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

Average Error: 10.4 → 5.9

Time: 15.7s

Precision: binary64

Cost: 7753

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot a - t\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{+144} \lor \neg \left(z \leq 2.3 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a} + \frac{\mathsf{fma}\left(\frac{-1}{z}, x, \frac{x}{z}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t_1} - \frac{x}{t_1}\\ \end{array} \]

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 (- (* z a) t)))
   (if (or (<= z -1.02e+144) (not (<= z 2.3e+152)))
     (+ (/ (- y (/ x z)) a) (/ (fma (/ -1.0 z) x (/ x z)) a))
     (- (/ (* z y) t_1) (/ x 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 = (z * a) - t;
	double tmp;
	if ((z <= -1.02e+144) || !(z <= 2.3e+152)) {
		tmp = ((y - (x / z)) / a) + (fma((-1.0 / z), x, (x / z)) / a);
	} else {
		tmp = ((z * y) / t_1) - (x / 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(z * a) - t)
	tmp = 0.0
	if ((z <= -1.02e+144) || !(z <= 2.3e+152))
		tmp = Float64(Float64(Float64(y - Float64(x / z)) / a) + Float64(fma(Float64(-1.0 / z), x, Float64(x / z)) / a));
	else
		tmp = Float64(Float64(Float64(z * y) / t_1) - Float64(x / t_1));
	end
	return 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[(z * a), $MachinePrecision] - t), $MachinePrecision]}, If[Or[LessEqual[z, -1.02e+144], N[Not[LessEqual[z, 2.3e+152]], $MachinePrecision]], N[(N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] + N[(N[(N[(-1.0 / z), $MachinePrecision] * x + N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Target

Original	10.4
Target	1.8
Herbie	5.9

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

Derivation

1. Split input into 2 regimes

2. if z < -1.02000000000000008e144 or 2.29999999999999985e152 < z

   1. Initial program 29.4

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

   2. Simplified 29.4

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

      [Start] 29.4	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 29.4	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 29.4	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 29.4	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 29.4	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 29.4	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 29.4	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 29.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 29.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 29.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 29.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 29.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 29.4	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 29.4	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 29.4	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 29.4	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 29.4	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Applied egg-rr 29.4

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

   4. Taylor expanded in a around inf 12.2

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

   5. Applied egg-rr 12.2

      \[\leadsto \color{blue}{\frac{y - \frac{x}{z}}{a} + {a}^{-1} \cdot \mathsf{fma}\left(-\frac{1}{z}, x, \frac{x}{z}\right)} \]

   6. Simplified 12.2

      \[\leadsto \color{blue}{\frac{y - \frac{x}{z}}{a} + \frac{1}{a} \cdot \mathsf{fma}\left(\frac{-1}{z}, x, \frac{x}{z}\right)} \]
      Proof

      [Start] 12.2	\[ \frac{y - \frac{x}{z}}{a} + {a}^{-1} \cdot \mathsf{fma}\left(-\frac{1}{z}, x, \frac{x}{z}\right) \]
      unpow-1 [=>] 12.2	\[ \frac{y - \frac{x}{z}}{a} + \color{blue}{\frac{1}{a}} \cdot \mathsf{fma}\left(-\frac{1}{z}, x, \frac{x}{z}\right) \]
      distribute-neg-frac [=>] 12.2	\[ \frac{y - \frac{x}{z}}{a} + \frac{1}{a} \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{z}}, x, \frac{x}{z}\right) \]
      metadata-eval [=>] 12.2	\[ \frac{y - \frac{x}{z}}{a} + \frac{1}{a} \cdot \mathsf{fma}\left(\frac{\color{blue}{-1}}{z}, x, \frac{x}{z}\right) \]

