Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A

Average Accuracy: 90.0% → 96.2%

Time: 4.0s

Precision: binary64

Cost: 1872

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ t_1 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{+107}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;t_0 \leq 10^{-278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)) (t_1 (/ x (/ z y))))
   (if (<= t_0 -5e+286)
     t_1
     (if (<= t_0 -2e+107)
       (* (* x y) (/ 1.0 z))
       (if (<= t_0 1e-278) t_1 (if (<= t_0 2e+305) t_0 (* x (/ y z))))))))

C

double code(double x, double y, double z) {
	return (x * y) / z;
}

↓

double code(double x, double y, double z) {
	double t_0 = (x * y) / z;
	double t_1 = x / (z / y);
	double tmp;
	if (t_0 <= -5e+286) {
		tmp = t_1;
	} else if (t_0 <= -2e+107) {
		tmp = (x * y) * (1.0 / z);
	} else if (t_0 <= 1e-278) {
		tmp = t_1;
	} else if (t_0 <= 2e+305) {
		tmp = t_0;
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / z
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * y) / z
    t_1 = x / (z / y)
    if (t_0 <= (-5d+286)) then
        tmp = t_1
    else if (t_0 <= (-2d+107)) then
        tmp = (x * y) * (1.0d0 / z)
    else if (t_0 <= 1d-278) then
        tmp = t_1
    else if (t_0 <= 2d+305) then
        tmp = t_0
    else
        tmp = x * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / z;
}

↓

public static double code(double x, double y, double z) {
	double t_0 = (x * y) / z;
	double t_1 = x / (z / y);
	double tmp;
	if (t_0 <= -5e+286) {
		tmp = t_1;
	} else if (t_0 <= -2e+107) {
		tmp = (x * y) * (1.0 / z);
	} else if (t_0 <= 1e-278) {
		tmp = t_1;
	} else if (t_0 <= 2e+305) {
		tmp = t_0;
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * y) / z

↓

def code(x, y, z):
	t_0 = (x * y) / z
	t_1 = x / (z / y)
	tmp = 0
	if t_0 <= -5e+286:
		tmp = t_1
	elif t_0 <= -2e+107:
		tmp = (x * y) * (1.0 / z)
	elif t_0 <= 1e-278:
		tmp = t_1
	elif t_0 <= 2e+305:
		tmp = t_0
	else:
		tmp = x * (y / z)
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / z)
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(x * y) / z)
	t_1 = Float64(x / Float64(z / y))
	tmp = 0.0
	if (t_0 <= -5e+286)
		tmp = t_1;
	elseif (t_0 <= -2e+107)
		tmp = Float64(Float64(x * y) * Float64(1.0 / z));
	elseif (t_0 <= 1e-278)
		tmp = t_1;
	elseif (t_0 <= 2e+305)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / z;
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (x * y) / z;
	t_1 = x / (z / y);
	tmp = 0.0;
	if (t_0 <= -5e+286)
		tmp = t_1;
	elseif (t_0 <= -2e+107)
		tmp = (x * y) * (1.0 / z);
	elseif (t_0 <= 1e-278)
		tmp = t_1;
	elseif (t_0 <= 2e+305)
		tmp = t_0;
	else
		tmp = x * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+286], t$95$1, If[LessEqual[t$95$0, -2e+107], N[(N[(x * y), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-278], t$95$1, If[LessEqual[t$95$0, 2e+305], t$95$0, N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	90.0%
Target	89.7%
Herbie	96.2%

\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x y) z) < -5.0000000000000004e286 or -1.9999999999999999e107 < (/.f64 (*.f64 x y) z) < 9.99999999999999938e-279

   1. Initial program 86.9%

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

   2. Simplified 93.7%

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

      [Start] 86.9	\[ \frac{x \cdot y}{z} \]
      associate-/l* [=>] 93.7	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

   if -5.0000000000000004e286 < (/.f64 (*.f64 x y) z) < -1.9999999999999999e107

   1. Initial program 99.6%

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

   2. Applied egg-rr 99.4%

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

      [Start] 99.6	\[ \frac{x \cdot y}{z} \]
      div-inv [=>] 99.4	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}} \]
      *-commutative [=>] 99.4	\[ \color{blue}{\frac{1}{z} \cdot \left(x \cdot y\right)} \]

   if 9.99999999999999938e-279 < (/.f64 (*.f64 x y) z) < 1.9999999999999999e305

   1. Initial program 99.2%

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

   if 1.9999999999999999e305 < (/.f64 (*.f64 x y) z)

   1. Initial program 4.1%

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

   2. Simplified 97.8%

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

      [Start] 4.1	\[ \frac{x \cdot y}{z} \]
      associate-*r/ [<=] 97.8	\[ \color{blue}{x \cdot \frac{y}{z}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -5 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq -2 \cdot 10^{+107}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 10^{-278}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.4%
Cost	1872

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 10^{-278}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	90.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-140} \lor \neg \left(z \leq 2.85 \cdot 10^{-202}\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Accuracy	90.1%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-238} \lor \neg \left(z \leq 4.6 \cdot 10^{-202}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Accuracy	90.0%
Cost	320

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

Reproduce

herbie shell --seed 2023135 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -4.262230790519429e-138) (/ (* x y) z) (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))