Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Average Error: 10.7 → 1.7

Time: 5.0s

Precision: binary64

Math

\[\frac{x - y \cdot z}{t - a \cdot z} \]

↓

\[\begin{array}{l} t_1 := \frac{x}{\mathsf{fma}\left(a, -z, t\right)} - \frac{y}{\frac{t}{z} - a}\\ t_2 := t - z \cdot a\\ \mathbf{if}\;z \leq -4.3 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{t_2} - \frac{z \cdot y}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (/ (- x (* y z)) (- t (* a z))))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (/ x (fma a (- z) t)) (/ y (- (/ t z) a))))
        (t_2 (- t (* z a))))
   (if (<= z -4.3e+21)
     t_1
     (if (<= z 4.1e+49) (- (/ x t_2) (/ (* z y) t_2)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	return (x - (y * z)) / (t - (a * z));
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x / fma(a, -z, t)) - (y / ((t / z) - a));
	double t_2 = t - (z * a);
	double tmp;
	if (z <= -4.3e+21) {
		tmp = t_1;
	} else if (z <= 4.1e+49) {
		tmp = (x / t_2) - ((z * y) / t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	return Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x / fma(a, Float64(-z), t)) - Float64(y / Float64(Float64(t / z) - a)))
	t_2 = Float64(t - Float64(z * a))
	tmp = 0.0
	if (z <= -4.3e+21)
		tmp = t_1;
	elseif (z <= 4.1e+49)
		tmp = Float64(Float64(x / t_2) - Float64(Float64(z * y) / t_2));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / N[(a * (-z) + t), $MachinePrecision]), $MachinePrecision] - N[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.3e+21], t$95$1, If[LessEqual[z, 4.1e+49], N[(N[(x / t$95$2), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.7
Target	1.8
Herbie	1.7

\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if z < -4.3e21 or 4.1e49 < z

   1. Initial program 23.4

      \[\frac{x - y \cdot z}{t - a \cdot z} \]

   2. Taylor expanded in x around 0 23.4

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}} \]

   3. Simplified 14.5

      \[\leadsto \color{blue}{\frac{x}{t - z \cdot a} - \frac{y}{\frac{t - z \cdot a}{z}}} \]

   4. Taylor expanded in t around 0 3.2

      \[\leadsto \frac{x}{t - z \cdot a} - \frac{y}{\color{blue}{\frac{t}{z} - a}} \]

   5. Taylor expanded in x around 0 3.2

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}} \]

   6. Simplified 3.2

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

   if -4.3e21 < z < 4.1e49

   1. Initial program 0.5

      \[\frac{x - y \cdot z}{t - a \cdot z} \]

   2. Taylor expanded in x around 0 0.5

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}} \]

   3. Simplified 2.8

      \[\leadsto \color{blue}{\frac{x}{t - z \cdot a} - \frac{y}{\frac{t - z \cdot a}{z}}} \]

   4. Taylor expanded in y around 0 0.5

      \[\leadsto \frac{x}{t - z \cdot a} - \color{blue}{\frac{y \cdot z}{t - a \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(a, -z, t\right)} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(a, -z, t\right)} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022165 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))