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

Average Error: 5.9 → 0.9

Time: 2.4s

Precision: binary64

Cost: 1360

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

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 y) z)))
   (if (<= (* x y) -2e+108)
     t_0
     (if (<= (* x y) -2e-182)
       t_1
       (if (<= (* x y) 5e-150)
         (/ x (/ z y))
         (if (<= (* x y) 2e+300) t_1 t_0))))))

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 * y) / z;
	double tmp;
	if ((x * y) <= -2e+108) {
		tmp = t_0;
	} else if ((x * y) <= -2e-182) {
		tmp = t_1;
	} else if ((x * y) <= 5e-150) {
		tmp = x / (z / y);
	} else if ((x * y) <= 2e+300) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 * y) / z
    if ((x * y) <= (-2d+108)) then
        tmp = t_0
    else if ((x * y) <= (-2d-182)) then
        tmp = t_1
    else if ((x * y) <= 5d-150) then
        tmp = x / (z / y)
    else if ((x * y) <= 2d+300) then
        tmp = t_1
    else
        tmp = t_0
    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 * y) / z;
	double tmp;
	if ((x * y) <= -2e+108) {
		tmp = t_0;
	} else if ((x * y) <= -2e-182) {
		tmp = t_1;
	} else if ((x * y) <= 5e-150) {
		tmp = x / (z / y);
	} else if ((x * y) <= 2e+300) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 * y) / z
	tmp = 0
	if (x * y) <= -2e+108:
		tmp = t_0
	elif (x * y) <= -2e-182:
		tmp = t_1
	elif (x * y) <= 5e-150:
		tmp = x / (z / y)
	elif (x * y) <= 2e+300:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(x * Float64(y / z))
	t_1 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= -2e+108)
		tmp = t_0;
	elseif (Float64(x * y) <= -2e-182)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e-150)
		tmp = Float64(x / Float64(z / y));
	elseif (Float64(x * y) <= 2e+300)
		tmp = t_1;
	else
		tmp = t_0;
	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 * y) / z;
	tmp = 0.0;
	if ((x * y) <= -2e+108)
		tmp = t_0;
	elseif ((x * y) <= -2e-182)
		tmp = t_1;
	elseif ((x * y) <= 5e-150)
		tmp = x / (z / y);
	elseif ((x * y) <= 2e+300)
		tmp = t_1;
	else
		tmp = t_0;
	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[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+108], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], -2e-182], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e-150], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+300], t$95$1, t$95$0]]]]]]

Target

Original	5.9
Target	5.9
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.0000000000000001e108 or 2.0000000000000001e300 < (*.f64 x y)

   1. Initial program 22.2

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

   2. Simplified 3.6

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

      [Start] 22.2	\[ \frac{x \cdot y}{z} \]
      associate-*r/ [<=] 3.6	\[ \color{blue}{x \cdot \frac{y}{z}} \]

   if -2.0000000000000001e108 < (*.f64 x y) < -2.0000000000000001e-182 or 4.9999999999999999e-150 < (*.f64 x y) < 2.0000000000000001e300

   1. Initial program 0.2

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

   if -2.0000000000000001e-182 < (*.f64 x y) < 4.9999999999999999e-150

   1. Initial program 8.9

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

   2. Simplified 0.8

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

      [Start] 8.9	\[ \frac{x \cdot y}{z} \]
      associate-/l* [=>] 0.8	\[ \color{blue}{\frac{x}{\frac{z}{y}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+108}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-182}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.2
Cost	850

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-211} \lor \neg \left(z \leq 1.9 \cdot 10^{-275} \lor \neg \left(z \leq 4.8 \cdot 10^{-46}\right) \land z \leq 5 \cdot 10^{+187}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Error	6.3
Cost	849

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{-215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-268}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 10^{-68} \lor \neg \left(z \leq 2.5 \cdot 10^{+186}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	6.0
Cost	848

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{-224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+182}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Error	6.2
Cost	320

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

Reproduce

herbie shell --seed 2023010 
(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))