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

Average Error: 16.8 → 6.5

Time: 7.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t_2}{\left(a + 1\right) + t_1}\\ t_4 := 1 + \left(a + t_1\right)\\ t_5 := \frac{x}{t_4}\\ \mathbf{if}\;t_3 \leq -2.552021906120745 \cdot 10^{-302}:\\ \;\;\;\;z \cdot \frac{y}{t \cdot t_4} + t_5\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, \frac{x}{b}, \frac{z}{b}\right) - \mathsf{fma}\left(\frac{t}{y}, \frac{z}{b \cdot b}, \frac{a \cdot \left(z \cdot t\right)}{y \cdot \left(b \cdot b\right)}\right)\\ \mathbf{elif}\;t_3 \leq 5.689827926640151 \cdot 10^{+266}:\\ \;\;\;\;\frac{t_2}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_5 + \frac{z}{b}\\ \end{array} \]

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 (/ (* y b) t))
        (t_2 (+ x (/ (* y z) t)))
        (t_3 (/ t_2 (+ (+ a 1.0) t_1)))
        (t_4 (+ 1.0 (+ a t_1)))
        (t_5 (/ x t_4)))
   (if (<= t_3 -2.552021906120745e-302)
     (+ (* z (/ y (* t t_4))) t_5)
     (if (<= t_3 0.0)
       (-
        (fma (/ t y) (/ x b) (/ z b))
        (fma (/ t y) (/ z (* b b)) (/ (* a (* z t)) (* y (* b b)))))
       (if (<= t_3 5.689827926640151e+266)
         (/ t_2 (+ (+ a 1.0) (* (* y b) (/ 1.0 t))))
         (+ t_5 (/ 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 = (y * b) / t;
	double t_2 = x + ((y * z) / t);
	double t_3 = t_2 / ((a + 1.0) + t_1);
	double t_4 = 1.0 + (a + t_1);
	double t_5 = x / t_4;
	double tmp;
	if (t_3 <= -2.552021906120745e-302) {
		tmp = (z * (y / (t * t_4))) + t_5;
	} else if (t_3 <= 0.0) {
		tmp = fma((t / y), (x / b), (z / b)) - fma((t / y), (z / (b * b)), ((a * (z * t)) / (y * (b * b))));
	} else if (t_3 <= 5.689827926640151e+266) {
		tmp = t_2 / ((a + 1.0) + ((y * b) * (1.0 / t)));
	} else {
		tmp = t_5 + (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(y * b) / t)
	t_2 = Float64(x + Float64(Float64(y * z) / t))
	t_3 = Float64(t_2 / Float64(Float64(a + 1.0) + t_1))
	t_4 = Float64(1.0 + Float64(a + t_1))
	t_5 = Float64(x / t_4)
	tmp = 0.0
	if (t_3 <= -2.552021906120745e-302)
		tmp = Float64(Float64(z * Float64(y / Float64(t * t_4))) + t_5);
	elseif (t_3 <= 0.0)
		tmp = Float64(fma(Float64(t / y), Float64(x / b), Float64(z / b)) - fma(Float64(t / y), Float64(z / Float64(b * b)), Float64(Float64(a * Float64(z * t)) / Float64(y * Float64(b * b)))));
	elseif (t_3 <= 5.689827926640151e+266)
		tmp = Float64(t_2 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) * Float64(1.0 / t))));
	else
		tmp = Float64(t_5 + Float64(z / b));
	end
	return 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[(y * b), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(a + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(x / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$3, -2.552021906120745e-302], N[(N[(z * N[(y / N[(t * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[(t / y), $MachinePrecision] * N[(x / b), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision] - N[(N[(t / y), $MachinePrecision] * N[(z / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(z * t), $MachinePrecision]), $MachinePrecision] / N[(y * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5.689827926640151e+266], N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$5 + N[(z / b), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	16.8
Target	13.0
Herbie	6.5

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

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))) < -2.5520219061207448e-302

   1. Initial program 7.6

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

   2. Taylor expanded in x around 0 6.0

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

   3. Applied egg-rr 3.0

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

   if -2.5520219061207448e-302 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 28.7

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

   2. Taylor expanded in y around inf 27.8

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

   3. Simplified 21.4

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 5.6898279266401508e266

   1. Initial program 0.4

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

   2. Applied egg-rr 0.5

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

   if 5.6898279266401508e266 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 58.7

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

   2. Taylor expanded in x around 0 51.4

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

   3. Taylor expanded in y around inf 11.3

      \[\leadsto \color{blue}{\frac{z}{b}} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -2.552021906120745 \cdot 10^{-302}:\\ \;\;\;\;z \cdot \frac{y}{t \cdot \left(1 + \left(a + \frac{y \cdot b}{t}\right)\right)} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, \frac{x}{b}, \frac{z}{b}\right) - \mathsf{fma}\left(\frac{t}{y}, \frac{z}{b \cdot b}, \frac{a \cdot \left(z \cdot t\right)}{y \cdot \left(b \cdot b\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 5.689827926640151 \cdot 10^{+266}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \left(y \cdot b\right) \cdot \frac{1}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)} + \frac{z}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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))))