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

Average Error: 10.5 → 3.0

Time: 7.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x - y \cdot z\\ t_2 := t - z \cdot a\\ t_3 := \frac{t_1}{t_2}\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{-248}:\\ \;\;\;\;\frac{x}{t_2} - z \cdot \frac{y}{t_2}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;{\left(\frac{t}{t_1} - z \cdot \frac{a}{t_1}\right)}^{-1}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+260}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]

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_2 (- t (* z a))) (t_3 (/ t_1 t_2)))
   (if (<= t_3 -5e-248)
     (- (/ x t_2) (* z (/ y t_2)))
     (if (<= t_3 0.0)
       (pow (- (/ t t_1) (* z (/ a t_1))) -1.0)
       (if (<= t_3 5e+260) t_3 (/ y (- a (/ t z))))))))

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);
	double t_2 = t - (z * a);
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -5e-248) {
		tmp = (x / t_2) - (z * (y / t_2));
	} else if (t_3 <= 0.0) {
		tmp = pow(((t / t_1) - (z * (a / t_1))), -1.0);
	} else if (t_3 <= 5e+260) {
		tmp = t_3;
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (x - (y * z)) / (t - (a * z))
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = x - (y * z)
    t_2 = t - (z * a)
    t_3 = t_1 / t_2
    if (t_3 <= (-5d-248)) then
        tmp = (x / t_2) - (z * (y / t_2))
    else if (t_3 <= 0.0d0) then
        tmp = ((t / t_1) - (z * (a / t_1))) ** (-1.0d0)
    else if (t_3 <= 5d+260) then
        tmp = t_3
    else
        tmp = y / (a - (t / z))
    end if
    code = tmp
end function

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);
	double t_2 = t - (z * a);
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -5e-248) {
		tmp = (x / t_2) - (z * (y / t_2));
	} else if (t_3 <= 0.0) {
		tmp = Math.pow(((t / t_1) - (z * (a / t_1))), -1.0);
	} else if (t_3 <= 5e+260) {
		tmp = t_3;
	} else {
		tmp = y / (a - (t / z));
	}
	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_2 = t - (z * a)
	t_3 = t_1 / t_2
	tmp = 0
	if t_3 <= -5e-248:
		tmp = (x / t_2) - (z * (y / t_2))
	elif t_3 <= 0.0:
		tmp = math.pow(((t / t_1) - (z * (a / t_1))), -1.0)
	elif t_3 <= 5e+260:
		tmp = t_3
	else:
		tmp = y / (a - (t / z))
	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(x - Float64(y * z))
	t_2 = Float64(t - Float64(z * a))
	t_3 = Float64(t_1 / t_2)
	tmp = 0.0
	if (t_3 <= -5e-248)
		tmp = Float64(Float64(x / t_2) - Float64(z * Float64(y / t_2)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(t / t_1) - Float64(z * Float64(a / t_1))) ^ -1.0;
	elseif (t_3 <= 5e+260)
		tmp = t_3;
	else
		tmp = Float64(y / Float64(a - Float64(t / z)));
	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_2 = t - (z * a);
	t_3 = t_1 / t_2;
	tmp = 0.0;
	if (t_3 <= -5e-248)
		tmp = (x / t_2) - (z * (y / t_2));
	elseif (t_3 <= 0.0)
		tmp = ((t / t_1) - (z * (a / t_1))) ^ -1.0;
	elseif (t_3 <= 5e+260)
		tmp = t_3;
	else
		tmp = y / (a - (t / z));
	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[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-248], N[(N[(x / t$95$2), $MachinePrecision] - N[(z * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[Power[N[(N[(t / t$95$1), $MachinePrecision] - N[(z * N[(a / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$3, 5e+260], t$95$3, N[(y / N[(a - N[(t / z), $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	10.5
Target	1.6
Herbie	3.0

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -5.0000000000000001e-248

   1. Initial program 5.0

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

   2. Applied egg-rr 5.4

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y \cdot z}\right)}^{2}, -\sqrt[3]{y \cdot z}, x\right)}}{t - a \cdot z} \]

   3. Taylor expanded in x around 0 5.0

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

   4. Simplified 3.5

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

   if -5.0000000000000001e-248 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 21.8

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

   2. Applied egg-rr 22.4

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

   3. Taylor expanded in t around 0 24.1

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

   4. Simplified 5.6

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.9999999999999996e260

   1. Initial program 0.2

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

   2. Applied egg-rr 0.6

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y \cdot z}\right)}^{2}, -\sqrt[3]{y \cdot z}, x\right)}}{t - a \cdot z} \]

   3. Taylor expanded in x around 0 0.2

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

   4. Simplified 0.2

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

   if 4.9999999999999996e260 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 53.3

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

   2. Applied egg-rr 53.4

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

   3. Taylor expanded in t around 0 53.4

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

   4. Simplified 54.9

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

   5. Taylor expanded in y around inf 7.2

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

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5 \cdot 10^{-248}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - z \cdot \frac{y}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;{\left(\frac{t}{x - y \cdot z} - z \cdot \frac{a}{x - y \cdot z}\right)}^{-1}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(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))))