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

Average Error: 17.0 → 5.3

Time: 7.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(a + 1\right) + \frac{y \cdot b}{t}\\ t_2 := 1 + \left(a + b \cdot \frac{y}{t}\right)\\ t_3 := \frac{x + \frac{y \cdot z}{t}}{t_1}\\ t_4 := 1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\frac{x}{t_4} + \frac{z}{t_4 \cdot \frac{t}{y}}\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{t_1}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+138}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, \frac{z}{t_2}, \frac{x}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z + x \cdot \frac{t}{y}}{b}\\ \end{array} \]

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 (+ 1.0 (+ a (* b (/ y t)))))
        (t_3 (/ (+ x (/ (* y z) t)) t_1))
        (t_4 (+ 1.0 (fma (/ y t) b a))))
   (if (<= t_3 (- INFINITY))
     (+ (/ x t_4) (/ z (* t_4 (/ t y))))
     (if (<= t_3 -1e-303)
       (/ (+ x (* (* y z) (/ 1.0 t))) t_1)
       (if (<= t_3 0.0)
         (/ (+ z (/ (* x t) y)) b)
         (if (<= t_3 5e+138)
           t_3
           (if (<= t_3 INFINITY)
             (fma (/ y t) (/ z t_2) (/ x t_2))
             (/ (+ z (* x (/ t y))) 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 = 1.0 + (a + (b * (y / t)));
	double t_3 = (x + ((y * z) / t)) / t_1;
	double t_4 = 1.0 + fma((y / t), b, a);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (x / t_4) + (z / (t_4 * (t / y)));
	} else if (t_3 <= -1e-303) {
		tmp = (x + ((y * z) * (1.0 / t))) / t_1;
	} else if (t_3 <= 0.0) {
		tmp = (z + ((x * t) / y)) / b;
	} else if (t_3 <= 5e+138) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma((y / t), (z / t_2), (x / t_2));
	} else {
		tmp = (z + (x * (t / y))) / 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(1.0 + Float64(a + Float64(b * Float64(y / t))))
	t_3 = Float64(Float64(x + Float64(Float64(y * z) / t)) / t_1)
	t_4 = Float64(1.0 + fma(Float64(y / t), b, a))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(x / t_4) + Float64(z / Float64(t_4 * Float64(t / y))));
	elseif (t_3 <= -1e-303)
		tmp = Float64(Float64(x + Float64(Float64(y * z) * Float64(1.0 / t))) / t_1);
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(z + Float64(Float64(x * t) / y)) / b);
	elseif (t_3 <= 5e+138)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = fma(Float64(y / t), Float64(z / t_2), Float64(x / t_2));
	else
		tmp = Float64(Float64(z + Float64(x * Float64(t / y))) / 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[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(a + N[(b * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(N[(y / t), $MachinePrecision] * b + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(x / t$95$4), $MachinePrecision] + N[(z / N[(t$95$4 * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-303], N[(N[(x + N[(N[(y * z), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(z + N[(N[(x * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$3, 5e+138], t$95$3, If[LessEqual[t$95$3, Infinity], N[(N[(y / t), $MachinePrecision] * N[(z / t$95$2), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(x * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]]]]]]

Target

Original	17.0
Target	13.4
Herbie	5.3

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

Derivation

1. Split input into 6 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 38.3

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

   3. Simplified 12.6

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

   4. Applied egg-rr 14.0

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -9.99999999999999931e-304

   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

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

   if -9.99999999999999931e-304 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 30.2

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

   2. Taylor expanded in x around 0 30.3

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

   3. Simplified 25.8

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

   4. Taylor expanded in b around inf 21.1

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 5.00000000000000016e138

   1. Initial program 0.5

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

   if 5.00000000000000016e138 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 25.7

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

   2. Taylor expanded in x around 0 15.8

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

   3. Simplified 6.8

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

   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 x around 0 61.9

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

   3. Simplified 55.0

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

   4. Taylor expanded in b around inf 4.5

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

   5. Simplified 1.7

      \[\leadsto \color{blue}{\frac{z + \frac{t}{y} \cdot x}{b}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)} + \frac{z}{\left(1 + \mathsf{fma}\left(\frac{y}{t}, b, a\right)\right) \cdot \frac{t}{y}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{z + \frac{x \cdot t}{y}}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 5 \cdot 10^{+138}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, \frac{z}{1 + \left(a + b \cdot \frac{y}{t}\right)}, \frac{x}{1 + \left(a + b \cdot \frac{y}{t}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z + x \cdot \frac{t}{y}}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022186 
(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))))