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

Average Error: 6.0 → 2.0

Time: 2.3s

Precision: binary64

Cost: 1612

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-316}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;t_0 \leq 10^{+295}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= t_0 -1e-316)
     t_0
     (if (<= t_0 0.0)
       (/ x (/ z y))
       (if (<= t_0 1e+295) t_0 (* y (* x (/ 1.0 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 tmp;
	if (t_0 <= -1e-316) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = x / (z / y);
	} else if (t_0 <= 1e+295) {
		tmp = t_0;
	} else {
		tmp = y * (x * (1.0 / 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) :: tmp
    t_0 = (x * y) / z
    if (t_0 <= (-1d-316)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = x / (z / y)
    else if (t_0 <= 1d+295) then
        tmp = t_0
    else
        tmp = y * (x * (1.0d0 / 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 tmp;
	if (t_0 <= -1e-316) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = x / (z / y);
	} else if (t_0 <= 1e+295) {
		tmp = t_0;
	} else {
		tmp = y * (x * (1.0 / z));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x * y) / z
	tmp = 0
	if t_0 <= -1e-316:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = x / (z / y)
	elif t_0 <= 1e+295:
		tmp = t_0
	else:
		tmp = y * (x * (1.0 / 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)
	tmp = 0.0
	if (t_0 <= -1e-316)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(x / Float64(z / y));
	elseif (t_0 <= 1e+295)
		tmp = t_0;
	else
		tmp = Float64(y * Float64(x * Float64(1.0 / 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;
	tmp = 0.0;
	if (t_0 <= -1e-316)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = x / (z / y);
	elseif (t_0 <= 1e+295)
		tmp = t_0;
	else
		tmp = y * (x * (1.0 / 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]}, If[LessEqual[t$95$0, -1e-316], t$95$0, If[LessEqual[t$95$0, 0.0], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+295], t$95$0, N[(y * N[(x * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	6.0
Target	6.3
Herbie	2.0

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x y) z) < -9.999999837e-317 or -0.0 < (/.f64 (*.f64 x y) z) < 9.9999999999999998e294

   1. Initial program 2.4

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

   if -9.999999837e-317 < (/.f64 (*.f64 x y) z) < -0.0

   1. Initial program 11.8

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
      Proof
      (/.f64 x (/.f64 z y)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x y) z)): 43 points increase in error, 48 points decrease in error

   if 9.9999999999999998e294 < (/.f64 (*.f64 x y) z)

   1. Initial program 52.2

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

   2. Applied egg-rr 4.8

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{1}{z}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -1 \cdot 10^{-316}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 0:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 10^{+295}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.5
Cost	840

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-210}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	6.3
Cost	320

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

Reproduce

herbie shell --seed 2022329 
(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))