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

Average Error: 16.9 → 5.4

Time: 19.3s

Precision: binary64

Cost: 6740

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := z \cdot \frac{y}{t + t \cdot \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-316}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t_1 \leq 10^{+300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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))))
        (t_2 (* z (/ y (+ t (* t (+ a (/ y (/ t b)))))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -5e-316)
       t_1
       (if (<= t_1 0.0)
         (/ (* y z) (+ (* y b) (* t (+ a 1.0))))
         (if (<= t_1 1e+300) t_1 (if (<= t_1 INFINITY) t_2 (/ 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 t_2 = z * (y / (t + (t * (a + (y / (t / b))))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -5e-316) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)));
	} else if (t_1 <= 1e+300) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} 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 t_2 = z * (y / (t + (t * (a + (y / (t / b))))));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 <= -5e-316) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)));
	} else if (t_1 <= 1e+300) {
		tmp = t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} 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))
	t_2 = z * (y / (t + (t * (a + (y / (t / b))))))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_2
	elif t_1 <= -5e-316:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)))
	elif t_1 <= 1e+300:
		tmp = t_1
	elif t_1 <= math.inf:
		tmp = t_2
	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)))
	t_2 = Float64(z * Float64(y / Float64(t + Float64(t * Float64(a + Float64(y / Float64(t / b)))))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -5e-316)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y * z) / Float64(Float64(y * b) + Float64(t * Float64(a + 1.0))));
	elseif (t_1 <= 1e+300)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	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));
	t_2 = z * (y / (t + (t * (a + (y / (t / b))))));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_2;
	elseif (t_1 <= -5e-316)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (y * z) / ((y * b) + (t * (a + 1.0)));
	elseif (t_1 <= 1e+300)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(z * N[(y / N[(t + N[(t * N[(a + N[(y / N[(t / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -5e-316], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(y * z), $MachinePrecision] / N[(N[(y * b), $MachinePrecision] + N[(t * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+300], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]

Target

Original	16.9
Target	13.4
Herbie	5.4

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -inf.0 or 1.0000000000000001e300 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 62.4

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

   2. Taylor expanded in x around 0 38.9

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

   3. Simplified 15.0

      \[\leadsto \color{blue}{\frac{y}{t + t \cdot \left(\frac{y}{\frac{t}{b}} + a\right)} \cdot z} \]
      Proof
      (*.f64 (/.f64 y (+.f64 t (*.f64 t (+.f64 (/.f64 y (/.f64 t b)) a)))) z): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y (+.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 t 1)) (*.f64 t (+.f64 (/.f64 y (/.f64 t b)) a)))) z): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y (+.f64 (*.f64 t 1) (*.f64 t (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y b) t)) a)))) z): 3 points increase in error, 8 points decrease in error
      (*.f64 (/.f64 y (Rewrite<= distribute-lft-in_binary64 (*.f64 t (+.f64 1 (+.f64 (/.f64 (*.f64 y b) t) a))))) z): 2 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 y (/.f64 (*.f64 t (+.f64 1 (+.f64 (/.f64 (*.f64 y b) t) a))) z))): 41 points increase in error, 22 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y z) (*.f64 t (+.f64 1 (+.f64 (/.f64 (*.f64 y b) t) a))))): 36 points increase in error, 33 points decrease in error

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -5.000000017e-316 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.0000000000000001e300

   1. Initial program 0.4

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

   if -5.000000017e-316 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 29.7

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

   2. Taylor expanded in x around 0 29.8

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

   3. Taylor expanded in t around 0 22.1

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

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

   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 y around inf 2.7

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t + t \cdot \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -5 \cdot 10^{-316}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+300}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{t + t \cdot \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.1
Cost	16072

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := z \cdot \frac{y}{t + t \cdot \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right) \cdot \frac{1}{a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 2
Error	22.7
Cost	1232

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{if}\;y \leq -8.288656587559581 \cdot 10^{+154}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -2.4053240005108682 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.634930832917584 \cdot 10^{+110}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 4.8151012081012854 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 3
Error	21.2
Cost	1232

\[\begin{array}{l} \mathbf{if}\;t \leq -2.006899282715869 \cdot 10^{-57}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \mathbf{elif}\;t \leq -3.327489954002317 \cdot 10^{-149}:\\ \;\;\;\;\frac{y \cdot z}{y \cdot b + t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t \leq -1.0384443237295707 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;t \leq 1.4578604210337144 \cdot 10^{-115}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \end{array} \]

Alternative 4
Error	28.9
Cost	972

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t_1}{a}\\ \mathbf{if}\;a \leq -25.281202353532876:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 5.895818998220034 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 255311114.6999046:\\ \;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	28.2
Cost	972

\[\begin{array}{l} t_1 := \frac{x + \frac{z}{\frac{t}{y}}}{a}\\ \mathbf{if}\;a \leq -25.281202353532876:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.895818998220034 \cdot 10^{-240}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{elif}\;a \leq 255311114.6999046:\\ \;\;\;\;\frac{x}{1 + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	23.3
Cost	968

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \mathbf{if}\;t \leq -1.0384443237295707 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4578604210337144 \cdot 10^{-115}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	20.3
Cost	968

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \mathbf{if}\;t \leq -6.23041294699088 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4578604210337144 \cdot 10^{-115}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	20.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -6.23041294699088 \cdot 10^{-36}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{a + 1}\\ \mathbf{elif}\;t \leq 1.4578604210337144 \cdot 10^{-115}:\\ \;\;\;\;\frac{z}{b} + \frac{x \cdot t}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{a + 1}\\ \end{array} \]

Alternative 9
Error	37.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;a \leq -25.281202353532876:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 8.546435848123383 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.2674761203755192 \cdot 10^{+190}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1.5508596556535804 \cdot 10^{+230}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 10
Error	37.1
Cost	720

\[\begin{array}{l} \mathbf{if}\;a \leq -25.281202353532876:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 8.546435848123383 \cdot 10^{-5}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{elif}\;a \leq 2.2674761203755192 \cdot 10^{+190}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1.5508596556535804 \cdot 10^{+230}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 11
Error	28.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5880265939445685 \cdot 10^{+20}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 4.8151012081012854 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternative 12
Error	36.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -25.281202353532876:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 8.546435848123383 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

Alternative 13
Error	50.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022310 
(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))))