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

Average Error: 17.0 → 7.2

Time: 9.4s

Precision: binary64

Cost: 5712

Math

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

↓

\[\begin{array}{l} t_1 := \left(a + 1\right) + \frac{y \cdot b}{t}\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t + t \cdot \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-321}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{t_1}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{t}{y} \cdot \frac{x + \frac{z}{\frac{t}{y}}}{b}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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 (+ (+ a 1.0) (/ (* y b) t))) (t_2 (/ (+ x (/ (* y z) t)) t_1)))
   (if (<= t_2 (- INFINITY))
     (* z (/ y (+ t (* t (+ a (/ y (/ t b)))))))
     (if (<= t_2 -2e-321)
       (/ (+ x (* (* y z) (/ 1.0 t))) t_1)
       (if (<= t_2 0.0)
         (* (/ t y) (/ (+ x (/ z (/ t y))) b))
         (if (<= t_2 2e+293) 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 = (a + 1.0) + ((y * b) / t);
	double t_2 = (x + ((y * z) / t)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = z * (y / (t + (t * (a + (y / (t / b))))));
	} else if (t_2 <= -2e-321) {
		tmp = (x + ((y * z) * (1.0 / t))) / t_1;
	} else if (t_2 <= 0.0) {
		tmp = (t / y) * ((x + (z / (t / y))) / b);
	} else if (t_2 <= 2e+293) {
		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 = (a + 1.0) + ((y * b) / t);
	double t_2 = (x + ((y * z) / t)) / t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = z * (y / (t + (t * (a + (y / (t / b))))));
	} else if (t_2 <= -2e-321) {
		tmp = (x + ((y * z) * (1.0 / t))) / t_1;
	} else if (t_2 <= 0.0) {
		tmp = (t / y) * ((x + (z / (t / y))) / b);
	} else if (t_2 <= 2e+293) {
		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 = (a + 1.0) + ((y * b) / t)
	t_2 = (x + ((y * z) / t)) / t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = z * (y / (t + (t * (a + (y / (t / b))))))
	elif t_2 <= -2e-321:
		tmp = (x + ((y * z) * (1.0 / t))) / t_1
	elif t_2 <= 0.0:
		tmp = (t / y) * ((x + (z / (t / y))) / b)
	elif t_2 <= 2e+293:
		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(a + 1.0) + Float64(Float64(y * b) / t))
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / Float64(t + Float64(t * Float64(a + Float64(y / Float64(t / b)))))));
	elseif (t_2 <= -2e-321)
		tmp = Float64(Float64(x + Float64(Float64(y * z) * Float64(1.0 / t))) / t_1);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(t / y) * Float64(Float64(x + Float64(z / Float64(t / y))) / b));
	elseif (t_2 <= 2e+293)
		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 = (a + 1.0) + ((y * b) / t);
	t_2 = (x + ((y * z) / t)) / t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = z * (y / (t + (t * (a + (y / (t / b))))));
	elseif (t_2 <= -2e-321)
		tmp = (x + ((y * z) * (1.0 / t))) / t_1;
	elseif (t_2 <= 0.0)
		tmp = (t / y) * ((x + (z / (t / y))) / b);
	elseif (t_2 <= 2e+293)
		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[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], 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$2, -2e-321], N[(N[(x + N[(N[(y * z), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(t / y), $MachinePrecision] * N[(N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+293], t$95$2, N[(z / b), $MachinePrecision]]]]]]]

Target

Original	17.0
Target	13.6
Herbie	7.2

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

Derivation

1. Split input into 5 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 39.9

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

   3. Simplified 13.2

      \[\leadsto \color{blue}{\frac{y}{t + t \cdot \left(\frac{y}{\frac{t}{b}} + a\right)} \cdot z} \]
      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): 7 points increase in error, 9 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, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 y (/.f64 (*.f64 t (+.f64 1 (+.f64 (/.f64 (*.f64 y b) t) a))) z))): 31 points increase in error, 29 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y z) (*.f64 t (+.f64 1 (+.f64 (/.f64 (*.f64 y b) t) a))))): 48 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))) < -2.00097e-321

   1. Initial program 0.5

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

   2. Applied egg-rr 0.6

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

   if -2.00097e-321 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 29.6

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

   2. Taylor expanded in b around inf 31.7

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

   3. Simplified 26.8

      \[\leadsto \color{blue}{\frac{t}{y} \cdot \frac{x + \frac{z}{\frac{t}{y}}}{b}} \]
      Proof
      (*.f64 (/.f64 t y) (/.f64 (+.f64 x (/.f64 z (/.f64 t y))) b)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 t y) (/.f64 (+.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z y) t))) b)): 8 points increase in error, 7 points decrease in error
      (*.f64 (/.f64 t y) (/.f64 (+.f64 x (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y z)) t)) b)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 t y) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 y z) t) x)) b)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t (+.f64 (/.f64 (*.f64 y z) t) x)) (*.f64 y b))): 53 points increase in error, 37 points decrease in error

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.9999999999999998e293

   1. Initial program 0.4

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

   if 1.9999999999999998e293 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 62.4

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

   2. Simplified 51.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)}} \]
      Proof
      (/.f64 (fma.f64 z (/.f64 y t) x) (+.f64 a (fma.f64 y (/.f64 b t) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (/.f64 y t)) x)) (+.f64 a (fma.f64 y (/.f64 b t) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 y t) z)) x) (+.f64 a (fma.f64 y (/.f64 b t) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y z) t)) x) (+.f64 a (fma.f64 y (/.f64 b t) 1))): 13 points increase in error, 20 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 y z) t))) (+.f64 a (fma.f64 y (/.f64 b t) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 a (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (/.f64 b t)) 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 a (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y b) t)) 1))): 17 points increase in error, 10 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 a (Rewrite<= +-commutative_binary64 (+.f64 1 (/.f64 (*.f64 y b) t))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 13.2

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t + t \cdot \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -2 \cdot 10^{-321}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{t}{y} \cdot \frac{x + \frac{z}{\frac{t}{y}}}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	51.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022291 
(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))))