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

Average Accuracy: 95.7% → 99.7%

Time: 10.1s

Precision: binary64

Cost: 968

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ (/ x (- z)) t)
   (if (<= (* z t) 2e+230) (/ x (- y (* z t))) (/ (/ (- x) t) z))))

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) <= -((double) INFINITY)) {
		tmp = (x / -z) / t;
	} else if ((z * t) <= 2e+230) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-x / t) / z;
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (x / -z) / t;
	} else if ((z * t) <= 2e+230) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-x / t) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x / (y - (z * t))

↓

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (x / -z) / t
	elif (z * t) <= 2e+230:
		tmp = x / (y - (z * t))
	else:
		tmp = (-x / t) / z
	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) <= Float64(-Inf))
		tmp = Float64(Float64(x / Float64(-z)) / t);
	elseif (Float64(z * t) <= 2e+230)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(Float64(-x) / t) / z);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
end

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = (x / -z) / t;
	elseif ((z * t) <= 2e+230)
		tmp = x / (y - (z * t));
	else
		tmp = (-x / t) / z;
	end
	tmp_2 = 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], (-Infinity)], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+230], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	95.7%
Target	96.7%
Herbie	99.7%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 58.2%

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

   2. Applied egg-rr 58.2%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\sqrt{y - z \cdot t}} \cdot \frac{\sqrt[3]{x}}{\sqrt{y - z \cdot t}}} \]
      Proof

      [Start] 58.2	\[ \frac{x}{y - z \cdot t} \]
      add-cube-cbrt [=>] 58.2	\[ \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{y - z \cdot t} \]
      add-sqr-sqrt [=>] 58.2	\[ \frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}{\color{blue}{\sqrt{y - z \cdot t} \cdot \sqrt{y - z \cdot t}}} \]
      times-frac [=>] 58.2	\[ \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{y - z \cdot t}} \cdot \frac{\sqrt[3]{x}}{\sqrt{y - z \cdot t}}} \]
      pow2 [=>] 58.2	\[ \frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{\sqrt{y - z \cdot t}} \cdot \frac{\sqrt[3]{x}}{\sqrt{y - z \cdot t}} \]

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{y - t \cdot z} \cdot \sqrt{y - t \cdot z}}} \]
      Proof

      [Start] 58.2	\[ \frac{{\left(\sqrt[3]{x}\right)}^{2}}{\sqrt{y - z \cdot t}} \cdot \frac{\sqrt[3]{x}}{\sqrt{y - z \cdot t}} \]
      associate-*l/ [=>] 58.2	\[ \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{\sqrt{y - z \cdot t}}}{\sqrt{y - z \cdot t}}} \]
      associate-*r/ [=>] 58.2	\[ \frac{\color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\sqrt{y - z \cdot t}}}}{\sqrt{y - z \cdot t}} \]
      unpow2 [=>] 58.2	\[ \frac{\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\sqrt{y - z \cdot t}}}{\sqrt{y - z \cdot t}} \]
      rem-3cbrt-lft [=>] 58.2	\[ \frac{\frac{\color{blue}{x}}{\sqrt{y - z \cdot t}}}{\sqrt{y - z \cdot t}} \]
      associate-/l/ [=>] 58.2	\[ \color{blue}{\frac{x}{\sqrt{y - z \cdot t} \cdot \sqrt{y - z \cdot t}}} \]
      *-commutative [=>] 58.2	\[ \frac{x}{\sqrt{y - \color{blue}{t \cdot z}} \cdot \sqrt{y - z \cdot t}} \]
      *-commutative [=>] 58.2	\[ \frac{x}{\sqrt{y - t \cdot z} \cdot \sqrt{y - \color{blue}{t \cdot z}}} \]

   4. Taylor expanded in y around 0 58.2%

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

   5. Simplified 99.8%

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

      [Start] 58.2	\[ -1 \cdot \frac{x}{t \cdot z} \]
      associate-*r/ [=>] 58.2	\[ \color{blue}{\frac{-1 \cdot x}{t \cdot z}} \]
      neg-mul-1 [<=] 58.2	\[ \frac{\color{blue}{-x}}{t \cdot z} \]
      associate-/r* [=>] 99.8	\[ \color{blue}{\frac{\frac{-x}{t}}{z}} \]

