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

Average Error: 4.7% → 0.72%

Time: 9.5s

Precision: binary64

Cost: 969

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+232} \lor \neg \left(z \cdot t \leq 10^{+201}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -1e+232) (not (<= (* z t) 1e+201)))
   (/ (/ (- x) t) z)
   (/ x (- y (* 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 tmp;
	if (((z * t) <= -1e+232) || !((z * t) <= 1e+201)) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / (y - (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) :: tmp
    if (((z * t) <= (-1d+232)) .or. (.not. ((z * t) <= 1d+201))) then
        tmp = (-x / t) / z
    else
        tmp = x / (y - (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 tmp;
	if (((z * t) <= -1e+232) || !((z * t) <= 1e+201)) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -1e+232) or not ((z * t) <= 1e+201):
		tmp = (-x / t) / z
	else:
		tmp = x / (y - (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)
	tmp = 0.0
	if ((Float64(z * t) <= -1e+232) || !(Float64(z * t) <= 1e+201))
		tmp = Float64(Float64(Float64(-x) / t) / z);
	else
		tmp = Float64(x / Float64(y - Float64(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)
	tmp = 0.0;
	if (((z * t) <= -1e+232) || ~(((z * t) <= 1e+201)))
		tmp = (-x / t) / z;
	else
		tmp = x / (y - (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_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -1e+232], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+201]], $MachinePrecision]], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	4.7%
Target	2.75%
Herbie	0.72%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000006e232 or 1.00000000000000004e201 < (*.f64 z t)

   1. Initial program 20.34

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

   2. Applied egg-rr 20.37

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

   3. Taylor expanded in y around 0 22.51

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

   4. Applied egg-rr 2.71

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

   if -1.00000000000000006e232 < (*.f64 z t) < 1.00000000000000004e201

   1. Initial program 0.15

      \[\frac{x}{y - z \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.72

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+232} \lor \neg \left(z \cdot t \leq 10^{+201}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Error	31.07%
Cost	649

\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-51} \lor \neg \left(t \leq 7.5 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	27.51%
Cost	649

\[\begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-63} \lor \neg \left(t \leq 6.3 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	43%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+255} \lor \neg \left(z \leq 4.4 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	46.97%
Cost	192

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

Reproduce

herbie shell --seed 2023121 
(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))))