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

Average Error: 16.3 → 7.7

Time: 21.1s

Precision: binary64

Cost: 3400

Math

\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \left(a + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+284}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0)))))
   (if (<= t_1 (- INFINITY))
     (* (/ y t) (/ z (+ 1.0 (+ a (* y (/ b t))))))
     (if (<= t_1 1e+284) t_1 (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / t) * (z / (1.0 + (a + (y * (b / t)))));
	} else if (t_1 <= 1e+284) {
		tmp = t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (y / t) * (z / (1.0 + (a + (y * (b / t)))));
	} else if (t_1 <= 1e+284) {
		tmp = t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

↓

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (y / t) * (z / (1.0 + (a + (y * (b / t)))))
	elif t_1 <= 1e+284:
		tmp = t_1
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / t) * Float64(z / Float64(1.0 + Float64(a + Float64(y * Float64(b / t))))));
	elseif (t_1 <= 1e+284)
		tmp = t_1;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

↓

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (y / t) * (z / (1.0 + (a + (y * (b / t)))));
	elseif (t_1 <= 1e+284)
		tmp = t_1;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / t), $MachinePrecision] * N[(z / N[(1.0 + N[(a + N[(y * N[(b / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+284], t$95$1, N[(z / b), $MachinePrecision]]]]

Target

Original	16.3
Target	13.3
Herbie	7.7

\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 64.0

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. Taylor expanded in x around 0 40.9

      \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(\frac{y \cdot b}{t} + a\right)\right)}} \]

   3. Simplified 20.7

      \[\leadsto \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(y \cdot \frac{b}{t} + a\right)}} \]
      Proof

      [Start] 40.9	\[ \frac{y \cdot z}{t \cdot \left(1 + \left(\frac{y \cdot b}{t} + a\right)\right)} \]
      times-frac [=>] 17.9	\[ \color{blue}{\frac{y}{t} \cdot \frac{z}{1 + \left(\frac{y \cdot b}{t} + a\right)}} \]
      associate-*r/ [<=] 20.7	\[ \frac{y}{t} \cdot \frac{z}{1 + \left(\color{blue}{y \cdot \frac{b}{t}} + a\right)} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.00000000000000008e284

   1. Initial program 5.9

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   if 1.00000000000000008e284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 61.4

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

   2. Simplified 49.9

      \[\leadsto \color{blue}{\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}} \]
      Proof

      [Start] 61.4	\[ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-/l* [=>] 55.1	\[ \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
      associate-+l+ [=>] 55.1	\[ \frac{x + \frac{y}{\frac{t}{z}}}{\color{blue}{a + \left(1 + \frac{y \cdot b}{t}\right)}} \]
      *-commutative [=>] 55.1	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{\color{blue}{b \cdot y}}{t}\right)} \]
      associate-/l* [=>] 49.9	\[ \frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \color{blue}{\frac{b}{\frac{t}{y}}}\right)} \]

   3. Taylor expanded in y around inf 14.1

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

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \left(a + y \cdot \frac{b}{t}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+284}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.4
Cost	1632

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{x}{1 + b \cdot \frac{y}{t}}\\ t_3 := \frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{if}\;a \leq -265000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -4.9 \cdot 10^{-33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -3.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-244}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.95 \cdot 10^{-30}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 0.5:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	26.7
Cost	1496

\[\begin{array}{l} t_1 := \frac{x}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{if}\;t \leq -2.1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-71}:\\ \;\;\;\;t \cdot \frac{\frac{z}{t} + \frac{x}{y}}{b}\\ \mathbf{elif}\;t \leq -5.9 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-114}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 1.24 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{y}{t}}{\frac{a + 1}{z}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	26.6
Cost	1496

\[\begin{array}{l} t_1 := \frac{x}{y \cdot \frac{b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t \leq -1.55:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-69}:\\ \;\;\;\;t \cdot \frac{\frac{z}{t} + \frac{x}{y}}{b}\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-125}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-114}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 1.52 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{y}{t}}{\frac{a + 1}{z}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	22.8
Cost	1492

\[\begin{array}{l} t_1 := y \cdot \frac{b}{t}\\ \mathbf{if}\;y \leq -8.4 \cdot 10^{+166}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -3.15 \cdot 10^{+145}:\\ \;\;\;\;\frac{x}{t_1 + \left(a + 1\right)}\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{+54}:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \left(a + t_1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x + \frac{\frac{z}{\frac{1}{y}}}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 5
Error	22.9
Cost	1364

\[\begin{array}{l} t_1 := \frac{x + y \cdot \frac{z}{t}}{a + 1}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-114}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+25}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{y \cdot \frac{b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	13.6
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-136} \lor \neg \left(t \leq 1.25 \cdot 10^{-114}\right):\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 7
Error	13.7
Cost	1352

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{t_1}{a + \left(1 + \frac{b}{\frac{t}{y}}\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-114}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a + \left(1 + \frac{y}{\frac{t}{b}}\right)}\\ \end{array} \]

Alternative 8
Error	24.4
Cost	1234

\[\begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+183} \lor \neg \left(b \leq 3500\right) \land \left(b \leq 7.1 \cdot 10^{+129} \lor \neg \left(b \leq 4.2 \cdot 10^{+191}\right)\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \end{array} \]

Alternative 9
Error	25.9
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+169}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{y \cdot \frac{b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-115}:\\ \;\;\;\;\frac{y}{t \cdot \frac{a + 1}{z}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 10
Error	24.3
Cost	1232

\[\begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+183}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq 31000:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \mathbf{elif}\;b \leq 7.1 \cdot 10^{+129}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+193}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 11
Error	29.8
Cost	972

\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-114}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 3.65 \cdot 10^{-9}:\\ \;\;\;\;z \cdot \frac{\frac{y}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+25}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	40.3
Cost	853

\[\begin{array}{l} \mathbf{if}\;a \leq -2.75 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-288}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-306}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;a \leq 0.002 \lor \neg \left(a \leq 8.3 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 13
Error	29.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-125} \lor \neg \left(t \leq 2.5 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 14
Error	37.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+63}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 15
Error	47.6
Cost	192

\[\frac{x}{a} \]

Reproduce

herbie shell --seed 2023011 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))