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

Average Error: 10.5 → 1.9

Time: 15.4s

Precision: binary64

Cost: 3794

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq -4 \cdot 10^{-295}\right) \land \left(t_1 \leq 0 \lor \neg \left(t_1 \leq 5 \cdot 10^{+285}\right)\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 -4e-295))
                (or (<= t_1 0.0) (not (<= t_1 5e+285)))))
     (/ 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 <= -4e-295) && ((t_1 <= 0.0) || !(t_1 <= 5e+285)))) {
		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 <= -4e-295) && ((t_1 <= 0.0) || !(t_1 <= 5e+285)))) {
		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 <= -4e-295) and ((t_1 <= 0.0) or not (t_1 <= 5e+285))):
		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 <= -4e-295) && ((t_1 <= 0.0) || !(t_1 <= 5e+285))))
		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 <= -4e-295)) && ((t_1 <= 0.0) || ~((t_1 <= 5e+285)))))
		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, -4e-295]], $MachinePrecision], Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 5e+285]], $MachinePrecision]]]], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Target

Original	10.5
Target	1.6
Herbie	1.9

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or -4.00000000000000024e-295 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0 or 5.00000000000000016e285 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 39.4

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

   2. Simplified 39.4

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

      [Start] 39.4	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 39.4	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 39.4	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 39.4	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 39.4	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 39.4	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 39.4	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 39.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 39.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 39.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 39.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 39.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 39.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 39.4	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 39.4	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 39.4	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 39.4	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Taylor expanded in y around inf 41.1

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

   4. Simplified 26.7

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

      [Start] 41.1	\[ \frac{y \cdot z}{a \cdot z - t} \]
      associate-/l* [=>] 26.7	\[ \color{blue}{\frac{y}{\frac{a \cdot z - t}{z}}} \]

   5. Taylor expanded in a around 0 6.8

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

   6. Simplified 6.8

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

      [Start] 6.8	\[ \frac{y}{a + -1 \cdot \frac{t}{z}} \]
      mul-1-neg [=>] 6.8	\[ \frac{y}{a + \color{blue}{\left(-\frac{t}{z}\right)}} \]
      distribute-neg-frac [=>] 6.8	\[ \frac{y}{a + \color{blue}{\frac{-t}{z}}} \]

   7. Taylor expanded in y around 0 6.8

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

   8. Simplified 6.8

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

      [Start] 6.8	\[ \frac{y}{a + -1 \cdot \frac{t}{z}} \]
      mul-1-neg [=>] 6.8	\[ \frac{y}{a + \color{blue}{\left(-\frac{t}{z}\right)}} \]
      unsub-neg [=>] 6.8	\[ \frac{y}{\color{blue}{a - \frac{t}{z}}} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.00000000000000024e-295 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5.00000000000000016e285

   1. Initial program 0.2

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq -4 \cdot 10^{-295}\right) \land \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 0 \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{+285}\right)\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \end{array} \]

Alternatives

Alternative 1
Error	38.9
Cost	1836

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot a}\\ t_2 := \frac{y}{t} \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -2.35 \cdot 10^{+198}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+72}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+138}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+223}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 2
Error	20.6
Cost	1572

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+57} \lor \neg \left(y \leq -5.4 \cdot 10^{-45}\right) \land \left(y \leq -4.2 \cdot 10^{-96} \lor \neg \left(y \leq 3.9 \cdot 10^{-165}\right) \land \left(y \leq 6.1 \cdot 10^{-142} \lor \neg \left(y \leq 1.05 \cdot 10^{-71}\right) \land \left(y \leq 0.000112 \lor \neg \left(y \leq 2.95 \cdot 10^{+34}\right)\right)\right)\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot a - t}\\ \end{array} \]

Alternative 3
Error	27.8
Cost	1108

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot a}\\ t_2 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{-99}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-204}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	30.3
Cost	1044

\[\begin{array}{l} t_1 := -\frac{y \cdot z}{t}\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{+111}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5200000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5
Error	36.3
Cost	1044

\[\begin{array}{l} t_1 := \frac{y}{t} \cdot \left(-z\right)\\ \mathbf{if}\;a \leq -9.6 \cdot 10^{+142}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq -3.8 \cdot 10^{-237}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;a \leq 5.3 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-56}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 6
Error	18.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -3.75 \cdot 10^{-115} \lor \neg \left(z \leq 32000000000\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \]

Alternative 7
Error	30.4
Cost	457

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+108} \lor \neg \left(z \leq 38000000000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 8
Error	42.4
Cost	192

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

Reproduce

herbie shell --seed 2023067 
(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))))