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

Average Accuracy: 95.5% → 99.8%

Time: 11.3s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \mathbf{elif}\;z \cdot t \leq 10^{+275}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{t}{x}}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ (/ -1.0 t) (/ z x))
   (if (<= (* z t) 1e+275) (/ x (- y (* z t))) (/ (/ -1.0 (/ t x)) 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 = (-1.0 / t) / (z / x);
	} else if ((z * t) <= 1e+275) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / (t / x)) / 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 = (-1.0 / t) / (z / x);
	} else if ((z * t) <= 1e+275) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / (t / x)) / 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 = (-1.0 / t) / (z / x)
	elif (z * t) <= 1e+275:
		tmp = x / (y - (z * t))
	else:
		tmp = (-1.0 / (t / x)) / 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(-1.0 / t) / Float64(z / x));
	elseif (Float64(z * t) <= 1e+275)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(-1.0 / Float64(t / x)) / 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 = (-1.0 / t) / (z / x);
	elseif ((z * t) <= 1e+275)
		tmp = x / (y - (z * t));
	else
		tmp = (-1.0 / (t / x)) / 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[(-1.0 / t), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+275], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(t / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	95.5%
Target	97.3%
Herbie	99.8%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.4%

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

   2. Applied egg-rr 66.4%

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

      [Start] 66.4	\[ \frac{x}{y - z \cdot t} \]
      div-inv [=>] 66.4	\[ \color{blue}{x \cdot \frac{1}{y - z \cdot t}} \]
      *-commutative [=>] 66.4	\[ \color{blue}{\frac{1}{y - z \cdot t} \cdot x} \]

   3. Taylor expanded in y around 0 66.4%

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

   4. Simplified 69.6%

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

      [Start] 66.4	\[ \frac{-1}{t \cdot z} \cdot x \]
      associate-/r* [=>] 69.6	\[ \color{blue}{\frac{\frac{-1}{t}}{z}} \cdot x \]

   5. Applied egg-rr 99.8%

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

      [Start] 69.6	\[ \frac{\frac{-1}{t}}{z} \cdot x \]
      associate-/r/ [<=] 99.8	\[ \color{blue}{\frac{\frac{-1}{t}}{\frac{z}{x}}} \]

   if -inf.0 < (*.f64 z t) < 9.9999999999999996e274

   1. Initial program 99.8%

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

   if 9.9999999999999996e274 < (*.f64 z t)

   1. Initial program 72.7%

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

   2. Applied egg-rr 72.7%

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

      [Start] 72.7	\[ \frac{x}{y - z \cdot t} \]
      div-inv [=>] 72.7	\[ \color{blue}{x \cdot \frac{1}{y - z \cdot t}} \]
      *-commutative [=>] 72.7	\[ \color{blue}{\frac{1}{y - z \cdot t} \cdot x} \]

   3. Taylor expanded in y around 0 72.0%

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

   4. Applied egg-rr 99.0%

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

      [Start] 72.0	\[ \frac{-1}{t \cdot z} \cdot x \]
      associate-/r* [=>] 74.9	\[ \color{blue}{\frac{\frac{-1}{t}}{z}} \cdot x \]
      associate-*l/ [=>] 99.0	\[ \color{blue}{\frac{\frac{-1}{t} \cdot x}{z}} \]

   5. Applied egg-rr 99.0%

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

      [Start] 99.0	\[ \frac{\frac{-1}{t} \cdot x}{z} \]
      associate-/r/ [<=] 99.0	\[ \frac{\color{blue}{\frac{-1}{\frac{t}{x}}}}{z} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \mathbf{elif}\;z \cdot t \leq 10^{+275}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{t}{x}}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+223}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+275}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \mathbf{elif}\;z \cdot t \leq 10^{+275}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 3
Accuracy	71.4%
Cost	913

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-110} \lor \neg \left(y \leq 230000000\right) \land y \leq 3.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Accuracy	72.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \]

Alternative 5
Accuracy	72.9%
Cost	649

\[\begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{-62} \lor \neg \left(t \leq 6.2 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6
Accuracy	57.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -50000000 \lor \neg \left(t \leq 1.1 \cdot 10^{+193}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7
Accuracy	57.7%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -50000000:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 8
Accuracy	52.6%
Cost	192

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

Reproduce

herbie shell --seed 2023135 
(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))))