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

Average Error: 2.6 → 0.2

Time: 9.7s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+244}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+273}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+244)
   (* (/ -1.0 t) (/ x z))
   (if (<= (* z t) 2e+273) (/ x (- y (* z t))) (* (/ -1.0 z) (/ x 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) <= -2e+244) {
		tmp = (-1.0 / t) * (x / z);
	} else if ((z * t) <= 2e+273) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / z) * (x / 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) <= (-2d+244)) then
        tmp = ((-1.0d0) / t) * (x / z)
    else if ((z * t) <= 2d+273) then
        tmp = x / (y - (z * t))
    else
        tmp = ((-1.0d0) / z) * (x / 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) <= -2e+244) {
		tmp = (-1.0 / t) * (x / z);
	} else if ((z * t) <= 2e+273) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / z) * (x / 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) <= -2e+244:
		tmp = (-1.0 / t) * (x / z)
	elif (z * t) <= 2e+273:
		tmp = x / (y - (z * t))
	else:
		tmp = (-1.0 / z) * (x / 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) <= -2e+244)
		tmp = Float64(Float64(-1.0 / t) * Float64(x / z));
	elseif (Float64(z * t) <= 2e+273)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(-1.0 / z) * Float64(x / 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) <= -2e+244)
		tmp = (-1.0 / t) * (x / z);
	elseif ((z * t) <= 2e+273)
		tmp = x / (y - (z * t));
	else
		tmp = (-1.0 / z) * (x / 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], -2e+244], N[(N[(-1.0 / t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+273], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] * N[(x / t), $MachinePrecision]), $MachinePrecision]]]

Target

Original	2.6
Target	1.7
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000015e244

   1. Initial program 15.3

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

   2. Taylor expanded in y around 0 15.8

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

   3. Simplified 15.8

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

      [Start] 15.8	\[ -1 \cdot \frac{x}{t \cdot z} \]
      *-commutative [<=] 15.8	\[ -1 \cdot \frac{x}{\color{blue}{z \cdot t}} \]
      associate-*r/ [=>] 15.8	\[ \color{blue}{\frac{-1 \cdot x}{z \cdot t}} \]
      neg-mul-1 [<=] 15.8	\[ \frac{\color{blue}{-x}}{z \cdot t} \]
      *-commutative [=>] 15.8	\[ \frac{-x}{\color{blue}{t \cdot z}} \]

   4. Applied egg-rr 0.8

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

   if -2.00000000000000015e244 < (*.f64 z t) < 1.99999999999999989e273

   1. Initial program 0.1

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

   if 1.99999999999999989e273 < (*.f64 z t)

   1. Initial program 14.1

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

   2. Applied egg-rr 14.5

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

   3. Taylor expanded in y around 0 14.5

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

   4. Simplified 0.6

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

      [Start] 14.5	\[ -1 \cdot \frac{x}{t \cdot z} \]
      *-commutative [<=] 14.5	\[ -1 \cdot \frac{x}{\color{blue}{z \cdot t}} \]
      associate-*r/ [=>] 14.5	\[ \color{blue}{\frac{-1 \cdot x}{z \cdot t}} \]
      neg-mul-1 [<=] 14.5	\[ \frac{\color{blue}{-x}}{z \cdot t} \]
      *-commutative [=>] 14.5	\[ \frac{-x}{\color{blue}{t \cdot z}} \]
      associate-/r* [=>] 0.6	\[ \color{blue}{\frac{\frac{-x}{t}}{z}} \]

   5. Applied egg-rr 0.7

      \[\leadsto \color{blue}{\frac{-1}{z} \cdot \frac{x}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+244}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+273}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.5
Cost	976

\[\begin{array}{l} t_1 := \frac{1}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	17.5
Cost	976

\[\begin{array}{l} t_1 := \frac{1}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{t}}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	17.5
Cost	912

\[\begin{array}{l} t_1 := \frac{1}{\frac{y}{x}}\\ t_2 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	17.5
Cost	912

\[\begin{array}{l} t_1 := \frac{1}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-78}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;\frac{-\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	28.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{+220}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 6
Error	30.1
Cost	320

\[\frac{1}{\frac{y}{x}} \]

Alternative 7
Error	29.9
Cost	192

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

Reproduce

herbie shell --seed 2023060 
(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))))