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

Average Error: 10.5 → 2.8

Time: 12.6s

Precision: binary64

Cost: 3794

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq -1 \cdot 10^{-321}\right) \land \left(t_1 \leq 4 \cdot 10^{-257} \lor \neg \left(t_1 \leq 10^{+275}\right)\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (or (<= t_1 (- INFINITY))
           (and (not (<= t_1 -1e-321))
                (or (<= t_1 4e-257) (not (<= t_1 1e+275)))))
     (/ y (- a (/ t z)))
     t_1)))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || (!(t_1 <= -1e-321) && ((t_1 <= 4e-257) || !(t_1 <= 1e+275)))) {
		tmp = y / (a - (t / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || (!(t_1 <= -1e-321) && ((t_1 <= 4e-257) || !(t_1 <= 1e+275)))) {
		tmp = y / (a - (t / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	return (x - (y * z)) / (t - (a * z))

↓

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if (t_1 <= -math.inf) or (not (t_1 <= -1e-321) and ((t_1 <= 4e-257) or not (t_1 <= 1e+275))):
		tmp = y / (a - (t / z))
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || (!(t_1 <= -1e-321) && ((t_1 <= 4e-257) || !(t_1 <= 1e+275))))
		tmp = Float64(y / Float64(a - Float64(t / z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (x - (y * z)) / (t - (a * z));
end

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (y * z)) / (t - (z * a));
	tmp = 0.0;
	if ((t_1 <= -Inf) || (~((t_1 <= -1e-321)) && ((t_1 <= 4e-257) || ~((t_1 <= 1e+275)))))
		tmp = y / (a - (t / z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], And[N[Not[LessEqual[t$95$1, -1e-321]], $MachinePrecision], Or[LessEqual[t$95$1, 4e-257], N[Not[LessEqual[t$95$1, 1e+275]], $MachinePrecision]]]], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Target

Original	10.5
Target	1.7
Herbie	2.8

\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or -9.98013e-322 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 3.9999999999999999e-257 or 9.9999999999999996e274 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 36.8

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

   2. Simplified 36.8

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

      [Start] 36.8	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 36.8	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 36.8	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 36.8	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 36.8	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 36.8	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 36.8	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 36.8	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 36.8	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 36.8	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 36.8	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 36.8	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 36.8	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 36.8	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 36.8	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 36.8	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 36.8	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Taylor expanded in y around inf 40.6

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

   4. Simplified 40.6

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

      [Start] 40.6	\[ \frac{y \cdot z}{a \cdot z - t} \]
      *-commutative [=>] 40.6	\[ \frac{\color{blue}{z \cdot y}}{a \cdot z - t} \]

   5. Taylor expanded in y around 0 40.6

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

   6. Simplified 9.4

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

      [Start] 40.6	\[ \frac{y \cdot z}{a \cdot z - t} \]
      associate-/l* [=>] 28.0	\[ \color{blue}{\frac{y}{\frac{a \cdot z - t}{z}}} \]
      *-commutative [=>] 28.0	\[ \frac{y}{\frac{\color{blue}{z \cdot a} - t}{z}} \]
      div-sub [=>] 28.0	\[ \frac{y}{\color{blue}{\frac{z \cdot a}{z} - \frac{t}{z}}} \]
      associate-/l* [=>] 13.0	\[ \frac{y}{\color{blue}{\frac{z}{\frac{z}{a}}} - \frac{t}{z}} \]
      associate-/r/ [=>] 9.4	\[ \frac{y}{\color{blue}{\frac{z}{z} \cdot a} - \frac{t}{z}} \]
      *-inverses [=>] 9.4	\[ \frac{y}{\color{blue}{1} \cdot a - \frac{t}{z}} \]
      *-commutative [<=] 9.4	\[ \frac{y}{\color{blue}{a \cdot 1} - \frac{t}{z}} \]
      *-rgt-identity [=>] 9.4	\[ \frac{y}{\color{blue}{a} - \frac{t}{z}} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -9.98013e-322 or 3.9999999999999999e-257 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 9.9999999999999996e274

   1. Initial program 0.2

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq -1 \cdot 10^{-321}\right) \land \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 4 \cdot 10^{-257} \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{+275}\right)\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.2
Cost	1241

\[\begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;t \leq -4.1 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+130} \lor \neg \left(t \leq 1.1 \cdot 10^{+213}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 2
Error	17.9
Cost	1240

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot a - t}\\ t_2 := \frac{y}{a - \frac{t}{z}}\\ t_3 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -5.6 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.13 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-43}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-186}:\\ \;\;\;\;\frac{y \cdot z - x}{-t}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+190}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	17.8
Cost	1240

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot a - t}\\ t_2 := \frac{y}{a - \frac{t}{z}}\\ t_3 := y \cdot z - x\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.13 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{t_3}{z \cdot a}\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-187}:\\ \;\;\;\;\frac{t_3}{-t}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+188}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 4
Error	25.3
Cost	1108

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{-a}\\ t_2 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.92 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	17.8
Cost	977

\[\begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -7.3 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{-x}{z \cdot a - t}\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+189} \lor \neg \left(z \leq 2.2 \cdot 10^{+291}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 6
Error	30.3
Cost	912

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{+46}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-35}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	30.2
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{-a}\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{+47}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Error	30.4
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9
Error	29.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 10
Error	42.0
Cost	192

\[\frac{x}{t} \]

Reproduce

herbie shell --seed 2023054 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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