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

Average Error: 3.1 → 1.6

Time: 7.8s

Precision: binary64

Cost: 836

\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := y - z \cdot t\\ \mathbf{if}\;t_1 \leq 10^{+308}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (- y (* z t))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- y (* z t)))) (if (<= t_1 1e+308) (/ x t_1) (/ (/ (- x) z) t))))

C

double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = y - (z * t);
	double tmp;
	if (t_1 <= 1e+308) {
		tmp = x / t_1;
	} else {
		tmp = (-x / z) / t;
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y - (z * t)
    if (t_1 <= 1d+308) then
        tmp = x / t_1
    else
        tmp = (-x / z) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = y - (z * t);
	double tmp;
	if (t_1 <= 1e+308) {
		tmp = x / t_1;
	} else {
		tmp = (-x / z) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x / (y - (z * t))

↓

def code(x, y, z, t):
	t_1 = y - (z * t)
	tmp = 0
	if t_1 <= 1e+308:
		tmp = x / t_1
	else:
		tmp = (-x / z) / t
	return tmp

Julia

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end

↓

function code(x, y, z, t)
	t_1 = Float64(y - Float64(z * t))
	tmp = 0.0
	if (t_1 <= 1e+308)
		tmp = Float64(x / t_1);
	else
		tmp = Float64(Float64(Float64(-x) / z) / t);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = y - (z * t);
	tmp = 0.0;
	if (t_1 <= 1e+308)
		tmp = x / t_1;
	else
		tmp = (-x / z) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+308], N[(x / t$95$1), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]]]

Target

Original	3.1
Target	1.8
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (-.f64 y (*.f64 z t)) < 1e308

   1. Initial program 1.7

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

   if 1e308 < (-.f64 y (*.f64 z t))

   1. Initial program 23.4

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

   2. Applied egg-rr 23.4

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

   3. Taylor expanded in y around 0 23.6

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

   4. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\frac{\frac{-x}{z}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y - z \cdot t \leq 10^{+308}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.0229049020860585 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.017572204679063067:\\ \;\;\;\;x \cdot \frac{\frac{-1}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	17.9
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -2.0229049020860585 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.017572204679063067:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	29.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 10^{-169}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	30.0
Cost	192

\[\frac{x}{y} \]

Reproduce

herbie shell --seed 2022295 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

  (/ x (- y (* z t))))