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

Average Accuracy: 95.9% → 97.5%

Time: 6.3s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+180) (/ (/ -1.0 t) (/ z x)) (/ 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) <= -2e+180) {
		tmp = (-1.0 / t) / (z / x);
	} 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) <= (-2d+180)) then
        tmp = ((-1.0d0) / t) / (z / x)
    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) <= -2e+180) {
		tmp = (-1.0 / t) / (z / x);
	} 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) <= -2e+180:
		tmp = (-1.0 / t) / (z / x)
	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) <= -2e+180)
		tmp = Float64(Float64(-1.0 / t) / Float64(z / x));
	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) <= -2e+180)
		tmp = (-1.0 / t) / (z / x);
	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], -2e+180], N[(N[(-1.0 / t), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	95.9%
Target	97.0%
Herbie	97.5%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e180

   1. Initial program 83.3%

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

   2. Applied egg-rr 83.3%

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

   3. Taylor expanded in y around 0 80.0%

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

   4. Simplified 81.7%

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

      [Start] 80.0	\[ \frac{-1}{t \cdot z} \cdot x \]
      associate-/r* [=>] 81.7	\[ \color{blue}{\frac{\frac{-1}{t}}{z}} \cdot x \]

   5. Applied egg-rr 95.7%

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

   if -2e180 < (*.f64 z t)

   1. Initial program 97.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	72.6%
Cost	914

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+78} \lor \neg \left(z \leq -3.7 \cdot 10^{+50} \lor \neg \left(z \leq -6 \cdot 10^{+19}\right) \land z \leq 3.3 \cdot 10^{-128}\right):\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Accuracy	72.5%
Cost	912

\[\begin{array}{l} t_1 := -\frac{\frac{x}{z}}{t}\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-128}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 3
Accuracy	97.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+241}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 4
Accuracy	56.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -7500000000 \lor \neg \left(t \leq 2.8 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Accuracy	52.0%
Cost	192

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

Reproduce

herbie shell --seed 2023129 
(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))))