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

Average Error: 16.4 → 5.7

Time: 9.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \mathsf{fma}\left(\frac{y}{\left(a + 1\right) - \frac{y \cdot b}{-t}}, \frac{z}{t}, \frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -4.472539530856903 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z + \frac{x}{\frac{y}{t}}}{b}\\ \mathbf{elif}\;t_1 \leq 1.7642758988097463 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z + x \cdot \frac{t}{y}}{b}\\ \end{array} \]

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 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2
         (fma
          (/ y (- (+ a 1.0) (/ (* y b) (- t))))
          (/ z t)
          (/ x (+ (+ a 1.0) (* b (/ y t)))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -4.472539530856903e-289)
       t_1
       (if (<= t_1 0.0)
         (/ (+ z (/ x (/ y t))) b)
         (if (<= t_1 1.7642758988097463e+126)
           t_1
           (if (<= t_1 INFINITY) 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 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = fma((y / ((a + 1.0) - ((y * b) / -t))), (z / t), (x / ((a + 1.0) + (b * (y / t)))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -4.472539530856903e-289) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z + (x / (y / t))) / b;
	} else if (t_1 <= 1.7642758988097463e+126) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 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(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = fma(Float64(y / Float64(Float64(a + 1.0) - Float64(Float64(y * b) / Float64(-t)))), Float64(z / t), Float64(x / Float64(Float64(a + 1.0) + Float64(b * Float64(y / t)))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -4.472539530856903e-289)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z + Float64(x / Float64(y / t))) / b);
	elseif (t_1 <= 1.7642758988097463e+126)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = 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[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(N[(a + 1.0), $MachinePrecision] - N[(N[(y * b), $MachinePrecision] / (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(x / N[(N[(a + 1.0), $MachinePrecision] + N[(b * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -4.472539530856903e-289], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(z + N[(x / N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 1.7642758988097463e+126], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(N[(z + N[(x * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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	16.4
Target	13.2
Herbie	5.7

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

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))) < -inf.0 or 1.7642758988097463e126 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 34.9

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

   2. Simplified 24.4

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

   3. Taylor expanded in z around 0 20.9

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

   4. Simplified 17.5

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

   5. Applied egg-rr 12.6

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -4.4725395308569033e-289 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 1.7642758988097463e126

   1. Initial program 0.5

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

   if -4.4725395308569033e-289 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 27.2

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

   2. Simplified 19.4

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

   3. Taylor expanded in z around 0 27.4

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

   4. Simplified 19.3

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

   5. Taylor expanded in b around inf 21.9

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

   6. Simplified 20.0

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

   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. Simplified 56.3

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

   3. Taylor expanded in z around 0 61.2

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

   4. Simplified 51.5

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

   5. Applied egg-rr 51.6

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(\sqrt[3]{y} \cdot \frac{1}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\right)}, \frac{z}{t}, \frac{x}{\left(1 + a\right) + \frac{y}{t} \cdot b}\right) \]

   6. Taylor expanded in b around inf 4.6

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

   7. Simplified 2.3

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

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\left(a + 1\right) - \frac{y \cdot b}{-t}}, \frac{z}{t}, \frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -4.472539530856903 \cdot 10^{-289}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{z + \frac{x}{\frac{y}{t}}}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 1.7642758988097463 \cdot 10^{+126}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\left(a + 1\right) - \frac{y \cdot b}{-t}}, \frac{z}{t}, \frac{x}{\left(a + 1\right) + b \cdot \frac{y}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z + x \cdot \frac{t}{y}}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022151 
(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))))