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

Average Error: 2.8 → 0.2

Time: 6.3s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+238}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+238)
   (/ (/ (- x) t) z)
   (if (<= (* z t) 2e+288) (/ 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) <= -2e+238) {
		tmp = (-x / t) / z;
	} else if ((z * t) <= 2e+288) {
		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) <= (-2d+238)) then
        tmp = (-x / t) / z
    else if ((z * t) <= 2d+288) 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) <= -2e+238) {
		tmp = (-x / t) / z;
	} else if ((z * t) <= 2e+288) {
		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) <= -2e+238:
		tmp = (-x / t) / z
	elif (z * t) <= 2e+288:
		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) <= -2e+238)
		tmp = Float64(Float64(Float64(-x) / t) / z);
	elseif (Float64(z * t) <= 2e+288)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / z) / Float64(-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+238)
		tmp = (-x / t) / z;
	elseif ((z * t) <= 2e+288)
		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], -2e+238], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+288], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision]]]

Target

Original	2.8
Target	1.8
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) < -2.0000000000000001e238

   1. Initial program 14.3

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

   2. Taylor expanded in y around 0 15.1

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

   3. Simplified 1.0

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

      [Start] 15.1	\[ -1 \cdot \frac{x}{t \cdot z} \]
      mul-1-neg [=>] 15.1	\[ \color{blue}{-\frac{x}{t \cdot z}} \]
      associate-/r* [=>] 1.0	\[ -\color{blue}{\frac{\frac{x}{t}}{z}} \]
      distribute-neg-frac [=>] 1.0	\[ \color{blue}{\frac{-\frac{x}{t}}{z}} \]

   if -2.0000000000000001e238 < (*.f64 z t) < 2e288

   1. Initial program 0.1

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

   if 2e288 < (*.f64 z t)

   1. Initial program 18.4

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

   2. Taylor expanded in y around 0 18.4

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

   3. Simplified 18.4

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

      [Start] 18.4	\[ -1 \cdot \frac{x}{t \cdot z} \]
      *-commutative [<=] 18.4	\[ -1 \cdot \frac{x}{\color{blue}{z \cdot t}} \]
      associate-*r/ [=>] 18.4	\[ \color{blue}{\frac{-1 \cdot x}{z \cdot t}} \]
      neg-mul-1 [<=] 18.4	\[ \frac{\color{blue}{-x}}{z \cdot t} \]
      *-commutative [=>] 18.4	\[ \frac{-x}{\color{blue}{t \cdot z}} \]

   4. Taylor expanded in x around 0 18.4

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

   5. Simplified 0.3

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

      [Start] 18.4	\[ -1 \cdot \frac{x}{t \cdot z} \]
      metadata-eval [<=] 18.4	\[ \color{blue}{\frac{1}{-1}} \cdot \frac{x}{t \cdot z} \]
      associate-/r* [=>] 0.1	\[ \frac{1}{-1} \cdot \color{blue}{\frac{\frac{x}{t}}{z}} \]
      times-frac [<=] 0.1	\[ \color{blue}{\frac{1 \cdot \frac{x}{t}}{-1 \cdot z}} \]
      *-lft-identity [=>] 0.1	\[ \frac{\color{blue}{\frac{x}{t}}}{-1 \cdot z} \]
      neg-mul-1 [<=] 0.1	\[ \frac{\frac{x}{t}}{\color{blue}{-z}} \]
      associate-/r* [<=] 18.4	\[ \color{blue}{\frac{x}{t \cdot \left(-z\right)}} \]
      distribute-rgt-neg-out [=>] 18.4	\[ \frac{x}{\color{blue}{-t \cdot z}} \]
      *-commutative [=>] 18.4	\[ \frac{x}{-\color{blue}{z \cdot t}} \]
      distribute-rgt-neg-in [=>] 18.4	\[ \frac{x}{\color{blue}{z \cdot \left(-t\right)}} \]
      associate-/r* [=>] 0.3	\[ \color{blue}{\frac{\frac{x}{z}}{-t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+238}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.3
Cost	1177

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ t_2 := \frac{\frac{-x}{t}}{z}\\ \mathbf{if}\;t \leq -108000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 1.12:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+218} \lor \neg \left(t \leq 2.8 \cdot 10^{+260}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	28.5
Cost	850

\[\begin{array}{l} \mathbf{if}\;z \leq -3.75 \cdot 10^{+286} \lor \neg \left(z \leq -7.5 \cdot 10^{+178}\right) \land \left(z \leq -2.9 \cdot 10^{+143} \lor \neg \left(z \leq 2.4 \cdot 10^{+18}\right)\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	17.7
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+45}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	27.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 5
Error	29.9
Cost	192

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

Reproduce

herbie shell --seed 2023056 
(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))))