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

Percentage Accurate: 95.9% → 97.6%

Time: 10.6s

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

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+214}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -4e+214) (/ (/ (- x) t) z) (/ x (fma (- z) t y))))

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) <= -4e+214) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / fma(-z, t, y);
	}
	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) <= -4e+214)
		tmp = Float64(Float64(Float64(-x) / t) / z);
	else
		tmp = Float64(x / fma(Float64(-z), t, y));
	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], -4e+214], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision]]

Target

Original	95.9%
Target	96.4%
Herbie	97.6%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -3.9999999999999998e214

   1. Initial program 69.8%

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

   2. Applied egg-rr 69.8%

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

      [Start] 69.8%	\[ \frac{x}{y - z \cdot t} \]
      clear-num [=>] 69.8%	\[ \color{blue}{\frac{1}{\frac{y - z \cdot t}{x}}} \]
      associate-/r/ [=>] 69.8%	\[ \color{blue}{\frac{1}{y - z \cdot t} \cdot x} \]

   3. Taylor expanded in y around 0 69.8%

      \[\leadsto \color{blue}{\frac{-1}{t \cdot z}} \cdot x \]

   4. Applied egg-rr 99.6%

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

      [Start] 69.8%	\[ \frac{-1}{t \cdot z} \cdot x \]
      associate-*l/ [=>] 69.8%	\[ \color{blue}{\frac{-1 \cdot x}{t \cdot z}} \]
      neg-mul-1 [<=] 69.8%	\[ \frac{\color{blue}{-x}}{t \cdot z} \]
      associate-/r* [=>] 99.6%	\[ \color{blue}{\frac{\frac{-x}{t}}{z}} \]

   if -3.9999999999999998e214 < (*.f64 z t)

   1. Initial program 98.4%

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

   2. Applied egg-rr 98.4%

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

      [Start] 98.4%	\[ \frac{x}{y - z \cdot t} \]
      sub-neg [=>] 98.4%	\[ \frac{x}{\color{blue}{y + \left(-z \cdot t\right)}} \]
      +-commutative [=>] 98.4%	\[ \frac{x}{\color{blue}{\left(-z \cdot t\right) + y}} \]
      distribute-lft-neg-in [=>] 98.4%	\[ \frac{x}{\color{blue}{\left(-z\right) \cdot t} + y} \]
      fma-def [=>] 98.4%	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(-z, t, y\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+214}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.6%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+214}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \end{array} \]

Alternative 2
Accuracy	77.4%
Cost	905

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-8} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Accuracy	78.3%
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 4
Accuracy	79.4%
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 5
Accuracy	63.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+152} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6
Accuracy	63.1%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \end{array} \]

Alternative 7
Accuracy	97.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+214}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 8
Accuracy	55.1%
Cost	192

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

Reproduce

herbie shell --seed 2023272 
(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))))