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

Average Error: 2.8 → 1.6

Time: 7.4s

Precision: binary64

Cost: 708

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY)) (/ -1.0 (* t (/ z x))) (/ 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 = -1.0 / (t * (z / x));
	} 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 = -1.0 / (t * (z / x));
	} 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 = -1.0 / (t * (z / x))
	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(-1.0 / Float64(t * Float64(z / x)));
	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 = -1.0 / (t * (z / x));
	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[(-1.0 / N[(t * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	2.8
Target	1.7
Herbie	1.6

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.1

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

   2. Taylor expanded in y around 0 19.1

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

   3. Simplified 19.1

      \[\leadsto \color{blue}{\frac{-x}{t \cdot z}} \]
      Proof
      (/.f64 (neg.f64 x) (*.f64 t z)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 x)) (*.f64 t z)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 x) (Rewrite<= *-commutative_binary64 (*.f64 z t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 x (*.f64 z t)))): 5 points increase in error, 0 points decrease in error
      (*.f64 -1 (/.f64 x (Rewrite=> *-commutative_binary64 (*.f64 t z)))): 0 points increase in error, 5 points decrease in error

   4. Applied egg-rr 0.2

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

   5. Applied egg-rr 1.1

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

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

   1. Initial program 1.7

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

4. Final simplification 1.6

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

Alternatives

Alternative 1
Error	17.1
Cost	914

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-17} \lor \neg \left(y \leq 1.8 \cdot 10^{-38}\right) \land \left(y \leq 12500000000000 \lor \neg \left(y \leq 3.8 \cdot 10^{+44}\right)\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 2
Error	24.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+128} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+111}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	30.0
Cost	192

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

Reproduce

herbie shell --seed 2022349 
(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))))