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

Average Error: 16.4 → 3.7

Time: 10.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := x + \frac{y}{\frac{t}{z}}\\ t_3 := \frac{a}{t_2} + \frac{1}{t_2}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}} \cdot \frac{z}{t}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-318}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;{\left(t_3 + \frac{y \cdot \frac{b}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{t}\right)}^{-1}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+299}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{\left(t_3 + \frac{b}{z}\right)}^{-1}\\ \end{array} \]

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 (+ x (/ y (/ t z))))
        (t_3 (+ (/ a t_2) (/ 1.0 t_2))))
   (if (<= t_1 (- INFINITY))
     (* (/ y (+ (+ a 1.0) (/ y (/ t b)))) (/ z t))
     (if (<= t_1 -1e-318)
       t_1
       (if (<= t_1 0.0)
         (pow (+ t_3 (/ (* y (/ b (fma y (/ z t) x))) t)) -1.0)
         (if (<= t_1 4e+299) t_1 (pow (+ t_3 (/ b z)) -1.0)))))))

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 = x + (y / (t / z));
	double t_3 = (a / t_2) + (1.0 / t_2);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / ((a + 1.0) + (y / (t / b)))) * (z / t);
	} else if (t_1 <= -1e-318) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = pow((t_3 + ((y * (b / fma(y, (z / t), x))) / t)), -1.0);
	} else if (t_1 <= 4e+299) {
		tmp = t_1;
	} else {
		tmp = pow((t_3 + (b / z)), -1.0);
	}
	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 = Float64(x + Float64(y / Float64(t / z)))
	t_3 = Float64(Float64(a / t_2) + Float64(1.0 / t_2))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / Float64(Float64(a + 1.0) + Float64(y / Float64(t / b)))) * Float64(z / t));
	elseif (t_1 <= -1e-318)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(t_3 + Float64(Float64(y * Float64(b / fma(y, Float64(z / t), x))) / t)) ^ -1.0;
	elseif (t_1 <= 4e+299)
		tmp = t_1;
	else
		tmp = Float64(t_3 + Float64(b / z)) ^ -1.0;
	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[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a / t$95$2), $MachinePrecision] + N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(N[(a + 1.0), $MachinePrecision] + N[(y / N[(t / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-318], t$95$1, If[LessEqual[t$95$1, 0.0], N[Power[N[(t$95$3 + N[(N[(y * N[(b / N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$1, 4e+299], t$95$1, N[Power[N[(t$95$3 + N[(b / z), $MachinePrecision]), $MachinePrecision], -1.0], $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	12.9
Herbie	3.7

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

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

   1. Initial program 64.0

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

   2. Simplified 38.7

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

   3. Applied egg-rr 38.7

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

   4. Taylor expanded in z around inf 38.6

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

   5. Simplified 21.6

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -9.9999875e-319 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 4.0000000000000002e299

   1. Initial program 0.4

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

   if -9.9999875e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 28.1

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

   2. Simplified 18.9

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

   3. Applied egg-rr 19.1

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

   4. Taylor expanded in b around 0 18.3

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

   5. Simplified 11.7

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

   6. Applied egg-rr 10.6

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

   if 4.0000000000000002e299 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

   1. Initial program 63.3

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

   2. Simplified 52.4

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

   3. Applied egg-rr 52.5

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

   4. Taylor expanded in b around 0 63.3

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

   5. Simplified 51.6

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

   6. Taylor expanded in y around inf 6.0

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\left(a + 1\right) + \frac{y}{\frac{t}{b}}} \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-318}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;{\left(\left(\frac{a}{x + \frac{y}{\frac{t}{z}}} + \frac{1}{x + \frac{y}{\frac{t}{z}}}\right) + \frac{y \cdot \frac{b}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}{t}\right)}^{-1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 4 \cdot 10^{+299}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{a}{x + \frac{y}{\frac{t}{z}}} + \frac{1}{x + \frac{y}{\frac{t}{z}}}\right) + \frac{b}{z}\right)}^{-1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022165 
(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))))