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

Average Accuracy: 95.7% → 99.7%

Time: 9.9s

Precision: binary64

Cost: 7817

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+285} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+250}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + \mathsf{fma}\left(-z, t, z \cdot t\right)\right) - z \cdot t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e+285) (not (<= (* z t) 2e+250)))
   (/ (/ x z) (- t))
   (/ x (- (+ y (fma (- z) t (* z t))) (* z t)))))

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+285) || !((z * t) <= 2e+250)) {
		tmp = (x / z) / -t;
	} else {
		tmp = x / ((y + fma(-z, t, (z * t))) - (z * t));
	}
	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+285) || !(Float64(z * t) <= 2e+250))
		tmp = Float64(Float64(x / z) / Float64(-t));
	else
		tmp = Float64(x / Float64(Float64(y + fma(Float64(-z), t, Float64(z * t))) - Float64(z * t)));
	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[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+285], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+250]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], N[(x / N[(N[(y + N[((-z) * t + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	95.7%
Target	97.2%
Herbie	99.7%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.00000000000000016e285 or 1.9999999999999998e250 < (*.f64 z t)

   1. Initial program 74.8%

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

   2. Taylor expanded in y around 0 73.8%

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

   3. Simplified 73.8%

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

      [Start] 73.8	\[ -1 \cdot \frac{x}{t \cdot z} \]
      *-commutative [<=] 73.8	\[ -1 \cdot \frac{x}{\color{blue}{z \cdot t}} \]
      associate-*r/ [=>] 73.8	\[ \color{blue}{\frac{-1 \cdot x}{z \cdot t}} \]
      neg-mul-1 [<=] 73.8	\[ \frac{\color{blue}{-x}}{z \cdot t} \]
      *-commutative [=>] 73.8	\[ \frac{-x}{\color{blue}{t \cdot z}} \]

   4. Taylor expanded in x around 0 73.8%

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

   5. Simplified 98.9%

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

      [Start] 73.8	\[ -1 \cdot \frac{x}{t \cdot z} \]
      metadata-eval [<=] 73.8	\[ \color{blue}{\frac{1}{-1}} \cdot \frac{x}{t \cdot z} \]
      associate-/r* [=>] 98.8	\[ \frac{1}{-1} \cdot \color{blue}{\frac{\frac{x}{t}}{z}} \]
      times-frac [<=] 98.8	\[ \color{blue}{\frac{1 \cdot \frac{x}{t}}{-1 \cdot z}} \]
      *-lft-identity [=>] 98.8	\[ \frac{\color{blue}{\frac{x}{t}}}{-1 \cdot z} \]
      neg-mul-1 [<=] 98.8	\[ \frac{\frac{x}{t}}{\color{blue}{-z}} \]
      associate-/r* [<=] 73.8	\[ \color{blue}{\frac{x}{t \cdot \left(-z\right)}} \]
      distribute-rgt-neg-out [=>] 73.8	\[ \frac{x}{\color{blue}{-t \cdot z}} \]
      *-commutative [=>] 73.8	\[ \frac{x}{-\color{blue}{z \cdot t}} \]
      distribute-rgt-neg-in [=>] 73.8	\[ \frac{x}{\color{blue}{z \cdot \left(-t\right)}} \]
      associate-/r* [=>] 98.9	\[ \color{blue}{\frac{\frac{x}{z}}{-t}} \]

   if -5.00000000000000016e285 < (*.f64 z t) < 1.9999999999999998e250

   1. Initial program 99.9%

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

   2. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+285} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+250}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + \mathsf{fma}\left(-z, t, z \cdot t\right)\right) - z \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+285} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+250}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 2
Accuracy	71.9%
Cost	913

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-95} \lor \neg \left(y \leq 5.3 \cdot 10^{-5}\right) \land y \leq 1.22 \cdot 10^{+41}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Accuracy	54.4%
Cost	717

\[\begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{+90} \lor \neg \left(t \leq 9.5 \cdot 10^{+102}\right) \land t \leq 10^{+206}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 4
Accuracy	54.3%
Cost	716

\[\begin{array}{l} \mathbf{if}\;t \leq 1.18 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{+207}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 5
Accuracy	71.3%
Cost	649

\[\begin{array}{l} \mathbf{if}\;t \leq -1.26 \cdot 10^{-74} \lor \neg \left(t \leq 1.6 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6
Accuracy	72.6%
Cost	649

\[\begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-73} \lor \neg \left(t \leq 1.25 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7
Accuracy	52.4%
Cost	192

\[\frac{x}{y} \]

Reproduce

herbie shell --seed 2023125 
(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))))