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

Average Error: 16.8 → 7.0

Time: 10.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t_2}{\left(a + 1\right) + t_1}\\ t_4 := 1 + \left(a + t_1\right)\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t_4}, \frac{z}{t}, \frac{x}{t_4}\right)\\ \mathbf{elif}\;t_3 \leq 1.9389044765786328 \cdot 10^{+301}:\\ \;\;\;\;\frac{t_2}{\left(a + 1\right) + \frac{b \cdot \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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 (/ (* 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))))
   (if (<= t_3 (- INFINITY))
     (fma (/ y t_4) (/ z t) (/ x t_4))
     (if (<= t_3 1.9389044765786328e+301)
       (/ t_2 (+ (+ a 1.0) (/ (* b (/ y (* (cbrt t) (cbrt t)))) (cbrt t))))
       (/ 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 tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma((y / t_4), (z / t), (x / t_4));
	} else if (t_3 <= 1.9389044765786328e+301) {
		tmp = t_2 / ((a + 1.0) + ((b * (y / (cbrt(t) * cbrt(t)))) / cbrt(t)));
	} 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(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))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(Float64(y / t_4), Float64(z / t), Float64(x / t_4));
	elseif (t_3 <= 1.9389044765786328e+301)
		tmp = Float64(t_2 / Float64(Float64(a + 1.0) + Float64(Float64(b * Float64(y / Float64(cbrt(t) * cbrt(t)))) / cbrt(t))));
	else
		tmp = 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]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(y / t$95$4), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(x / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1.9389044765786328e+301], N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(b * N[(y / N[(N[Power[t, 1/3], $MachinePrecision] * N[Power[t, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[t, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $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	13.1
Herbie	7.0

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

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 39.2

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

   3. Simplified 18.5

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.9389044765786328e301

   1. Initial program 6.2

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

   2. Applied add-cube-cbrt_binary64 6.4

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}} \]

   3. Applied associate-/r*_binary64 6.4

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \color{blue}{\frac{\frac{y \cdot b}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}} \]

   4. Simplified 5.5

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{\color{blue}{\frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot b}}{\sqrt[3]{t}}} \]

   if 1.9389044765786328e301 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 63.6

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

   2. Taylor expanded in y around inf 12.3

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

4. Final simplification 7.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + \left(a + \frac{y \cdot b}{t}\right)}, \frac{z}{t}, \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 1.9389044765786328 \cdot 10^{+301}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{b \cdot \frac{y}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))