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

Average Error: 10.8 → 1.7

Time: 15.5s

Precision: binary64

Cost: 3793

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-316} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]

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 (<= t_1 (- INFINITY))
     (* z (/ y (- (* z a) t)))
     (if (or (<= t_1 -1e-316) (and (not (<= t_1 0.0)) (<= t_1 2e+304)))
       t_1
       (/ y (- a (/ t z)))))))

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)) {
		tmp = z * (y / ((z * a) - t));
	} else if ((t_1 <= -1e-316) || (!(t_1 <= 0.0) && (t_1 <= 2e+304))) {
		tmp = t_1;
	} else {
		tmp = y / (a - (t / z));
	}
	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) {
		tmp = z * (y / ((z * a) - t));
	} else if ((t_1 <= -1e-316) || (!(t_1 <= 0.0) && (t_1 <= 2e+304))) {
		tmp = t_1;
	} else {
		tmp = y / (a - (t / z));
	}
	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:
		tmp = z * (y / ((z * a) - t))
	elif (t_1 <= -1e-316) or (not (t_1 <= 0.0) and (t_1 <= 2e+304)):
		tmp = t_1
	else:
		tmp = y / (a - (t / z))
	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))
		tmp = Float64(z * Float64(y / Float64(Float64(z * a) - t)));
	elseif ((t_1 <= -1e-316) || (!(t_1 <= 0.0) && (t_1 <= 2e+304)))
		tmp = t_1;
	else
		tmp = Float64(y / Float64(a - Float64(t / z)));
	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)
		tmp = z * (y / ((z * a) - t));
	elseif ((t_1 <= -1e-316) || (~((t_1 <= 0.0)) && (t_1 <= 2e+304)))
		tmp = t_1;
	else
		tmp = y / (a - (t / z));
	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[LessEqual[t$95$1, (-Infinity)], N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, -1e-316], And[N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision], LessEqual[t$95$1, 2e+304]]], t$95$1, N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	10.8
Target	1.7
Herbie	1.7

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{y \cdot z - x}{z \cdot a - t}} \]
      Proof
      (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 z a) t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 a z)) t)): 0 points increase in error, 17 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 a z) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (*.f64 y z) x)) (*.f64 -1 (-.f64 (*.f64 a z) t)))): 17 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 y z) x))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 17 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 y z) (neg.f64 x)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 17 points decrease in error
      (/.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 y z)) (neg.f64 (neg.f64 x)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (neg.f64 x)) (neg.f64 (*.f64 y z)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> remove-double-neg_binary64 x) (neg.f64 (*.f64 y z))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (*.f64 y z))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 a z) t)))): 17 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 a z) (neg.f64 t))))): 0 points increase in error, 17 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 a z)) (neg.f64 (neg.f64 t))))): 17 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (+.f64 (neg.f64 (*.f64 a z)) (Rewrite=> remove-double-neg_binary64 t))): 0 points increase in error, 17 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 (*.f64 a z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 a z)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 64.0

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

   4. Simplified 0.2

      \[\leadsto \color{blue}{\frac{y}{\frac{a \cdot z - t}{z}}} \]
      Proof
      (/.f64 y (/.f64 (-.f64 (*.f64 a z) t) z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y z) (-.f64 (*.f64 a z) t))): 2 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.3

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -9.999999837e-317 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.9999999999999999e304

   1. Initial program 0.2

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

   if -9.999999837e-317 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0 or 1.9999999999999999e304 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 39.7

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

   2. Simplified 39.7

      \[\leadsto \color{blue}{\frac{y \cdot z - x}{z \cdot a - t}} \]
      Proof
      (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 z a) t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 a z)) t)): 0 points increase in error, 17 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 a z) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (*.f64 y z) x)) (*.f64 -1 (-.f64 (*.f64 a z) t)))): 17 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 y z) x))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 17 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 y z) (neg.f64 x)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 17 points decrease in error
      (/.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 y z)) (neg.f64 (neg.f64 x)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (neg.f64 x)) (neg.f64 (*.f64 y z)))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> remove-double-neg_binary64 x) (neg.f64 (*.f64 y z))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (*.f64 y z))) (*.f64 -1 (-.f64 (*.f64 a z) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 a z) t)))): 17 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (neg.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 a z) (neg.f64 t))))): 0 points increase in error, 17 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 a z)) (neg.f64 (neg.f64 t))))): 17 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (+.f64 (neg.f64 (*.f64 a z)) (Rewrite=> remove-double-neg_binary64 t))): 0 points increase in error, 17 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 (*.f64 a z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (*.f64 y z)) (Rewrite<= sub-neg_binary64 (-.f64 t (*.f64 a z)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 40.2

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

   4. Simplified 30.7

      \[\leadsto \color{blue}{\frac{y}{\frac{a \cdot z - t}{z}}} \]
      Proof
      (/.f64 y (/.f64 (-.f64 (*.f64 a z) t) z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y z) (-.f64 (*.f64 a z) t))): 2 points increase in error, 0 points decrease in error

   5. Taylor expanded in a around 0 7.0

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

   6. Simplified 7.0

      \[\leadsto \frac{y}{\color{blue}{a + \frac{-t}{z}}} \]
      Proof
      (/.f64 y (+.f64 a (/.f64 (neg.f64 t) z))): 0 points increase in error, 0 points decrease in error
      (/.f64 y (+.f64 a (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 t z))))): 3 points increase in error, 0 points decrease in error
      (/.f64 y (+.f64 a (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 t z))))): 0 points increase in error, 3 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1 \cdot 10^{-316} \lor \neg \left(\frac{x - y \cdot z}{t - z \cdot a} \leq 0\right) \land \frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.3
Cost	3405

\[\begin{array}{l} t_1 := z \cdot a - t\\ t_2 := \frac{x}{t_1}\\ t_3 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-316}:\\ \;\;\;\;y \cdot \frac{z}{t_1} - t_2\\ \mathbf{elif}\;t_3 \leq 0 \lor \neg \left(t_3 \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t_1} - t_2\\ \end{array} \]

Alternative 2
Error	2.3
Cost	3021

\[\begin{array}{l} t_1 := z \cdot a - t\\ t_2 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-316}:\\ \;\;\;\;y \cdot \frac{z}{t_1} - \frac{x}{t_1}\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	1.7
Cost	2760

\[\begin{array}{l} t_1 := z \cdot a - t\\ t_2 := \frac{x - y \cdot z}{t - z \cdot a}\\ t_3 := y \cdot z - x\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-154}:\\ \;\;\;\;y \cdot \frac{z}{t_1} - \frac{x}{t_1}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{a \cdot \frac{z}{t_3} - \frac{t}{t_3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]

Alternative 4
Error	15.4
Cost	1232

\[\begin{array}{l} t_1 := \frac{y}{a} - \frac{x}{z \cdot a - t}\\ \mathbf{if}\;a \leq -5.2 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{t} - \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;a \leq 15200000:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	30.4
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-144}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 6
Error	30.4
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.14 \cdot 10^{-144}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	18.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-36} \lor \neg \left(z \leq 700\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \]

Alternative 8
Error	24.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-31}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9
Error	18.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{-30}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+17}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]

Alternative 10
Error	29.7
Cost	457

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-34} \lor \neg \left(z \leq 4.5 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 11
Error	42.7
Cost	192

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

Reproduce

herbie shell --seed 2022343 
(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))))