   7. Applied egg-rr 12.2

      \[\leadsto \frac{y - \frac{x}{z}}{a} + \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{z}, x, \frac{x}{z}\right)}{a}} \]

   if -1.02000000000000008e144 < z < 2.29999999999999985e152

   1. Initial program 3.7

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

   2. Simplified 3.7

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

      [Start] 3.7	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 3.7	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{t - a \cdot z} \]
      remove-double-neg [<=] 3.7	\[ \frac{\color{blue}{\left(-\left(-x\right)\right)} + \left(-y \cdot z\right)}{t - a \cdot z} \]
      distribute-neg-in [<=] 3.7	\[ \frac{\color{blue}{-\left(\left(-x\right) + y \cdot z\right)}}{t - a \cdot z} \]
      +-commutative [<=] 3.7	\[ \frac{-\color{blue}{\left(y \cdot z + \left(-x\right)\right)}}{t - a \cdot z} \]
      sub-neg [<=] 3.7	\[ \frac{-\color{blue}{\left(y \cdot z - x\right)}}{t - a \cdot z} \]
      neg-mul-1 [=>] 3.7	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{t - a \cdot z} \]
      sub-neg [=>] 3.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      remove-double-neg [<=] 3.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{\left(-\left(-t\right)\right)} + \left(-a \cdot z\right)} \]
      distribute-neg-in [<=] 3.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-\left(\left(-t\right) + a \cdot z\right)}} \]
      +-commutative [<=] 3.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z + \left(-t\right)\right)}} \]
      sub-neg [<=] 3.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{-\color{blue}{\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 3.7	\[ \frac{-1 \cdot \left(y \cdot z - x\right)}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      times-frac [=>] 3.7	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 3.7	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 3.7	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 3.7	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Applied egg-rr 3.7

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+144} \lor \neg \left(z \leq 2.3 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a} + \frac{\mathsf{fma}\left(\frac{-1}{z}, x, \frac{x}{z}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{z \cdot a - t} - \frac{x}{z \cdot a - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.4
Cost	1368

\[\begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -580000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 155000000000:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	18.5
Cost	1368

\[\begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -1750000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-90}:\\ \;\;\;\;\frac{y}{a} - \frac{x}{z \cdot a}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-158}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 280000000000:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	18.6
Cost	1368

\[\begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -165000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-90}:\\ \;\;\;\;\frac{y}{a} - \frac{x}{z \cdot a}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 300000000000:\\ \;\;\;\;\frac{y}{\frac{z \cdot a - t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	6.2
Cost	1353

\[\begin{array}{l} t_1 := z \cdot a - t\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+183} \lor \neg \left(z \leq 2.8 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t_1} - \frac{x}{t_1}\\ \end{array} \]

Alternative 5
Error	19.0
Cost	1242

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -180000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-89} \lor \neg \left(z \leq 6.2 \cdot 10^{-157} \lor \neg \left(z \leq 1.52 \cdot 10^{-134}\right) \land z \leq 210000000000\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \end{array} \]

Alternative 6
Error	31.5
Cost	1044

\[\begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -205000:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-168}:\\ \;\;\;\;\frac{-z}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	31.4
Cost	1044

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -5500:\\ \;\;\;\;\frac{\frac{-x}{a}}{z}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-168}:\\ \;\;\;\;\frac{z \cdot \left(-y\right)}{t}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Error	24.1
Cost	978

\[\begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+105} \lor \neg \left(z \leq 9.8 \cdot 10^{+30}\right) \land \left(z \leq 4 \cdot 10^{+92} \lor \neg \left(z \leq 4.3 \cdot 10^{+155}\right)\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 9
Error	6.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+183} \lor \neg \left(z \leq 2.5 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \]

Alternative 10
Error	31.7
Cost	912

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-168}:\\ \;\;\;\;\frac{-z}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 11
Error	18.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -22500 \lor \neg \left(z \leq 6.7 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 12
Error	30.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -10200000:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 13
Error	42.7
Cost	192

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

Reproduce

herbie shell --seed 2023041 
(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))))