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

Average Error: 10.3 → 5.7

Time: 13.3s

Precision: binary64

Cost: 2248

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+267}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{z} - y}{a}\\ \end{array} \]

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 5e+267) t_1 (- (/ (- (/ x z) 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 <= 5e+267) {
		tmp = t_1;
	} else {
		tmp = -(((x / z) - 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 <= 5e+267) {
		tmp = t_1;
	} else {
		tmp = -(((x / z) - 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 <= 5e+267:
		tmp = t_1
	else:
		tmp = -(((x / z) - 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 <= 5e+267)
		tmp = t_1;
	else
		tmp = Float64(-Float64(Float64(Float64(x / z) - 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 <= 5e+267)
		tmp = t_1;
	else
		tmp = -(((x / z) - 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, 5e+267], t$95$1, (-N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / a), $MachinePrecision])]]]

Target

Original	10.3
Target	1.7
Herbie	5.7

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

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

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

      [Start] 64.0	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 64.0	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Taylor expanded in z around inf 22.4

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.9999999999999999e267

   1. Initial program 4.2

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

   if 4.9999999999999999e267 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 53.2

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

   2. Simplified 53.2

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

      [Start] 53.2	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 53.2	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Taylor expanded in x around 0 53.2

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

   4. Simplified 53.2

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

      [Start] 53.2	\[ \frac{x}{t - a \cdot z} + -1 \cdot \frac{y \cdot z}{t - a \cdot z} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 53.2	\[ \color{blue}{-1 \cdot \frac{y \cdot z}{t - a \cdot z} + \frac{x}{t - a \cdot z}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 53.2	\[ \color{blue}{\frac{y \cdot z}{t - a \cdot z} \cdot -1} + \frac{x}{t - a \cdot z} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 53.2	\[ \color{blue}{\left(-\frac{y \cdot z}{t - a \cdot z}\right)} + \frac{x}{t - a \cdot z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 53.2	\[ \left(-\frac{y \cdot z}{t - \color{blue}{z \cdot a}}\right) + \frac{x}{t - a \cdot z} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 53.2	\[ \left(-\frac{y \cdot z}{t - z \cdot a}\right) + \frac{x}{t - \color{blue}{z \cdot a}} \]

   5. Taylor expanded in a around -inf 15.2

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

   6. Simplified 15.2

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

      [Start] 15.2	\[ -1 \cdot \frac{\frac{x}{z} - y}{a} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 15.2	\[ \color{blue}{\frac{\frac{x}{z} - y}{a} \cdot -1} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 15.2	\[ \color{blue}{-\frac{\frac{x}{z} - y}{a}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 5 \cdot 10^{+267}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{z} - y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	19.7
Cost	1496

\[\begin{array}{l} t_1 := \frac{x}{t} + z \cdot \left(-1 \cdot \frac{y}{t}\right)\\ t_2 := -\frac{\frac{x}{z} - y}{a}\\ t_3 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-150}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-254}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 1.48 \cdot 10^{-148}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	20.1
Cost	1304

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ t_2 := -\frac{\frac{x}{z} - y}{a}\\ t_3 := t - z \cdot a\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.75 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{t_3}\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{+29}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{t_3}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	20.3
Cost	1040

\[\begin{array}{l} t_1 := -\frac{\frac{x}{z} - y}{a}\\ t_2 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -470000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	23.1
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+145}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-152}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5
Error	31.3
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+145}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 6
Error	22.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+145}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	31.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+145}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Error	41.8
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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))))