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

Average Error: 10.2 → 3.8

Time: 6.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x}{t_1}\\ t_3 := \frac{x - y \cdot z}{t_1}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-111}:\\ \;\;\;\;t_2 - \frac{y}{\frac{t_1}{z}}\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_3\right)\right)\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t_2 - \frac{y \cdot z}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{x}{z \cdot a}\\ \end{array} \]

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 (- t (* z a))) (t_2 (/ x t_1)) (t_3 (/ (- x (* y z)) t_1)))
   (if (<= t_3 -1e-111)
     (- t_2 (/ y (/ t_1 z)))
     (if (<= t_3 -2e-310)
       (log1p (expm1 t_3))
       (if (<= t_3 0.0)
         (/ (- y (/ x z)) a)
         (if (<= t_3 5e+298)
           (- t_2 (/ (* y z) t_1))
           (- (/ y a) (/ x (* z 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 = t - (z * a);
	double t_2 = x / t_1;
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -1e-111) {
		tmp = t_2 - (y / (t_1 / z));
	} else if (t_3 <= -2e-310) {
		tmp = log1p(expm1(t_3));
	} else if (t_3 <= 0.0) {
		tmp = (y - (x / z)) / a;
	} else if (t_3 <= 5e+298) {
		tmp = t_2 - ((y * z) / t_1);
	} else {
		tmp = (y / a) - (x / (z * 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 = t - (z * a);
	double t_2 = x / t_1;
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -1e-111) {
		tmp = t_2 - (y / (t_1 / z));
	} else if (t_3 <= -2e-310) {
		tmp = Math.log1p(Math.expm1(t_3));
	} else if (t_3 <= 0.0) {
		tmp = (y - (x / z)) / a;
	} else if (t_3 <= 5e+298) {
		tmp = t_2 - ((y * z) / t_1);
	} else {
		tmp = (y / a) - (x / (z * 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 = t - (z * a)
	t_2 = x / t_1
	t_3 = (x - (y * z)) / t_1
	tmp = 0
	if t_3 <= -1e-111:
		tmp = t_2 - (y / (t_1 / z))
	elif t_3 <= -2e-310:
		tmp = math.log1p(math.expm1(t_3))
	elif t_3 <= 0.0:
		tmp = (y - (x / z)) / a
	elif t_3 <= 5e+298:
		tmp = t_2 - ((y * z) / t_1)
	else:
		tmp = (y / a) - (x / (z * 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(t - Float64(z * a))
	t_2 = Float64(x / t_1)
	t_3 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_3 <= -1e-111)
		tmp = Float64(t_2 - Float64(y / Float64(t_1 / z)));
	elseif (t_3 <= -2e-310)
		tmp = log1p(expm1(t_3));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (t_3 <= 5e+298)
		tmp = Float64(t_2 - Float64(Float64(y * z) / t_1));
	else
		tmp = Float64(Float64(y / a) - Float64(x / Float64(z * a)));
	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[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-111], N[(t$95$2 - N[(y / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-310], N[Log[1 + N[(Exp[t$95$3] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$3, 5e+298], N[(t$95$2 - N[(N[(y * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] - N[(x / N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.2
Target	1.6
Herbie	3.8

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

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.00000000000000009e-111

   1. Initial program 5.7

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

   2. Taylor expanded in x around 0 5.7

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

   3. Simplified 0.8

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

   if -1.00000000000000009e-111 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.999999999999994e-310

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

   3. Taylor expanded in x around 0 0.2

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

   4. Simplified 7.7

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

   5. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y \cdot z}{t - z \cdot a}\right)\right)} \]

   if -1.999999999999994e-310 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 24.4

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

   2. Applied egg-rr 24.4

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

   3. Taylor expanded in x around 0 24.4

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

   4. Simplified 24.5

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

   5. Taylor expanded in t around 0 27.2

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

   6. Simplified 17.7

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

   if -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5.0000000000000003e298

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

   3. Taylor expanded in x around 0 0.2

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

   4. Simplified 4.3

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

   5. Applied egg-rr 0.2

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

   if 5.0000000000000003e298 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 61.8

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

   2. Applied egg-rr 61.8

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

   3. Taylor expanded in x around 0 61.8

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

   4. Simplified 37.0

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

   5. Taylor expanded in a around inf 12.1

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

   6. Simplified 11.9

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{x}{z \cdot a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t - z \cdot a}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y \cdot z}{t - z \cdot a}\right)\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{x}{z \cdot a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022159 
(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))))