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

Average Error: 2.6 → 1.4

Time: 9.4s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+244}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -1e+244) (* (/ -1.0 z) (/ x t)) (/ 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+244) {
		tmp = (-1.0 / z) * (x / t);
	} 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+244)) then
        tmp = ((-1.0d0) / z) * (x / t)
    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+244) {
		tmp = (-1.0 / z) * (x / t);
	} 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+244:
		tmp = (-1.0 / z) * (x / t)
	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+244)
		tmp = Float64(Float64(-1.0 / z) * Float64(x / t));
	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+244)
		tmp = (-1.0 / z) * (x / t);
	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[LessEqual[N[(z * t), $MachinePrecision], -1e+244], N[(N[(-1.0 / z), $MachinePrecision] * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	2.6
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) < -1.00000000000000007e244

   1. Initial program 13.6

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

   2. Applied egg-rr 13.8

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

   3. Taylor expanded in y around 0 14.1

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

   4. Simplified 0.7

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

      [Start] 14.1	\[ -1 \cdot \frac{x}{t \cdot z} \]
      *-commutative [<=] 14.1	\[ -1 \cdot \frac{x}{\color{blue}{z \cdot t}} \]
      associate-*r/ [=>] 14.1	\[ \color{blue}{\frac{-1 \cdot x}{z \cdot t}} \]
      neg-mul-1 [<=] 14.1	\[ \frac{\color{blue}{-x}}{z \cdot t} \]
      *-commutative [=>] 14.1	\[ \frac{-x}{\color{blue}{t \cdot z}} \]
      associate-/r* [=>] 0.7	\[ \color{blue}{\frac{\frac{-x}{t}}{z}} \]

   5. Applied egg-rr 0.7

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

   if -1.00000000000000007e244 < (*.f64 z t)

   1. Initial program 1.4

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+244}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.3
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \]

Alternative 2
Error	18.8
Cost	912

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \]

Alternative 3
Error	18.3
Cost	912

\[\begin{array}{l} \mathbf{if}\;t \leq -7.1 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \]

Alternative 4
Error	16.7
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{-45}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Error	26.9
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+55} \lor \neg \left(t \leq 4.2 \cdot 10^{+197}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6
Error	26.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{+197}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 7
Error	29.6
Cost	192

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

Reproduce

herbie shell --seed 2023046 
(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))))