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

Average Accuracy: 95.7% → 99.1%

Time: 10.2s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+248}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{\frac{x}{t}}}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+262}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -5e+248)
   (/ 1.0 (- (/ y x) (/ z (/ x t))))
   (if (<= (* z t) 2e+262) (/ 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) <= -5e+248) {
		tmp = 1.0 / ((y / x) - (z / (x / t)));
	} else if ((z * t) <= 2e+262) {
		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) <= (-5d+248)) then
        tmp = 1.0d0 / ((y / x) - (z / (x / t)))
    else if ((z * t) <= 2d+262) 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) <= -5e+248) {
		tmp = 1.0 / ((y / x) - (z / (x / t)));
	} else if ((z * t) <= 2e+262) {
		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) <= -5e+248:
		tmp = 1.0 / ((y / x) - (z / (x / t)))
	elif (z * t) <= 2e+262:
		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) <= -5e+248)
		tmp = Float64(1.0 / Float64(Float64(y / x) - Float64(z / Float64(x / t))));
	elseif (Float64(z * t) <= 2e+262)
		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) <= -5e+248)
		tmp = 1.0 / ((y / x) - (z / (x / t)));
	elseif ((z * t) <= 2e+262)
		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], -5e+248], N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(z / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+262], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	95.7%
Target	97.8%
Herbie	99.1%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4.9999999999999996e248

   1. Initial program 79.2%

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

   2. Applied egg-rr 79.2%

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

      [Start] 79.2	\[ \frac{x}{y - z \cdot t} \]
      div-inv [=>] 79.2	\[ \color{blue}{x \cdot \frac{1}{y - z \cdot t}} \]
      *-commutative [=>] 79.2	\[ \color{blue}{\frac{1}{y - z \cdot t} \cdot x} \]

   3. Applied egg-rr 78.8%

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

      [Start] 79.2	\[ \frac{1}{y - z \cdot t} \cdot x \]
      associate-/r/ [<=] 78.8	\[ \color{blue}{\frac{1}{\frac{y - z \cdot t}{x}}} \]

   4. Applied egg-rr 92.6%

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

      [Start] 78.8	\[ \frac{1}{\frac{y - z \cdot t}{x}} \]
      div-sub [=>] 72.5	\[ \frac{1}{\color{blue}{\frac{y}{x} - \frac{z \cdot t}{x}}} \]
      associate-/l* [=>] 92.6	\[ \frac{1}{\frac{y}{x} - \color{blue}{\frac{z}{\frac{x}{t}}}} \]

   if -4.9999999999999996e248 < (*.f64 z t) < 2e262

   1. Initial program 99.9%

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

   if 2e262 < (*.f64 z t)

   1. Initial program 75.8%

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

   2. Applied egg-rr 75.6%

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

      [Start] 75.8	\[ \frac{x}{y - z \cdot t} \]
      clear-num [=>] 75.6	\[ \color{blue}{\frac{1}{\frac{y - z \cdot t}{x}}} \]
      inv-pow [=>] 75.6	\[ \color{blue}{{\left(\frac{y - z \cdot t}{x}\right)}^{-1}} \]

   3. Taylor expanded in y around 0 75.1%

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

   4. Simplified 99.1%

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

      [Start] 75.1	\[ -1 \cdot \frac{x}{t \cdot z} \]
      *-commutative [<=] 75.1	\[ -1 \cdot \frac{x}{\color{blue}{z \cdot t}} \]
      associate-*r/ [=>] 75.1	\[ \color{blue}{\frac{-1 \cdot x}{z \cdot t}} \]
      neg-mul-1 [<=] 75.1	\[ \frac{\color{blue}{-x}}{z \cdot t} \]
      *-commutative [=>] 75.1	\[ \frac{-x}{\color{blue}{t \cdot z}} \]
      associate-/r* [=>] 99.1	\[ \color{blue}{\frac{\frac{-x}{t}}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+248}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{\frac{x}{t}}}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+262}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.3%
Cost	1441

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{-t}\\ t_2 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-124}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 34000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+83} \lor \neg \left(y \leq 6 \cdot 10^{+115}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	69.3%
Cost	1441

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{-t}\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+81} \lor \neg \left(y \leq 6 \cdot 10^{+115}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	1288

\[\begin{array}{l} t_1 := y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+306}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-z}{\frac{x}{t}}}\\ \end{array} \]

Alternative 4
Accuracy	70.7%
Cost	1244

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;y \leq -4 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+278}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Accuracy	99.7%
Cost	1224

\[\begin{array}{l} t_1 := y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+306}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]

Alternative 6
Accuracy	59.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+190} \lor \neg \left(z \leq 7 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7
Accuracy	54.2%
Cost	192

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

Reproduce

herbie shell --seed 2023130 
(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))))