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

Average Error: 16.14% → 6.4%

Time: 13.4s

Precision: binary64

Cost: 2892

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot \frac{y}{z \cdot a - t}\\ t_2 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

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 (* z (/ y (- (* z a) t)))) (t_2 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 2e+292) t_2 (if (<= t_2 INFINITY) t_1 (/ 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 = z * (y / ((z * a) - t));
	double t_2 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+292) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} 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 = z * (y / ((z * a) - t));
	double t_2 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 2e+292) {
		tmp = t_2;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} 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 = z * (y / ((z * a) - t))
	t_2 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 2e+292:
		tmp = t_2
	elif t_2 <= math.inf:
		tmp = t_1
	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(z * Float64(y / Float64(Float64(z * a) - t)))
	t_2 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+292)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = t_1;
	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 = z * (y / ((z * a) - t));
	t_2 = (x - (y * z)) / (t - (z * a));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 2e+292)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = t_1;
	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[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+292], t$95$2, If[LessEqual[t$95$2, Infinity], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Target

Original	16.14%
Target	2.68%
Herbie	6.4%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or 2e292 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 89.43

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

   2. Simplified 89.43

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

      [Start] 89.43	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 89.43	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 89.43	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 89.43	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 89.43	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 89.43	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 89.43	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 89.43	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 89.43	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 89.43	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 89.43	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 89.43	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 89.43	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 89.43	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 89.43	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 89.43	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 89.43	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Taylor expanded in y around inf 96.06

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

   4. Simplified 9.05

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

      [Start] 96.06	\[ \frac{y \cdot z}{a \cdot z - t} \]
      associate-/l* [=>] 6.88	\[ \color{blue}{\frac{y}{\frac{a \cdot z - t}{z}}} \]
      *-commutative [=>] 6.88	\[ \frac{y}{\frac{\color{blue}{z \cdot a} - t}{z}} \]
      associate-/r/ [=>] 9.05	\[ \color{blue}{\frac{y}{z \cdot a - t} \cdot z} \]
      *-commutative [<=] 9.05	\[ \frac{y}{\color{blue}{a \cdot z} - t} \cdot z \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2e292

   1. Initial program 6.52

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

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

   1. Initial program 100

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

   2. Simplified 100

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

      [Start] 100	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 100	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 100	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 100	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 100	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 100	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 100	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 100	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 100	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 100	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 100	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 100	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 100	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 100	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 100	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 100	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 100	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Taylor expanded in z around inf 0

      \[\leadsto \color{blue}{\frac{y}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	30.26%
Cost	1372

\[\begin{array}{l} t_1 := z \cdot a - t\\ t_2 := z \cdot \frac{y}{t_1}\\ t_3 := \frac{y - \frac{x}{z}}{a}\\ t_4 := \frac{-x}{t_1}\\ t_5 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+145}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-150}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{-254}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-100}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+56}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	41.37%
Cost	1240

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+280}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -8 \cdot 10^{+215}:\\ \;\;\;\;\frac{-y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{+145}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{-x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 3
Error	52.32%
Cost	1176

\[\begin{array}{l} t_1 := \frac{-y}{\frac{t}{z}}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+280}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{+214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+145}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.56 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 4
Error	52.17%
Cost	1176

\[\begin{array}{l} t_1 := \frac{-y}{\frac{t}{z}}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+280}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -8 \cdot 10^{+215}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+145}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -7.8 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-20}:\\ \;\;\;\;\frac{y \cdot z}{-t}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5
Error	31.68%
Cost	978

\[\begin{array}{l} \mathbf{if}\;t \leq -1.72 \cdot 10^{+62} \lor \neg \left(t \leq -1 \cdot 10^{+29}\right) \land \left(t \leq -14500000 \lor \neg \left(t \leq 1.6 \cdot 10^{-42}\right)\right):\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 6
Error	49.28%
Cost	912

\[\begin{array}{l} t_1 := \frac{y}{t} \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	48.5%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+145}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Error	65.35%
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))))