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

Average Error: 2.7 → 0.4

Time: 9.9s

Precision: binary64

Cost: 7816

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 1.5 \cdot 10^{+187}:\\ \;\;\;\;\frac{x}{\left(y + \mathsf{fma}\left(-z, t, z \cdot t\right)\right) - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ (/ (- x) t) z)
   (if (<= (* z t) 1.5e+187)
     (/ x (- (+ y (fma (- z) t (* z t))) (* z t)))
     (/ (/ x 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) <= -((double) INFINITY)) {
		tmp = (-x / t) / z;
	} else if ((z * t) <= 1.5e+187) {
		tmp = x / ((y + fma(-z, t, (z * t))) - (z * t));
	} else {
		tmp = (x / 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) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) / t) / z);
	elseif (Float64(z * t) <= 1.5e+187)
		tmp = Float64(x / Float64(Float64(y + fma(Float64(-z), t, Float64(z * t))) - Float64(z * t)));
	else
		tmp = Float64(Float64(x / z) / Float64(-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[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1.5e+187], N[(x / N[(N[(y + N[((-z) * t + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision]]]

Target

Original	2.7
Target	1.7
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 19.1

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

   2. Applied egg-rr 19.1

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

   3. Taylor expanded in y around 0 19.1

      \[\leadsto \color{blue}{-1 \cdot \left({1}^{0.3333333333333333} \cdot \frac{x}{t \cdot z}\right)} \]

   4. Simplified 0.1

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

      [Start] 19.1	\[ -1 \cdot \left({1}^{0.3333333333333333} \cdot \frac{x}{t \cdot z}\right) \]
      mul-1-neg [=>] 19.1	\[ \color{blue}{-{1}^{0.3333333333333333} \cdot \frac{x}{t \cdot z}} \]
      *-commutative [=>] 19.1	\[ -{1}^{0.3333333333333333} \cdot \frac{x}{\color{blue}{z \cdot t}} \]
      associate-*r/ [=>] 19.1	\[ -\color{blue}{\frac{{1}^{0.3333333333333333} \cdot x}{z \cdot t}} \]
      pow-base-1 [=>] 19.1	\[ -\frac{\color{blue}{1} \cdot x}{z \cdot t} \]
      *-lft-identity [=>] 19.1	\[ -\frac{\color{blue}{x}}{z \cdot t} \]
      associate-/l/ [<=] 0.1	\[ -\color{blue}{\frac{\frac{x}{t}}{z}} \]

   if -inf.0 < (*.f64 z t) < 1.5e187

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   if 1.5e187 < (*.f64 z t)

   1. Initial program 11.0

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

   2. Taylor expanded in y around 0 12.9

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

   3. Simplified 12.9

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

      [Start] 12.9	\[ -1 \cdot \frac{x}{t \cdot z} \]
      *-commutative [<=] 12.9	\[ -1 \cdot \frac{x}{\color{blue}{z \cdot t}} \]
      associate-*r/ [=>] 12.9	\[ \color{blue}{\frac{-1 \cdot x}{z \cdot t}} \]
      neg-mul-1 [<=] 12.9	\[ \frac{\color{blue}{-x}}{z \cdot t} \]
      *-commutative [=>] 12.9	\[ \frac{-x}{\color{blue}{t \cdot z}} \]

   4. Taylor expanded in x around 0 12.9

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

   5. Simplified 2.5

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

      [Start] 12.9	\[ -1 \cdot \frac{x}{t \cdot z} \]
      metadata-eval [<=] 12.9	\[ \color{blue}{\frac{1}{-1}} \cdot \frac{x}{t \cdot z} \]
      associate-/r* [=>] 2.9	\[ \frac{1}{-1} \cdot \color{blue}{\frac{\frac{x}{t}}{z}} \]
      times-frac [<=] 2.9	\[ \color{blue}{\frac{1 \cdot \frac{x}{t}}{-1 \cdot z}} \]
      *-lft-identity [=>] 2.9	\[ \frac{\color{blue}{\frac{x}{t}}}{-1 \cdot z} \]
      neg-mul-1 [<=] 2.9	\[ \frac{\frac{x}{t}}{\color{blue}{-z}} \]
      associate-/r* [<=] 12.9	\[ \color{blue}{\frac{x}{t \cdot \left(-z\right)}} \]
      distribute-rgt-neg-out [=>] 12.9	\[ \frac{x}{\color{blue}{-t \cdot z}} \]
      *-commutative [=>] 12.9	\[ \frac{x}{-\color{blue}{z \cdot t}} \]
      distribute-rgt-neg-in [=>] 12.9	\[ \frac{x}{\color{blue}{z \cdot \left(-t\right)}} \]
      associate-/r* [=>] 2.5	\[ \color{blue}{\frac{\frac{x}{z}}{-t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 1.5 \cdot 10^{+187}:\\ \;\;\;\;\frac{x}{\left(y + \mathsf{fma}\left(-z, t, z \cdot t\right)\right) - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 1.5 \cdot 10^{+187}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]

Alternative 2
Error	19.1
Cost	914

\[\begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+141} \lor \neg \left(z \leq -1.45 \cdot 10^{+86} \lor \neg \left(z \leq -165000000\right) \land z \leq 1.22 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	19.1
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{-x}{t}}{z}\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{+86}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -17000:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	18.5
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{-t}\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -12500000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-120}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 5
Error	27.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+179} \lor \neg \left(z \leq 135\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6
Error	30.1
Cost	192

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

Reproduce

herbie shell --seed 2023059 
(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))))