   6. Taylor expanded in x around 0 58.2%

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

   7. Simplified 99.8%

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

      [Start] 58.2	\[ -1 \cdot \frac{x}{t \cdot z} \]
      associate-/r* [=>] 99.8	\[ -1 \cdot \color{blue}{\frac{\frac{x}{t}}{z}} \]
      metadata-eval [<=] 99.8	\[ \color{blue}{\frac{1}{-1}} \cdot \frac{\frac{x}{t}}{z} \]
      times-frac [<=] 99.8	\[ \color{blue}{\frac{1 \cdot \frac{x}{t}}{-1 \cdot z}} \]
      neg-mul-1 [<=] 99.8	\[ \frac{1 \cdot \frac{x}{t}}{\color{blue}{-z}} \]
      associate-*l/ [<=] 99.7	\[ \color{blue}{\frac{1}{-z} \cdot \frac{x}{t}} \]
      associate-*r/ [=>] 99.7	\[ \color{blue}{\frac{\frac{1}{-z} \cdot x}{t}} \]
      associate-*l/ [=>] 99.8	\[ \frac{\color{blue}{\frac{1 \cdot x}{-z}}}{t} \]
      *-lft-identity [=>] 99.8	\[ \frac{\frac{\color{blue}{x}}{-z}}{t} \]

   if -inf.0 < (*.f64 z t) < 2.0000000000000002e230

   1. Initial program 99.9%

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

   if 2.0000000000000002e230 < (*.f64 z t)

   1. Initial program 72.0%

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

   2. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\sqrt{y - z \cdot t}} \cdot \frac{\sqrt[3]{x}}{\sqrt{y - z \cdot t}}} \]
      Proof

      [Start] 72.0	\[ \frac{x}{y - z \cdot t} \]
      add-cube-cbrt [=>] 71.5	\[ \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{y - z \cdot t} \]
      add-sqr-sqrt [=>] 0.0	\[ \frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}{\color{blue}{\sqrt{y - z \cdot t} \cdot \sqrt{y - z \cdot t}}} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{y - z \cdot t}} \cdot \frac{\sqrt[3]{x}}{\sqrt{y - z \cdot t}}} \]
      pow2 [=>] 0.0	\[ \frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{\sqrt{y - z \cdot t}} \cdot \frac{\sqrt[3]{x}}{\sqrt{y - z \cdot t}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{y - t \cdot z} \cdot \sqrt{y - t \cdot z}}} \]
      Proof

      [Start] 0.0	\[ \frac{{\left(\sqrt[3]{x}\right)}^{2}}{\sqrt{y - z \cdot t}} \cdot \frac{\sqrt[3]{x}}{\sqrt{y - z \cdot t}} \]
      associate-*l/ [=>] 0.0	\[ \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{\sqrt[3]{x}}{\sqrt{y - z \cdot t}}}{\sqrt{y - z \cdot t}}} \]
      associate-*r/ [=>] 0.0	\[ \frac{\color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}}{\sqrt{y - z \cdot t}}}}{\sqrt{y - z \cdot t}} \]
      unpow2 [=>] 0.0	\[ \frac{\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{x}}{\sqrt{y - z \cdot t}}}{\sqrt{y - z \cdot t}} \]
      rem-3cbrt-lft [=>] 0.0	\[ \frac{\frac{\color{blue}{x}}{\sqrt{y - z \cdot t}}}{\sqrt{y - z \cdot t}} \]
      associate-/l/ [=>] 0.0	\[ \color{blue}{\frac{x}{\sqrt{y - z \cdot t} \cdot \sqrt{y - z \cdot t}}} \]
      *-commutative [=>] 0.0	\[ \frac{x}{\sqrt{y - \color{blue}{t \cdot z}} \cdot \sqrt{y - z \cdot t}} \]
      *-commutative [=>] 0.0	\[ \frac{x}{\sqrt{y - t \cdot z} \cdot \sqrt{y - \color{blue}{t \cdot z}}} \]

   4. Taylor expanded in y around 0 72.0%

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

   5. Simplified 99.7%

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

      [Start] 72.0	\[ -1 \cdot \frac{x}{t \cdot z} \]
      associate-*r/ [=>] 72.0	\[ \color{blue}{\frac{-1 \cdot x}{t \cdot z}} \]
      neg-mul-1 [<=] 72.0	\[ \frac{\color{blue}{-x}}{t \cdot z} \]
      associate-/r* [=>] 99.7	\[ \color{blue}{\frac{\frac{-x}{t}}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	73.1%
Cost	1176

\[\begin{array}{l} t_1 := \frac{\frac{x}{-z}}{t}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	73.4%
Cost	1176

\[\begin{array}{l} t_1 := \frac{\frac{x}{-z}}{t}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 3
Accuracy	70.8%
Cost	913

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -9.6 \cdot 10^{-43} \lor \neg \left(y \leq -1.85 \cdot 10^{-142}\right) \land y \leq 5 \cdot 10^{-67}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Accuracy	58.1%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+37} \lor \neg \left(t \leq 8.5 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Accuracy	58.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+171}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 6
Accuracy	54.0%
Cost	192

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

Reproduce

herbie shell --seed 2023160 
(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))))