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

Average Error: 2.6 → 1.6

Time: 3.3s

Precision: binary64

\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+227}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\mathsf{fma}\left(\frac{\frac{y}{t}}{t}, \frac{x}{z}, \frac{x}{t}\right) \cdot \frac{1}{z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 5e+227)
   (/ x (- y (* z t)))
   (* -1.0 (* (fma (/ (/ y t) t) (/ x z) (/ x t)) (/ 1.0 z)))))

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) <= 5e+227) {
		tmp = x / (y - (z * t));
	} else {
		tmp = -1.0 * (fma(((y / t) / t), (x / z), (x / t)) * (1.0 / z));
	}
	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) <= 5e+227)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(-1.0 * Float64(fma(Float64(Float64(y / t) / t), Float64(x / z), Float64(x / t)) * Float64(1.0 / z)));
	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], 5e+227], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[(N[(y / t), $MachinePrecision] / t), $MachinePrecision] * N[(x / z), $MachinePrecision] + N[(x / t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	2.6
Target	1.7
Herbie	1.6

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 4.9999999999999996e227

   1. Initial program 1.6

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

   if 4.9999999999999996e227 < (*.f64 z t)

   1. Initial program 11.6

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

   2. Taylor expanded in y around 0 15.0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot x}{{t}^{2} \cdot {z}^{2}} + -1 \cdot \frac{x}{t \cdot z}} \]

   3. Simplified 1.8

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

   4. Applied egg-rr 1.6

      \[\leadsto -1 \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\frac{y}{t}}{t}, \frac{x}{z}, \frac{x}{t}\right) \cdot \frac{1}{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+227}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\mathsf{fma}\left(\frac{\frac{y}{t}}{t}, \frac{x}{z}, \frac{x}{t}\right) \cdot \frac{1}{z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022203 
(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))))