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

Average Error: 4.25% → 2.81%

Time: 9.4s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+179}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 1e+179) (/ x (- y (* z t))) (/ (/ (- 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 tmp;
	if ((z * t) <= 1e+179) {
		tmp = x / (y - (z * t));
	} 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) :: tmp
    if ((z * t) <= 1d+179) then
        tmp = x / (y - (z * t))
    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 tmp;
	if ((z * t) <= 1e+179) {
		tmp = x / (y - (z * t));
	} 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):
	tmp = 0
	if (z * t) <= 1e+179:
		tmp = x / (y - (z * t))
	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)
	tmp = 0.0
	if (Float64(z * t) <= 1e+179)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	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)
	tmp = 0.0;
	if ((z * t) <= 1e+179)
		tmp = x / (y - (z * t));
	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_] := If[LessEqual[N[(z * t), $MachinePrecision], 1e+179], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]]

Target

Original	4.25%
Target	2.76%
Herbie	2.81%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 9.9999999999999998e178

   1. Initial program 2.55

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

   if 9.9999999999999998e178 < (*.f64 z t)

   1. Initial program 15.25

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

   2. Applied egg-rr 15.92

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

   3. Taylor expanded in y around 0 18.71

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

   4. Simplified 4.78

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

      [Start] 18.71	\[ -1 \cdot \frac{x}{t \cdot z} \]
      *-commutative [<=] 18.71	\[ -1 \cdot \frac{x}{\color{blue}{z \cdot t}} \]
      associate-*r/ [=>] 18.71	\[ \color{blue}{\frac{-1 \cdot x}{z \cdot t}} \]
      neg-mul-1 [<=] 18.71	\[ \frac{\color{blue}{-x}}{z \cdot t} \]
      *-commutative [=>] 18.71	\[ \frac{-x}{\color{blue}{t \cdot z}} \]
      associate-/r* [=>] 4.78	\[ \color{blue}{\frac{\frac{-x}{t}}{z}} \]

   5. Taylor expanded in x around 0 18.71

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

   6. Simplified 4.48

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

      [Start] 18.71	\[ -1 \cdot \frac{x}{t \cdot z} \]
      associate-*r/ [=>] 18.71	\[ \color{blue}{\frac{-1 \cdot x}{t \cdot z}} \]
      associate-/l/ [<=] 4.48	\[ \color{blue}{\frac{\frac{-1 \cdot x}{z}}{t}} \]
      mul-1-neg [=>] 4.48	\[ \frac{\frac{\color{blue}{-x}}{z}}{t} \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.81

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+179}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.39%
Cost	1426

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+64} \lor \neg \left(z \cdot t \leq 4 \cdot 10^{-132}\right) \land \left(z \cdot t \leq 10^{-79} \lor \neg \left(z \cdot t \leq 10^{+31}\right)\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	21.49%
Cost	1424

\[\begin{array}{l} t_1 := -\frac{\frac{x}{t}}{z}\\ \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{-79}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	21.51%
Cost	1424

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{-79}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 4
Error	21.51%
Cost	1424

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{-79}:\\ \;\;\;\;x \cdot \frac{-1}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 5
Error	21.51%
Cost	1424

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{-79}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 10^{+31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 6
Error	36.63%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+130} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7
Error	46.98%
Cost	192

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

Reproduce

herbie shell --seed 2023089 
(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))))