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

Average Error: 2.6 → 0.3

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;{\left(z \cdot \frac{-t}{x}\right)}^{-1}\\ \mathbf{elif}\;z \cdot t \leq 1.2348795495565224 \cdot 10^{+217}:\\ \;\;\;\;\frac{-x}{z \cdot t - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (pow (* z (/ (- t) x)) -1.0)
   (if (<= (* z t) 1.2348795495565224e+217)
     (/ (- x) (- (* z t) y))
     (/ (/ (- x) 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 = pow((z * (-t / x)), -1.0);
	} else if ((z * t) <= 1.2348795495565224e+217) {
		tmp = -x / ((z * t) - y);
	} else {
		tmp = (-x / 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 = Math.pow((z * (-t / x)), -1.0);
	} else if ((z * t) <= 1.2348795495565224e+217) {
		tmp = -x / ((z * t) - y);
	} else {
		tmp = (-x / 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 = math.pow((z * (-t / x)), -1.0)
	elif (z * t) <= 1.2348795495565224e+217:
		tmp = -x / ((z * t) - y)
	else:
		tmp = (-x / 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(z * Float64(Float64(-t) / x)) ^ -1.0;
	elseif (Float64(z * t) <= 1.2348795495565224e+217)
		tmp = Float64(Float64(-x) / Float64(Float64(z * t) - y));
	else
		tmp = Float64(Float64(Float64(-x) / 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 = (z * (-t / x)) ^ -1.0;
	elseif ((z * t) <= 1.2348795495565224e+217)
		tmp = -x / ((z * t) - y);
	else
		tmp = (-x / 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[Power[N[(z * N[((-t) / x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1.2348795495565224e+217], N[((-x) / N[(N[(z * t), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.6
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 23.2

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

   2. Applied egg-rr 23.2

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

   3. Taylor expanded in y around 0 23.2

      \[\leadsto {\color{blue}{\left(-1 \cdot \frac{t \cdot z}{x}\right)}}^{-1} \]

   4. Simplified 0.7

      \[\leadsto {\color{blue}{\left(\left(-\frac{t}{x}\right) \cdot z\right)}}^{-1} \]

   if -inf.0 < (*.f64 z t) < 1.2348795495565224e217

   1. Initial program 0.1

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

   2. Applied egg-rr 0.7

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

   3. Taylor expanded in x around -inf 0.1

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

   4. Simplified 0.1

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

   if 1.2348795495565224e217 < (*.f64 z t)

   1. Initial program 10.9

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

   2. Taylor expanded in y around 0 12.3

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

   3. Simplified 1.8

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;{\left(z \cdot \frac{-t}{x}\right)}^{-1}\\ \mathbf{elif}\;z \cdot t \leq 1.2348795495565224 \cdot 10^{+217}:\\ \;\;\;\;\frac{-x}{z \cdot t - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))))