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

Average Error: 10.9 → 2.1

Time: 14.8s

Precision: binary64

Cost: 3792

Math

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

↓

\[\begin{array}{l} t_1 := -\frac{y}{\left(-a\right) + \frac{t}{z}}\\ t_2 := \frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-312}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{+276}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (- (/ y (+ (- a) (/ t z)))))
        (t_2 (/ (- x (* y z)) (- t (* a z)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e-312)
       t_2
       (if (<= t_2 0.0) t_1 (if (<= t_2 1e+276) t_2 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 = -(y / (-a + (t / z)));
	double t_2 = (x - (y * z)) / (t - (a * z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e-312) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+276) {
		tmp = t_2;
	} 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 = -(y / (-a + (t / z)));
	double t_2 = (x - (y * z)) / (t - (a * z));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -1e-312) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+276) {
		tmp = t_2;
	} 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 = -(y / (-a + (t / z)))
	t_2 = (x - (y * z)) / (t - (a * z))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -1e-312:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = t_1
	elif t_2 <= 1e+276:
		tmp = t_2
	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(y / Float64(Float64(-a) + Float64(t / z))))
	t_2 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -1e-312)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 1e+276)
		tmp = t_2;
	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 = -(y / (-a + (t / z)));
	t_2 = (x - (y * z)) / (t - (a * z));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -1e-312)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 1e+276)
		tmp = t_2;
	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[(y / N[((-a) + N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e-312], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 1e+276], t$95$2, t$95$1]]]]]]

Target

Original	10.9
Target	1.7
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or -9.9999999999847e-313 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0 or 1.0000000000000001e276 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 41.1

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

   2. Simplified 41.1

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

      [Start] 41.1	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational.json-simplify-2 [=>] 41.1	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Taylor expanded in x around 0 42.9

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

   4. Simplified 27.3

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

      [Start] 42.9	\[ -1 \cdot \frac{y \cdot z}{t - a \cdot z} \]
      rational.json-simplify-2 [=>] 42.9	\[ \color{blue}{\frac{y \cdot z}{t - a \cdot z} \cdot -1} \]
      rational.json-simplify-9 [=>] 42.9	\[ \color{blue}{-\frac{y \cdot z}{t - a \cdot z}} \]
      rational.json-simplify-2 [=>] 42.9	\[ -\frac{\color{blue}{z \cdot y}}{t - a \cdot z} \]
      rational.json-simplify-2 [<=] 42.9	\[ -\frac{z \cdot y}{t - \color{blue}{z \cdot a}} \]
      rational.json-simplify-49 [=>] 27.3	\[ -\color{blue}{y \cdot \frac{z}{t - z \cdot a}} \]

   5. Applied egg-rr 27.3

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

   6. Taylor expanded in t around 0 7.3

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

   7. Simplified 7.3

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

      [Start] 7.3	\[ -\frac{y}{\frac{t}{z} + -1 \cdot a} \]
      rational.json-simplify-1 [=>] 7.3	\[ -\frac{y}{\color{blue}{-1 \cdot a + \frac{t}{z}}} \]
      rational.json-simplify-2 [=>] 7.3	\[ -\frac{y}{\color{blue}{a \cdot -1} + \frac{t}{z}} \]
      rational.json-simplify-9 [=>] 7.3	\[ -\frac{y}{\color{blue}{\left(-a\right)} + \frac{t}{z}} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -9.9999999999847e-313 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.0000000000000001e276

   1. Initial program 0.2

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -\infty:\\ \;\;\;\;-\frac{y}{\left(-a\right) + \frac{t}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -1 \cdot 10^{-312}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 0:\\ \;\;\;\;-\frac{y}{\left(-a\right) + \frac{t}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 10^{+276}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-\frac{y}{\left(-a\right) + \frac{t}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	25.9
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-18}:\\ \;\;\;\;-y \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-201}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+85}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 2
Error	25.7
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+205}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-18}:\\ \;\;\;\;-y \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 3
Error	18.2
Cost	844

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -650:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 6400:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	18.0
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-19}:\\ \;\;\;\;-\frac{y}{\left(-a\right) + \frac{t}{z}}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 0.94:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 5
Error	31.8
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+205}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-19}:\\ \;\;\;\;-y \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 5500:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 6
Error	29.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+56}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 490:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	42.5
Cost	192

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

Reproduce

herbie shell --seed 2023073 
(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))))