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

Average Error: 2.6 → 1.3

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY)) (/ (/ (- x) z) t) (/ x (fma z (- t) y))))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / fma(z, -t, y);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	else
		tmp = Float64(x / fma(z, Float64(-t), y));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.6
Target	1.7
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 20.0

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

   2. Taylor expanded in y around 0 20.0

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

   3. Simplified 0.1

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

   if -inf.0 < (*.f64 z t)

   1. Initial program 1.4

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

   2. Applied egg-rr 1.4

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(z, -t, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022146 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

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