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

Average Error: 10.0 → 4.1

Time: 7.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t_1}\\ \mathbf{if}\;t_2 \leq -4 \cdot 10^{-314}:\\ \;\;\;\;\frac{x}{t_1} - \frac{y}{\frac{t_1}{z}}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{y}{a} + \frac{t \cdot \frac{y}{a \cdot a} - \frac{x}{a}}{z}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

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 (* y z)) t_1)))
   (if (<= t_2 -4e-314)
     (- (/ x t_1) (/ y (/ t_1 z)))
     (if (<= t_2 0.0)
       (+ (/ y a) (/ (- (* t (/ y (* a a))) (/ x a)) z))
       (if (<= t_2 2e+305) t_2 (/ (- y (/ 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 - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -4e-314) {
		tmp = (x / t_1) - (y / (t_1 / z));
	} else if (t_2 <= 0.0) {
		tmp = (y / a) + (((t * (y / (a * a))) - (x / a)) / z);
	} else if (t_2 <= 2e+305) {
		tmp = t_2;
	} else {
		tmp = (y - (x / z)) / a;
	}
	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) :: tmp
    t_1 = t - (z * a)
    t_2 = (x - (y * z)) / t_1
    if (t_2 <= (-4d-314)) then
        tmp = (x / t_1) - (y / (t_1 / z))
    else if (t_2 <= 0.0d0) then
        tmp = (y / a) + (((t * (y / (a * a))) - (x / a)) / z)
    else if (t_2 <= 2d+305) then
        tmp = t_2
    else
        tmp = (y - (x / z)) / a
    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 - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -4e-314) {
		tmp = (x / t_1) - (y / (t_1 / z));
	} else if (t_2 <= 0.0) {
		tmp = (y / a) + (((t * (y / (a * a))) - (x / a)) / z);
	} else if (t_2 <= 2e+305) {
		tmp = t_2;
	} else {
		tmp = (y - (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 - (y * z)) / t_1
	tmp = 0
	if t_2 <= -4e-314:
		tmp = (x / t_1) - (y / (t_1 / z))
	elif t_2 <= 0.0:
		tmp = (y / a) + (((t * (y / (a * a))) - (x / a)) / z)
	elif t_2 <= 2e+305:
		tmp = t_2
	else:
		tmp = (y - (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(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= -4e-314)
		tmp = Float64(Float64(x / t_1) - Float64(y / Float64(t_1 / z)));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(y / a) + Float64(Float64(Float64(t * Float64(y / Float64(a * a))) - Float64(x / a)) / z));
	elseif (t_2 <= 2e+305)
		tmp = t_2;
	else
		tmp = Float64(Float64(y - Float64(x / z)) / 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 = t - (z * a);
	t_2 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_2 <= -4e-314)
		tmp = (x / t_1) - (y / (t_1 / z));
	elseif (t_2 <= 0.0)
		tmp = (y / a) + (((t * (y / (a * a))) - (x / a)) / z);
	elseif (t_2 <= 2e+305)
		tmp = t_2;
	else
		tmp = (y - (x / z)) / 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[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-314], N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(y / a), $MachinePrecision] + N[(N[(N[(t * N[(y / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+305], t$95$2, N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]]

Target

Original	10.0
Target	1.6
Herbie	4.1

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -3.9999999999e-314

   1. Initial program 4.8

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

   2. Taylor expanded in x around 0 4.8

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

   3. Simplified 4.3

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

   4. Applied egg-rr 1.9

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

   if -3.9999999999e-314 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 23.0

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

   2. Taylor expanded in z around inf 28.2

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{a \cdot z} + \frac{y}{a}\right) - -1 \cdot \frac{y \cdot t}{{a}^{2} \cdot z}} \]

   3. Simplified 16.8

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

   if -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.9999999999999999e305

   1. Initial program 0.2

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

   2. Taylor expanded in x around 0 0.2

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

   3. Simplified 4.2

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

   4. Applied egg-rr 0.2

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

   if 1.9999999999999999e305 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 63.4

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

   2. Taylor expanded in x around 0 63.4

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

   3. Simplified 38.2

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

   4. Applied egg-rr 38.0

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

   5. Taylor expanded in t around 0 10.6

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

   6. Simplified 10.6

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -4 \cdot 10^{-314}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t - z \cdot a}{z}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y}{a} + \frac{t \cdot \frac{y}{a \cdot a} - \frac{x}{a}}{z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(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))))