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

Average Error: 2.9 → 0.1

Time: 29.2s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \end{array}\\ \mathbf{elif}\;z \cdot t \leq 10^{+282}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (if (!= x 0.0) (/ (/ -1.0 t) (/ z x)) (- (/ (/ x z) t)))
   (if (<= (* z t) 1e+282) (/ x (- y (* z t))) (* (/ x t) (/ -1.0 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_1;
	if ((z * t) <= -((double) INFINITY)) {
		double tmp_2;
		if (x != 0.0) {
			tmp_2 = (-1.0 / t) / (z / x);
		} else {
			tmp_2 = -((x / z) / t);
		}
		tmp_1 = tmp_2;
	} else if ((z * t) <= 1e+282) {
		tmp_1 = x / (y - (z * t));
	} else {
		tmp_1 = (x / t) * (-1.0 / z);
	}
	return tmp_1;
}

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_1;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		double tmp_2;
		if (x != 0.0) {
			tmp_2 = (-1.0 / t) / (z / x);
		} else {
			tmp_2 = -((x / z) / t);
		}
		tmp_1 = tmp_2;
	} else if ((z * t) <= 1e+282) {
		tmp_1 = x / (y - (z * t));
	} else {
		tmp_1 = (x / t) * (-1.0 / z);
	}
	return tmp_1;
}

Python

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

↓

def code(x, y, z, t):
	tmp_1 = 0
	if (z * t) <= -math.inf:
		tmp_2 = 0
		if x != 0.0:
			tmp_2 = (-1.0 / t) / (z / x)
		else:
			tmp_2 = -((x / z) / t)
		tmp_1 = tmp_2
	elif (z * t) <= 1e+282:
		tmp_1 = x / (y - (z * t))
	else:
		tmp_1 = (x / t) * (-1.0 / z)
	return tmp_1

Julia

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end

↓

function code(x, y, z, t)
	tmp_1 = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp_2 = 0.0
		if (x != 0.0)
			tmp_2 = Float64(Float64(-1.0 / t) / Float64(z / x));
		else
			tmp_2 = Float64(-Float64(Float64(x / z) / t));
		end
		tmp_1 = tmp_2;
	elseif (Float64(z * t) <= 1e+282)
		tmp_1 = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp_1 = Float64(Float64(x / t) * Float64(-1.0 / z));
	end
	return tmp_1
end

MATLAB

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

↓

function tmp_4 = code(x, y, z, t)
	tmp_2 = 0.0;
	if ((z * t) <= -Inf)
		tmp_3 = 0.0;
		if (x ~= 0.0)
			tmp_3 = (-1.0 / t) / (z / x);
		else
			tmp_3 = -((x / z) / t);
		end
		tmp_2 = tmp_3;
	elseif ((z * t) <= 1e+282)
		tmp_2 = x / (y - (z * t));
	else
		tmp_2 = (x / t) * (-1.0 / z);
	end
	tmp_4 = tmp_2;
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)], If[Unequal[x, 0.0], N[(N[(-1.0 / t), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], (-N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision])], If[LessEqual[N[(z * t), $MachinePrecision], 1e+282], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	2.9
Target	1.6
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 22.5

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

   2. Applied egg-rr 22.5

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(t, -z, y\right)} \cdot x} \]

   3. Taylor expanded in t around inf 22.5

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

   4. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ } \end{array}} \]

   if -inf.0 < (*.f64 z t) < 1.00000000000000003e282

   1. Initial program 0.1

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

   if 1.00000000000000003e282 < (*.f64 z t)

   1. Initial program 16.9

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

   2. Applied egg-rr 16.9

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(t, -z, y\right)} \cdot x} \]

   3. Taylor expanded in t around inf 16.9

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

   4. Applied egg-rr 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	968

\[\begin{array}{l} t_1 := \frac{x}{t} \cdot \frac{-1}{z}\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+282}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	18.0
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{-t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	18.7
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	29.7
Cost	192

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

Reproduce

herbie shell --seed 2023010 
(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))))