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

Average Error: 3.0 → 0.2

Time: 7.6s

Precision: binary64

Cost: 969

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+226}\right):\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 1e+226)))
   (/ (/ x (- t)) z)
   (/ 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)) || !((z * t) <= 1e+226)) {
		tmp = (x / -t) / z;
	} 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) || !((z * t) <= 1e+226)) {
		tmp = (x / -t) / z;
	} 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) or not ((z * t) <= 1e+226):
		tmp = (x / -t) / z
	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)) || !(Float64(z * t) <= 1e+226))
		tmp = Float64(Float64(x / Float64(-t)) / z);
	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) || ~(((z * t) <= 1e+226)))
		tmp = (x / -t) / z;
	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[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+226]], $MachinePrecision]], N[(N[(x / (-t)), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	3.0
Target	1.8
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 9.99999999999999961e225 < (*.f64 z t)

   1. Initial program 16.4

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

   2. Applied egg-rr 49.8

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

   3. Simplified 49.8

      \[\leadsto \frac{x}{\color{blue}{\left(y - t \cdot z\right) + 2 \cdot \mathsf{fma}\left(-t, z, t \cdot z\right)}} \]
      Proof

      [Start] 49.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right)} \]
      associate-+r+ [=>] 49.8	\[ \frac{x}{\color{blue}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)}} \]
      fma-udef [=>] 49.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)}} \]
      neg-mul-1 [=>] 49.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-1 \cdot t\right)} \cdot z + z \cdot t\right)} \]
      associate-*r* [<=] 49.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{-1 \cdot \left(t \cdot z\right)} + z \cdot t\right)} \]
      *-commutative [<=] 49.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(-1 \cdot \color{blue}{\left(z \cdot t\right)} + z \cdot t\right)} \]
      mul-1-neg [=>] 49.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right)} + z \cdot t\right)} \]
      *-rgt-identity [<=] 49.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + z \cdot t\right)} \]
      fma-udef [<=] 49.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)}} \]
      associate-+r+ [<=] 49.8	\[ \frac{x}{\color{blue}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]
      fma-udef [=>] 49.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)} + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      distribute-lft-neg-in [<=] 49.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\color{blue}{\left(-t \cdot z\right)} + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      *-commutative [<=] 49.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\left(-\color{blue}{z \cdot t}\right) + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      associate-+l+ [=>] 49.8	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(-z \cdot t\right) + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)}} \]
      *-rgt-identity [<=] 49.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)} \]
      associate-+l+ [<=] 49.8	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(\left(-z \cdot t\right) \cdot 1 + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]

   4. Taylor expanded in y around 0 50.4

      \[\leadsto \color{blue}{\frac{x}{2 \cdot \left(-1 \cdot \left(t \cdot z\right) + t \cdot z\right) - t \cdot z}} \]

   5. Simplified 1.0

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

      [Start] 50.4	\[ \frac{x}{2 \cdot \left(-1 \cdot \left(t \cdot z\right) + t \cdot z\right) - t \cdot z} \]
      +-commutative [=>] 50.4	\[ \frac{x}{2 \cdot \color{blue}{\left(t \cdot z + -1 \cdot \left(t \cdot z\right)\right)} - t \cdot z} \]
      mul-1-neg [=>] 50.4	\[ \frac{x}{2 \cdot \left(t \cdot z + \color{blue}{\left(-t \cdot z\right)}\right) - t \cdot z} \]
      sub-neg [<=] 50.4	\[ \frac{x}{2 \cdot \color{blue}{\left(t \cdot z - t \cdot z\right)} - t \cdot z} \]
      +-inverses [=>] 17.1	\[ \frac{x}{2 \cdot \color{blue}{0} - t \cdot z} \]
      metadata-eval [=>] 17.1	\[ \frac{x}{\color{blue}{0} - t \cdot z} \]
      sub0-neg [=>] 17.1	\[ \frac{x}{\color{blue}{-t \cdot z}} \]
      *-commutative [=>] 17.1	\[ \frac{x}{-\color{blue}{z \cdot t}} \]
      distribute-rgt-neg-in [=>] 17.1	\[ \frac{x}{\color{blue}{z \cdot \left(-t\right)}} \]
      neg-sub0 [=>] 17.1	\[ \frac{x}{z \cdot \color{blue}{\left(0 - t\right)}} \]
      metadata-eval [<=] 17.1	\[ \frac{x}{z \cdot \left(\color{blue}{2 \cdot 0} - t\right)} \]
      mul0-lft [<=] 17.1	\[ \frac{x}{z \cdot \left(2 \cdot \color{blue}{\left(0 \cdot t\right)} - t\right)} \]
      metadata-eval [<=] 17.1	\[ \frac{x}{z \cdot \left(2 \cdot \left(\color{blue}{\left(-1 + 1\right)} \cdot t\right) - t\right)} \]
      distribute-lft1-in [<=] 17.1	\[ \frac{x}{z \cdot \left(2 \cdot \color{blue}{\left(-1 \cdot t + t\right)} - t\right)} \]
      associate-/l/ [<=] 1.0	\[ \color{blue}{\frac{\frac{x}{2 \cdot \left(-1 \cdot t + t\right) - t}}{z}} \]

   if -inf.0 < (*.f64 z t) < 9.99999999999999961e225

   1. Initial program 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+226}\right):\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.8
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-89}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	28.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	30.3
Cost	192

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

Reproduce

herbie shell --seed 2023067 
(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))))