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

Average Error: 2.7 → 1.4

Time: 13.2s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 2e+303) (/ x (- y (* z t))) (/ (/ (- x) t) z)))

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) <= 2e+303) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-x / t) / z;
	}
	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) <= 2d+303) then
        tmp = x / (y - (z * t))
    else
        tmp = (-x / t) / z
    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) <= 2e+303) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-x / t) / z;
	}
	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) <= 2e+303:
		tmp = x / (y - (z * t))
	else:
		tmp = (-x / t) / z
	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) <= 2e+303)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(Float64(-x) / t) / z);
	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) <= 2e+303)
		tmp = x / (y - (z * t));
	else
		tmp = (-x / t) / z;
	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[LessEqual[N[(z * t), $MachinePrecision], 2e+303], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	2.7
Target	1.8
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 2e303

   1. Initial program 1.4

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

   if 2e303 < (*.f64 z t)

   1. Initial program 20.1

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

   2. Applied egg-rr 20.1

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

   3. Taylor expanded in y around 0 20.2

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

   4. Applied egg-rr 0.2

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.9
Cost	912

\[\begin{array}{l} t_1 := -\frac{x}{t \cdot z}\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 125:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	18.2
Cost	648

\[\begin{array}{l} t_1 := \frac{\frac{-x}{t}}{z}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	17.2
Cost	648

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	30.0
Cost	192

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

Reproduce

herbie shell --seed 2023064 
(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))))