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

Average Error: 3.0 → 0.3

Time: 6.7s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 10^{+195}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ (/ (- x) t) z)
   (if (<= (* z t) 1e+195) (/ x (- y (* z t))) (* (/ -1.0 z) (/ x 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 / t) / z;
	} else if ((z * t) <= 1e+195) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / z) * (x / 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 / t) / z;
	} else if ((z * t) <= 1e+195) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / z) * (x / 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 / t) / z
	elif (z * t) <= 1e+195:
		tmp = x / (y - (z * t))
	else:
		tmp = (-1.0 / z) * (x / 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) / t) / z);
	elseif (Float64(z * t) <= 1e+195)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(-1.0 / z) * Float64(x / 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 / t) / z;
	elseif ((z * t) <= 1e+195)
		tmp = x / (y - (z * t));
	else
		tmp = (-1.0 / z) * (x / 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) / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+195], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] * N[(x / t), $MachinePrecision]), $MachinePrecision]]]

Target

Original	3.0
Target	1.7
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 23.3

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

   2. Taylor expanded in y around 0 23.3

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{-x}{t}}{z}} \]
      Proof
      (/.f64 (/.f64 (neg.f64 x) t) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x)) t) z): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 -1 x) (*.f64 t z))): 56 points increase in error, 43 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 x (*.f64 t z)))): 0 points increase in error, 0 points decrease in error

   if -inf.0 < (*.f64 z t) < 9.99999999999999977e194

   1. Initial program 0.1

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

   if 9.99999999999999977e194 < (*.f64 z t)

   1. Initial program 11.7

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

   2. Taylor expanded in y around 0 12.9

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

   3. Simplified 1.4

      \[\leadsto \color{blue}{\frac{\frac{-x}{t}}{z}} \]
      Proof
      (/.f64 (/.f64 (neg.f64 x) t) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x)) t) z): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 -1 x) (*.f64 t z))): 56 points increase in error, 43 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 x (*.f64 t z)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 1.5

      \[\leadsto \color{blue}{\frac{-1}{z} \cdot \frac{x}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 10^{+195}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.8
Cost	912

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;y \leq -7.038440341044346 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-188}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.00035187784391656787:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	13.4
Cost	904

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	30.0
Cost	192

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

Reproduce

herbie shell --seed 2022298 
(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))))