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

Average Error: 10.1 → 3.1

Time: 15.1s

Precision: binary64

Cost: 3792

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{a \cdot z - t}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-314}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{y \cdot z - x}{a} \cdot \frac{1}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+287}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

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 (/ (- x (* y z)) (- t (* a z)))))
   (if (<= t_1 (- INFINITY))
     (* y (/ z (- (* a z) t)))
     (if (<= t_1 -1e-314)
       t_1
       (if (<= t_1 0.0)
         (* (/ (- (* y z) x) a) (/ 1.0 z))
         (if (<= t_1 1e+287) 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 = (x - (y * z)) / (t - (a * z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (z / ((a * z) - t));
	} else if (t_1 <= -1e-314) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (((y * z) - x) / a) * (1.0 / z);
	} else if (t_1 <= 1e+287) {
		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 = (x - (y * z)) / (t - (a * z));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (z / ((a * z) - t));
	} else if (t_1 <= -1e-314) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (((y * z) - x) / a) * (1.0 / z);
	} else if (t_1 <= 1e+287) {
		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 = (x - (y * z)) / (t - (a * z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * (z / ((a * z) - t))
	elif t_1 <= -1e-314:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (((y * z) - x) / a) * (1.0 / z)
	elif t_1 <= 1e+287:
		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(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / Float64(Float64(a * z) - t)));
	elseif (t_1 <= -1e-314)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(y * z) - x) / a) * Float64(1.0 / z));
	elseif (t_1 <= 1e+287)
		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 = (x - (y * z)) / (t - (a * z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * (z / ((a * z) - t));
	elseif (t_1 <= -1e-314)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (((y * z) - x) / a) * (1.0 / z);
	elseif (t_1 <= 1e+287)
		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[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(z / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-314], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / a), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+287], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Target

Original	10.1
Target	1.6
Herbie	3.1

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

Derivation

1. Split input into 4 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.json-simplify-2 [=>] 64.0	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Applied egg-rr 64.0

      \[\leadsto \color{blue}{\frac{1}{z \cdot a - t} \cdot \left(y \cdot z - x\right)} \]

   4. Taylor expanded in y around inf 64.0

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

   5. Simplified 0.6

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

      [Start] 64.0	\[ \frac{y \cdot z}{a \cdot z - t} \]
      rational.json-simplify-2 [<=] 64.0	\[ \frac{\color{blue}{z \cdot y}}{a \cdot z - t} \]
      rational.json-simplify-2 [<=] 64.0	\[ \frac{z \cdot y}{\color{blue}{z \cdot a} - t} \]
      rational.json-simplify-49 [=>] 0.6	\[ \color{blue}{y \cdot \frac{z}{z \cdot a - t}} \]
      rational.json-simplify-2 [=>] 0.6	\[ y \cdot \frac{z}{\color{blue}{a \cdot z} - t} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -9.9999999996e-315 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.0000000000000001e287

   1. Initial program 0.2

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

   if -9.9999999996e-315 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 25.0

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

   2. Simplified 25.0

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

      [Start] 25.0	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational.json-simplify-2 [=>] 25.0	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Taylor expanded in t around 0 41.1

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

   4. Simplified 41.1

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

      [Start] 41.1	\[ \frac{x - y \cdot z}{-1 \cdot \left(a \cdot z\right)} \]
      rational.json-simplify-2 [<=] 41.1	\[ \frac{x - y \cdot z}{-1 \cdot \color{blue}{\left(z \cdot a\right)}} \]
      rational.json-simplify-43 [=>] 41.1	\[ \frac{x - y \cdot z}{\color{blue}{z \cdot \left(a \cdot -1\right)}} \]
      rational.json-simplify-9 [=>] 41.1	\[ \frac{x - y \cdot z}{z \cdot \color{blue}{\left(-a\right)}} \]

   5. Applied egg-rr 13.1

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

   if 1.0000000000000001e287 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 59.1

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

   2. Simplified 59.1

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

      [Start] 59.1	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      rational.json-simplify-2 [=>] 59.1	\[ \frac{x - y \cdot z}{t - \color{blue}{z \cdot a}} \]

   3. Taylor expanded in z around inf 13.0

      \[\leadsto \color{blue}{\frac{y}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{a \cdot z - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -1 \cdot 10^{-314}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{y \cdot z - x}{a} \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 10^{+287}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.0
Cost	776

\[\begin{array}{l} t_1 := \frac{y + \frac{x}{-z}}{a}\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	29.8
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+107}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{x}{-z}}{a}\\ \mathbf{elif}\;z \leq -5.7 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 3
Error	23.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+109}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 4
Error	29.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5
Error	42.3
Cost	192

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

Reproduce

herbie shell --seed 2023068 
(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))))