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

Average Error: 10.4 → 5.2

Time: 18.7s

Precision: binary64

Cost: 9804

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-256}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{y}{a} + \left(-\frac{\frac{x}{a} - \frac{y \cdot t}{{a}^{2}}}{z}\right)\\ \mathbf{elif}\;t_1 \leq 10^{+307}:\\ \;\;\;\;\frac{\left(x + x\right) + \left(-\left(y \cdot z + x\right)\right)}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

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 (* a z)))))
   (if (<= t_1 (- INFINITY))
     (/ y a)
     (if (<= t_1 -1e-256)
       t_1
       (if (<= t_1 0.0)
         (+ (/ y a) (- (/ (- (/ x a) (/ (* y t) (pow a 2.0))) z)))
         (if (<= t_1 1e+307)
           (/ (+ (+ x x) (- (+ (* y z) x))) (- t (* z a)))
           (/ y a)))))))

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 - (a * z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y / a;
	} else if (t_1 <= -1e-256) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y / a) + -(((x / a) - ((y * t) / pow(a, 2.0))) / z);
	} else if (t_1 <= 1e+307) {
		tmp = ((x + x) + -((y * z) + x)) / (t - (z * a));
	} else {
		tmp = y / a;
	}
	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 - (a * z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y / a;
	} else if (t_1 <= -1e-256) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y / a) + -(((x / a) - ((y * t) / Math.pow(a, 2.0))) / z);
	} else if (t_1 <= 1e+307) {
		tmp = ((x + x) + -((y * z) + x)) / (t - (z * a));
	} else {
		tmp = y / a;
	}
	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 - (a * z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y / a
	elif t_1 <= -1e-256:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (y / a) + -(((x / a) - ((y * t) / math.pow(a, 2.0))) / z)
	elif t_1 <= 1e+307:
		tmp = ((x + x) + -((y * z) + x)) / (t - (z * a))
	else:
		tmp = y / a
	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(a * z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y / a);
	elseif (t_1 <= -1e-256)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y / a) + Float64(-Float64(Float64(Float64(x / a) - Float64(Float64(y * t) / (a ^ 2.0))) / z)));
	elseif (t_1 <= 1e+307)
		tmp = Float64(Float64(Float64(x + x) + Float64(-Float64(Float64(y * z) + x))) / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / a);
	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 - (a * z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y / a;
	elseif (t_1 <= -1e-256)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (y / a) + -(((x / a) - ((y * t) / (a ^ 2.0))) / z);
	elseif (t_1 <= 1e+307)
		tmp = ((x + x) + -((y * z) + x)) / (t - (z * a));
	else
		tmp = y / a;
	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[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y / a), $MachinePrecision], If[LessEqual[t$95$1, -1e-256], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(y / a), $MachinePrecision] + (-N[(N[(N[(x / a), $MachinePrecision] - N[(N[(y * t), $MachinePrecision] / N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$1, 1e+307], N[(N[(N[(x + x), $MachinePrecision] + (-N[(N[(y * z), $MachinePrecision] + x), $MachinePrecision])), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]

Target

Original	10.4
Target	1.7
Herbie	5.2

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or 9.99999999999999986e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 63.8

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

   2. Simplified 63.8

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

      [Start] 63.8	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational_best-simplify-2 [=>] 63.8	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Taylor expanded in z around inf 14.3

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -9.99999999999999977e-257

   1. Initial program 0.2

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

   if -9.99999999999999977e-257 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 22.8

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

   2. Simplified 22.8

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

      [Start] 22.8	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational_best-simplify-2 [=>] 22.8	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Taylor expanded in z around -inf 21.3

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{y \cdot t}{{a}^{2}}}{z} + \frac{y}{a}} \]

   4. Simplified 21.3

      \[\leadsto \color{blue}{\frac{y}{a} + \left(-\frac{\frac{x}{a} - \frac{y \cdot t}{{a}^{2}}}{z}\right)} \]
      Proof

      [Start] 21.3	\[ -1 \cdot \frac{\frac{x}{a} - \frac{y \cdot t}{{a}^{2}}}{z} + \frac{y}{a} \]
      rational_best-simplify-1 [=>] 21.3	\[ \color{blue}{\frac{y}{a} + -1 \cdot \frac{\frac{x}{a} - \frac{y \cdot t}{{a}^{2}}}{z}} \]
      rational_best-simplify-2 [=>] 21.3	\[ \frac{y}{a} + \color{blue}{\frac{\frac{x}{a} - \frac{y \cdot t}{{a}^{2}}}{z} \cdot -1} \]
      rational_best-simplify-12 [=>] 21.3	\[ \frac{y}{a} + \color{blue}{\left(-\frac{\frac{x}{a} - \frac{y \cdot t}{{a}^{2}}}{z}\right)} \]

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 9.99999999999999986e306

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational_best-simplify-2 [=>] 0.2	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Applied egg-rr 0.2

      \[\leadsto \frac{\color{blue}{\left(x + x\right) + \left(0 - \left(x + y \cdot z\right)\right)}}{t - z \cdot a} \]

   4. Simplified 0.2

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

      [Start] 0.2	\[ \frac{\left(x + x\right) + \left(0 - \left(x + y \cdot z\right)\right)}{t - z \cdot a} \]
      rational_best-simplify-10 [=>] 0.2	\[ \frac{\left(x + x\right) + \color{blue}{\left(-\left(x + y \cdot z\right)\right)}}{t - z \cdot a} \]
      rational_best-simplify-1 [=>] 0.2	\[ \frac{\left(x + x\right) + \left(-\color{blue}{\left(y \cdot z + x\right)}\right)}{t - z \cdot a} \]

3. Recombined 4 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -1 \cdot 10^{-256}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{y}{a} + \left(-\frac{\frac{x}{a} - \frac{y \cdot t}{{a}^{2}}}{z}\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 10^{+307}:\\ \;\;\;\;\frac{\left(x + x\right) + \left(-\left(y \cdot z + x\right)\right)}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.4
Cost	2568

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 10^{+307}:\\ \;\;\;\;\frac{\left(x + x\right) + \left(-\left(y \cdot z + x\right)\right)}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 2
Error	5.4
Cost	2248

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 3
Error	20.6
Cost	1168

\[\begin{array}{l} t_1 := \frac{y}{a} + \left(-\frac{x}{a \cdot z}\right)\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{y \cdot z}{t}\right) + \frac{x}{t}\\ \end{array} \]

Alternative 4
Error	26.0
Cost	1108

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{+223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{+161}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5
Error	26.0
Cost	1108

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+161}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq -1.38 \cdot 10^{-101}:\\ \;\;\;\;\left(-\frac{y \cdot z}{t}\right) + \frac{x}{t}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 6
Error	24.8
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.3 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	30.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Error	42.2
Cost	192

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

Reproduce

herbie shell --seed 2023092 
(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))))