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

Average Accuracy: 95.8% → 99.2%

Time: 9.9s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{-1}{t} \cdot x}{z}\\ \mathbf{elif}\;z \cdot t \leq 10^{+182}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(-\frac{t}{\frac{x}{z}}\right)}^{-1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -5e+261)
   (/ (* (/ -1.0 t) x) z)
   (if (<= (* z t) 1e+182) (/ x (- y (* z t))) (pow (- (/ t (/ x z))) -1.0))))

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+261) {
		tmp = ((-1.0 / t) * x) / z;
	} else if ((z * t) <= 1e+182) {
		tmp = x / (y - (z * t));
	} else {
		tmp = pow(-(t / (x / z)), -1.0);
	}
	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+261)) then
        tmp = (((-1.0d0) / t) * x) / z
    else if ((z * t) <= 1d+182) then
        tmp = x / (y - (z * t))
    else
        tmp = -(t / (x / z)) ** (-1.0d0)
    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+261) {
		tmp = ((-1.0 / t) * x) / z;
	} else if ((z * t) <= 1e+182) {
		tmp = x / (y - (z * t));
	} else {
		tmp = Math.pow(-(t / (x / z)), -1.0);
	}
	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+261:
		tmp = ((-1.0 / t) * x) / z
	elif (z * t) <= 1e+182:
		tmp = x / (y - (z * t))
	else:
		tmp = math.pow(-(t / (x / z)), -1.0)
	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+261)
		tmp = Float64(Float64(Float64(-1.0 / t) * x) / z);
	elseif (Float64(z * t) <= 1e+182)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(-Float64(t / Float64(x / z))) ^ -1.0;
	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+261)
		tmp = ((-1.0 / t) * x) / z;
	elseif ((z * t) <= 1e+182)
		tmp = x / (y - (z * t));
	else
		tmp = -(t / (x / z)) ^ -1.0;
	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+261], N[(N[(N[(-1.0 / t), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+182], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[(-N[(t / N[(x / z), $MachinePrecision]), $MachinePrecision]), -1.0], $MachinePrecision]]]

Target

Original	95.8%
Target	97.1%
Herbie	99.2%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000001e261

   1. Initial program 74.6%

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

   2. Applied egg-rr 74.6%

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

      [Start] 74.6	\[ \frac{x}{y - z \cdot t} \]
      clear-num [=>] 74.1	\[ \color{blue}{\frac{1}{\frac{y - z \cdot t}{x}}} \]
      associate-/r/ [=>] 74.6	\[ \color{blue}{\frac{1}{y - z \cdot t} \cdot x} \]

   3. Taylor expanded in y around 0 74.2%

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

   4. Applied egg-rr 99.3%

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

      [Start] 74.2	\[ \frac{-1}{t \cdot z} \cdot x \]
      associate-/r* [=>] 76.1	\[ \color{blue}{\frac{\frac{-1}{t}}{z}} \cdot x \]
      associate-*l/ [=>] 99.3	\[ \color{blue}{\frac{\frac{-1}{t} \cdot x}{z}} \]

   if -5.0000000000000001e261 < (*.f64 z t) < 1.0000000000000001e182

   1. Initial program 99.9%

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

   if 1.0000000000000001e182 < (*.f64 z t)

   1. Initial program 84.4%

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

   2. Applied egg-rr 84.1%

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

      [Start] 84.4	\[ \frac{x}{y - z \cdot t} \]
      clear-num [=>] 84.1	\[ \color{blue}{\frac{1}{\frac{y - z \cdot t}{x}}} \]
      inv-pow [=>] 84.1	\[ \color{blue}{{\left(\frac{y - z \cdot t}{x}\right)}^{-1}} \]

   3. Taylor expanded in y around 0 80.6%

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

   4. Simplified 95.1%

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

      [Start] 80.6	\[ {\left(-1 \cdot \frac{t \cdot z}{x}\right)}^{-1} \]
      mul-1-neg [=>] 80.6	\[ {\color{blue}{\left(-\frac{t \cdot z}{x}\right)}}^{-1} \]
      associate-/l* [=>] 95.1	\[ {\left(-\color{blue}{\frac{t}{\frac{x}{z}}}\right)}^{-1} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{-1}{t} \cdot x}{z}\\ \mathbf{elif}\;z \cdot t \leq 10^{+182}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(-\frac{t}{\frac{x}{z}}\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+253} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{-1}{t} \cdot x}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 3
Accuracy	71.8%
Cost	649

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-97} \lor \neg \left(y \leq 6.2 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 4
Accuracy	53.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+26} \lor \neg \left(t \leq 1.8 \cdot 10^{+243}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Accuracy	53.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+191}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 6
Accuracy	52.5%
Cost	320

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

Alternative 7
Accuracy	52.9%
Cost	192

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

Reproduce

herbie shell --seed 2023142 
(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))))