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

Average Error: 10.7 → 4.8

Time: 6.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x}{t_1}\\ t_3 := \frac{x - y \cdot z}{t_1}\\ \mathbf{if}\;t_3 \leq -1.8021950710243513 \cdot 10^{-25}:\\ \;\;\;\;t_2 - y \cdot \frac{z}{t_1}\\ \mathbf{elif}\;t_3 \leq 4.1429951425011623 \cdot 10^{+307}:\\ \;\;\;\;t_2 - \frac{y \cdot z}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{t_1}, \frac{y}{a}\right)\\ \end{array} \]

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 (- t (* z a))) (t_2 (/ x t_1)) (t_3 (/ (- x (* y z)) t_1)))
   (if (<= t_3 -1.8021950710243513e-25)
     (- t_2 (* y (/ z t_1)))
     (if (<= t_3 4.1429951425011623e+307)
       (- t_2 (/ (* y z) t_1))
       (fma x (/ 1.0 t_1) (/ y a))))))

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 = t - (z * a);
	double t_2 = x / t_1;
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -1.8021950710243513e-25) {
		tmp = t_2 - (y * (z / t_1));
	} else if (t_3 <= 4.1429951425011623e+307) {
		tmp = t_2 - ((y * z) / t_1);
	} else {
		tmp = fma(x, (1.0 / t_1), (y / a));
	}
	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(t - Float64(z * a))
	t_2 = Float64(x / t_1)
	t_3 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_3 <= -1.8021950710243513e-25)
		tmp = Float64(t_2 - Float64(y * Float64(z / t_1)));
	elseif (t_3 <= 4.1429951425011623e+307)
		tmp = Float64(t_2 - Float64(Float64(y * z) / t_1));
	else
		tmp = fma(x, Float64(1.0 / t_1), Float64(y / a));
	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[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -1.8021950710243513e-25], N[(t$95$2 - N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4.1429951425011623e+307], N[(t$95$2 - N[(N[(y * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 / t$95$1), $MachinePrecision] + N[(y / a), $MachinePrecision]), $MachinePrecision]]]]]]

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	1.9
Herbie	4.8

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.80219507102435134e-25

   1. Initial program 8.8

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

   2. Taylor expanded in x around 0 8.8

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

   3. Applied egg-rr 0.2

      \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{\frac{y}{1} \cdot \frac{z}{t - z \cdot a}} \]

   if -1.80219507102435134e-25 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.14299514250116234e307

   1. Initial program 5.9

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

   2. Taylor expanded in x around 0 5.9

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

   if 4.14299514250116234e307 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 63.9

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

   2. Taylor expanded in x around 0 63.9

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

   3. Applied egg-rr 63.9

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

   4. Taylor expanded in z around inf 10.4

      \[\leadsto \mathsf{fma}\left(x, \frac{1}{t - z \cdot a}, \color{blue}{\frac{y}{a}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.8021950710243513 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - y \cdot \frac{z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 4.1429951425011623 \cdot 10^{+307}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{t - z \cdot a}, \frac{y}{a}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))))