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

Average Accuracy: 95.5% → 97.8%

Time: 7.8s

Precision: binary64

Cost: 708

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY)) (/ (- (/ x z)) t) (/ x (- y (* 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) <= -((double) INFINITY)) {
		tmp = -(x / z) / t;
	} else {
		tmp = x / (y - (z * t));
	}
	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 {
		tmp = x / (y - (z * t));
	}
	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
	else:
		tmp = x / (y - (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) <= Float64(-Inf))
		tmp = Float64(Float64(-Float64(x / z)) / t);
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	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;
	else
		tmp = x / (y - (z * t));
	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], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	95.5%
Target	97.4%
Herbie	97.8%

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.2%

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

   2. Taylor expanded in y around 0 64.2%

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

   3. Simplified 99.8%

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

      [Start] 64.2	\[ -1 \cdot \frac{x}{t \cdot z} \]
      mul-1-neg [=>] 64.2	\[ \color{blue}{-\frac{x}{t \cdot z}} \]
      associate-/r* [=>] 99.8	\[ -\color{blue}{\frac{\frac{x}{t}}{z}} \]
      distribute-neg-frac [=>] 99.8	\[ \color{blue}{\frac{-\frac{x}{t}}{z}} \]

   4. Taylor expanded in x around 0 64.2%

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

   5. Simplified 99.9%

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

      [Start] 64.2	\[ -1 \cdot \frac{x}{t \cdot z} \]
      mul-1-neg [=>] 64.2	\[ \color{blue}{-\frac{x}{t \cdot z}} \]
      associate-/l/ [<=] 99.9	\[ -\color{blue}{\frac{\frac{x}{z}}{t}} \]

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

   1. Initial program 97.6%

      \[\frac{x}{y - z \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

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

Alternatives

Alternative 1
Accuracy	69.5%
Cost	914

\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-68} \lor \neg \left(t \leq 4.5 \cdot 10^{-16}\right) \land \left(t \leq 80 \lor \neg \left(t \leq 1.75 \cdot 10^{+68}\right)\right):\\ \;\;\;\;\frac{-\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Accuracy	70.0%
Cost	912

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{-\frac{x}{z}}{t}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 23:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \]

Alternative 3
Accuracy	72.3%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1250:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]

Alternative 4
Accuracy	54.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{+195}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 5
Accuracy	53.1%
Cost	192

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

Reproduce

herbie shell --seed 2023133 
(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))))