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

Average Error: 2.8 → 1.5

Time: 3.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+302}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{t \cdot t}, \frac{x}{z \cdot z}, \frac{\frac{x}{z}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+302)
   (- (fma (/ y (* t t)) (/ x (* z z)) (/ (/ 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) <= -2e+302) {
		tmp = -fma((y / (t * t)), (x / (z * z)), ((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) <= -2e+302)
		tmp = Float64(-fma(Float64(y / Float64(t * t)), Float64(x / Float64(z * z)), 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], -2e+302], (-N[(N[(y / N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision]]

Target

Original	2.8
Target	1.8
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000002e302

   1. Initial program 18.9

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

   2. Applied egg-rr 18.9

      \[\leadsto \color{blue}{\sqrt[3]{\frac{x}{y - z \cdot t}} \cdot {\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{2}} \]

   3. Taylor expanded in y around 0 20.6

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

   4. Simplified 0.1

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

   if -2.0000000000000002e302 < (*.f64 z t)

   1. Initial program 1.6

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

   2. Applied egg-rr 1.7

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

   3. Taylor expanded in x around 0 1.6

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

   4. Applied egg-rr 1.6

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+302}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{t \cdot t}, \frac{x}{z \cdot z}, \frac{\frac{x}{z}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(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))))