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

Average Accuracy: 90.3% → 99.3%

Time: 3.3s

Precision: binary64

Cost: 1361

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+176}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-235} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-204}\right) \land x \cdot y \leq 10^{+219}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -5e+176)
   (* x (/ y z))
   (if (or (<= (* x y) -2e-235)
           (and (not (<= (* x y) 2e-204)) (<= (* x y) 1e+219)))
     (/ (* x y) z)
     (* y (/ x z)))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -5e+176) {
		tmp = x * (y / z);
	} else if (((x * y) <= -2e-235) || (!((x * y) <= 2e-204) && ((x * y) <= 1e+219))) {
		tmp = (x * y) / z;
	} else {
		tmp = y * (x / 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) :: tmp
    if ((x * y) <= (-5d+176)) then
        tmp = x * (y / z)
    else if (((x * y) <= (-2d-235)) .or. (.not. ((x * y) <= 2d-204)) .and. ((x * y) <= 1d+219)) then
        tmp = (x * y) / z
    else
        tmp = y * (x / 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 tmp;
	if ((x * y) <= -5e+176) {
		tmp = x * (y / z);
	} else if (((x * y) <= -2e-235) || (!((x * y) <= 2e-204) && ((x * y) <= 1e+219))) {
		tmp = (x * y) / z;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (x * y) <= -5e+176:
		tmp = x * (y / z)
	elif ((x * y) <= -2e-235) or (not ((x * y) <= 2e-204) and ((x * y) <= 1e+219)):
		tmp = (x * y) / z
	else:
		tmp = y * (x / z)
	return tmp

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= -5e+176)
		tmp = Float64(x * Float64(y / z));
	elseif ((Float64(x * y) <= -2e-235) || (!(Float64(x * y) <= 2e-204) && (Float64(x * y) <= 1e+219)))
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * y) <= -5e+176)
		tmp = x * (y / z);
	elseif (((x * y) <= -2e-235) || (~(((x * y) <= 2e-204)) && ((x * y) <= 1e+219)))
		tmp = (x * y) / z;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+176], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e-235], And[N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e-204]], $MachinePrecision], LessEqual[N[(x * y), $MachinePrecision], 1e+219]]], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	90.3%
Target	90.5%
Herbie	99.3%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5e176

   1. Initial program 63.5%

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

   2. Simplified 96.9%

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

      [Start] 63.5	\[ \frac{x \cdot y}{z} \]
      associate-*r/ [<=] 96.9	\[ \color{blue}{x \cdot \frac{y}{z}} \]

   if -5e176 < (*.f64 x y) < -1.9999999999999999e-235 or 2e-204 < (*.f64 x y) < 9.99999999999999965e218

   1. Initial program 99.6%

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

   if -1.9999999999999999e-235 < (*.f64 x y) < 2e-204 or 9.99999999999999965e218 < (*.f64 x y)

   1. Initial program 77.1%

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

   2. Simplified 99.4%

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

      [Start] 77.1	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 99.4	\[ \color{blue}{\frac{x}{z} \cdot y} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+176}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-235} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-204}\right) \land x \cdot y \leq 10^{+219}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.1%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{-204} \lor \neg \left(z \leq 1.35 \cdot 10^{+139}\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	90.0%
Cost	320

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

Reproduce

herbie shell --seed 2023133 
(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))