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

Average Error: 17.0 → 6.3

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t_2}{\left(a + 1\right) + t_1}\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{-318}:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)} + \frac{x}{1 + \left(a + \left(y \cdot b\right) \cdot \frac{1}{t}\right)}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, \frac{x}{b}, \frac{z}{b}\right) - \mathsf{fma}\left(\frac{a}{y}, \frac{z}{b} \cdot \frac{t}{b}, \frac{\frac{\frac{z}{\frac{y}{t}}}{b}}{b}\right)\\ \mathbf{elif}\;t_3 \leq 10^{+275}:\\ \;\;\;\;\frac{t_2}{\left(a + 1\right) - \frac{y \cdot b}{-t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{x}{1 + \left(a + t_1\right)}\\ \end{array} \]

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))))
   (if (<= t_3 -5e-318)
     (+
      (* z (/ y (fma y b (fma t a t))))
      (/ x (+ 1.0 (+ a (* (* y b) (/ 1.0 t))))))
     (if (<= t_3 0.0)
       (-
        (fma (/ t y) (/ x b) (/ z b))
        (fma (/ a y) (* (/ z b) (/ t b)) (/ (/ (/ z (/ y t)) b) b)))
       (if (<= t_3 1e+275)
         (/ t_2 (- (+ a 1.0) (/ (* y b) (- t))))
         (+ (/ z b) (/ x (+ 1.0 (+ a t_1)))))))))

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 tmp;
	if (t_3 <= -5e-318) {
		tmp = (z * (y / fma(y, b, fma(t, a, t)))) + (x / (1.0 + (a + ((y * b) * (1.0 / t)))));
	} else if (t_3 <= 0.0) {
		tmp = fma((t / y), (x / b), (z / b)) - fma((a / y), ((z / b) * (t / b)), (((z / (y / t)) / b) / b));
	} else if (t_3 <= 1e+275) {
		tmp = t_2 / ((a + 1.0) - ((y * b) / -t));
	} else {
		tmp = (z / b) + (x / (1.0 + (a + t_1)));
	}
	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))
	tmp = 0.0
	if (t_3 <= -5e-318)
		tmp = Float64(Float64(z * Float64(y / fma(y, b, fma(t, a, t)))) + Float64(x / Float64(1.0 + Float64(a + Float64(Float64(y * b) * Float64(1.0 / t))))));
	elseif (t_3 <= 0.0)
		tmp = Float64(fma(Float64(t / y), Float64(x / b), Float64(z / b)) - fma(Float64(a / y), Float64(Float64(z / b) * Float64(t / b)), Float64(Float64(Float64(z / Float64(y / t)) / b) / b)));
	elseif (t_3 <= 1e+275)
		tmp = Float64(t_2 / Float64(Float64(a + 1.0) - Float64(Float64(y * b) / Float64(-t))));
	else
		tmp = Float64(Float64(z / b) + Float64(x / Float64(1.0 + Float64(a + t_1))));
	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]}, If[LessEqual[t$95$3, -5e-318], N[(N[(z * N[(y / N[(y * b + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 + N[(a + N[(N[(y * b), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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[(a / y), $MachinePrecision] * N[(N[(z / b), $MachinePrecision] * N[(t / b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z / N[(y / t), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+275], N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] - N[(N[(y * b), $MachinePrecision] / (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / b), $MachinePrecision] + N[(x / N[(1.0 + N[(a + t$95$1), $MachinePrecision]), $MachinePrecision]), $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	17.0
Target	13.4
Herbie	6.3

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

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))) < -4.9999987e-318

   1. Initial program 8.2

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

   2. Taylor expanded in x around 0 6.4

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

   3. Taylor expanded in z around inf 6.4

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

   4. Simplified 4.2

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

   5. Taylor expanded in z around 0 6.4

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

   6. Simplified 3.1

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

   7. Applied egg-rr 3.1

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

   if -4.9999987e-318 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 29.0

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

   2. Taylor expanded in y around inf 26.7

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

   3. Simplified 20.2

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 9.9999999999999996e274

   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 0.5

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

   if 9.9999999999999996e274 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 60.8

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

   2. Taylor expanded in x around 0 52.8

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

   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 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -5 \cdot 10^{-318}:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(t, a, t\right)\right)} + \frac{x}{1 + \left(a + \left(y \cdot b\right) \cdot \frac{1}{t}\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, \frac{x}{b}, \frac{z}{b}\right) - \mathsf{fma}\left(\frac{a}{y}, \frac{z}{b} \cdot \frac{t}{b}, \frac{\frac{\frac{z}{\frac{y}{t}}}{b}}{b}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+275}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) - \frac{y \cdot b}{-t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))