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

Average Error: 10.8 → 2.3

Time: 5.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x}{t_1}\\ t_3 := t_2 - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z \leq -1.22074817042731 \cdot 10^{+59}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 3.163945609906825 \cdot 10^{-168}:\\ \;\;\;\;t_2 - {\left(\frac{t_1}{z \cdot y}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 (- t_2 (/ y (- (/ t z) a)))))
   (if (<= z -1.22074817042731e+59)
     t_3
     (if (<= z 3.163945609906825e-168)
       (- t_2 (pow (/ t_1 (* z y)) -1.0))
       t_3))))

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 = t_2 - (y / ((t / z) - a));
	double tmp;
	if (z <= -1.22074817042731e+59) {
		tmp = t_3;
	} else if (z <= 3.163945609906825e-168) {
		tmp = t_2 - pow((t_1 / (z * y)), -1.0);
	} else {
		tmp = t_3;
	}
	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 = t - (z * a)
    t_2 = x / t_1
    t_3 = t_2 - (y / ((t / z) - a))
    if (z <= (-1.22074817042731d+59)) then
        tmp = t_3
    else if (z <= 3.163945609906825d-168) then
        tmp = t_2 - ((t_1 / (z * y)) ** (-1.0d0))
    else
        tmp = t_3
    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 = t - (z * a);
	double t_2 = x / t_1;
	double t_3 = t_2 - (y / ((t / z) - a));
	double tmp;
	if (z <= -1.22074817042731e+59) {
		tmp = t_3;
	} else if (z <= 3.163945609906825e-168) {
		tmp = t_2 - Math.pow((t_1 / (z * y)), -1.0);
	} else {
		tmp = t_3;
	}
	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 = t_2 - (y / ((t / z) - a))
	tmp = 0
	if z <= -1.22074817042731e+59:
		tmp = t_3
	elif z <= 3.163945609906825e-168:
		tmp = t_2 - math.pow((t_1 / (z * y)), -1.0)
	else:
		tmp = t_3
	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(t_2 - Float64(y / Float64(Float64(t / z) - a)))
	tmp = 0.0
	if (z <= -1.22074817042731e+59)
		tmp = t_3;
	elseif (z <= 3.163945609906825e-168)
		tmp = Float64(t_2 - (Float64(t_1 / Float64(z * y)) ^ -1.0));
	else
		tmp = t_3;
	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 = t - (z * a);
	t_2 = x / t_1;
	t_3 = t_2 - (y / ((t / z) - a));
	tmp = 0.0;
	if (z <= -1.22074817042731e+59)
		tmp = t_3;
	elseif (z <= 3.163945609906825e-168)
		tmp = t_2 - ((t_1 / (z * y)) ^ -1.0);
	else
		tmp = t_3;
	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[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.22074817042731e+59], t$95$3, If[LessEqual[z, 3.163945609906825e-168], N[(t$95$2 - N[Power[N[(t$95$1 / N[(z * y), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.8
Target	1.9
Herbie	2.3

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

Derivation

1. Split input into 2 regimes

2. if z < -1.22074817042731008e59 or 3.163945609906825e-168 < z

   1. Initial program 17.8

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

   2. Applied egg-rr 17.8

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

   3. Applied egg-rr 12.2

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

   4. Applied egg-rr 12.3

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

   5. Taylor expanded in y around 0 17.8

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

   6. Simplified 3.3

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

   if -1.22074817042731008e59 < z < 3.163945609906825e-168

   1. Initial program 0.6

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

   2. Applied egg-rr 0.6

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

   3. Applied egg-rr 0.8

      \[\leadsto \frac{x}{t - z \cdot a} + \left(-\color{blue}{{\left(\frac{t - z \cdot a}{y \cdot z}\right)}^{-1}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22074817042731 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \leq 3.163945609906825 \cdot 10^{-168}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - {\left(\frac{t - z \cdot a}{z \cdot y}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))))