8%	\[ {\left(-\color{blue}{\frac{t}{\frac{x}{z}}}\right)}^{-1} \]
      distribute-neg-frac [=>] 96.8%	\[ {\color{blue}{\left(\frac{-t}{\frac{x}{z}}\right)}}^{-1} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{-x}{z}}{t}} \]
      Step-by-step derivation

      [Start] 96.8%	\[ {\left(\frac{-t}{\frac{x}{z}}\right)}^{-1} \]
      unpow-1 [=>] 96.8%	\[ \color{blue}{\frac{1}{\frac{-t}{\frac{x}{z}}}} \]
      clear-num [<=] 99.8%	\[ \color{blue}{\frac{\frac{x}{z}}{-t}} \]
      frac-2neg [=>] 99.8%	\[ \color{blue}{\frac{-\frac{x}{z}}{-\left(-t\right)}} \]
      distribute-neg-frac [=>] 99.8%	\[ \frac{\color{blue}{\frac{-x}{z}}}{-\left(-t\right)} \]
      remove-double-neg [=>] 99.8%	\[ \frac{\frac{-x}{z}}{\color{blue}{t}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+266}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.0%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+266}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 2
Accuracy	77.4%
Cost	1425

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq -20000000000000:\\ \;\;\;\;x \cdot \frac{1}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-9} \lor \neg \left(z \cdot t \leq 1000000\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Accuracy	80.6%
Cost	1228

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 1000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	80.6%
Cost	1164

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 1000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+186}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	79.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 1000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \end{array} \]

Alternative 6
Accuracy	63.2%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -3 \cdot 10^{+152} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7
Accuracy	98.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+266}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 8
Accuracy	54.3%
Cost	192

\[\frac{x}{y} \]

Reproduce

herbie shell --seed 2023178 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

  (/ x (- y (* z t))))