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

Average Error: 5.9 → 1.0

Time: 7.4s

Precision: binary64

Cost: 1360

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{z} \cdot y\\ t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+108}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+194}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x z) y)) (t_1 (/ (* x y) z)))
   (if (<= (* x y) -2e+108)
     (* (/ y z) x)
     (if (<= (* x y) -1e-207)
       t_1
       (if (<= (* x y) 2e-118) t_0 (if (<= (* x y) 4e+194) 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 / z) * y;
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -2e+108) {
		tmp = (y / z) * x;
	} else if ((x * y) <= -1e-207) {
		tmp = t_1;
	} else if ((x * y) <= 2e-118) {
		tmp = t_0;
	} else if ((x * y) <= 4e+194) {
		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 / z) * y
    t_1 = (x * y) / z
    if ((x * y) <= (-2d+108)) then
        tmp = (y / z) * x
    else if ((x * y) <= (-1d-207)) then
        tmp = t_1
    else if ((x * y) <= 2d-118) then
        tmp = t_0
    else if ((x * y) <= 4d+194) 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 / z) * y;
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -2e+108) {
		tmp = (y / z) * x;
	} else if ((x * y) <= -1e-207) {
		tmp = t_1;
	} else if ((x * y) <= 2e-118) {
		tmp = t_0;
	} else if ((x * y) <= 4e+194) {
		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 / z) * y
	t_1 = (x * y) / z
	tmp = 0
	if (x * y) <= -2e+108:
		tmp = (y / z) * x
	elif (x * y) <= -1e-207:
		tmp = t_1
	elif (x * y) <= 2e-118:
		tmp = t_0
	elif (x * y) <= 4e+194:
		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(Float64(x / z) * y)
	t_1 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= -2e+108)
		tmp = Float64(Float64(y / z) * x);
	elseif (Float64(x * y) <= -1e-207)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e-118)
		tmp = t_0;
	elseif (Float64(x * y) <= 4e+194)
		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 / z) * y;
	t_1 = (x * y) / z;
	tmp = 0.0;
	if ((x * y) <= -2e+108)
		tmp = (y / z) * x;
	elseif ((x * y) <= -1e-207)
		tmp = t_1;
	elseif ((x * y) <= 2e-118)
		tmp = t_0;
	elseif ((x * y) <= 4e+194)
		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[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+108], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-207], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e-118], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 4e+194], t$95$1, t$95$0]]]]]]

Target

Original	5.9
Target	6.0
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 13.7

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

   2. Applied egg-rr 4.2

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

   if -2.0000000000000001e108 < (*.f64 x y) < -9.99999999999999925e-208 or 1.99999999999999997e-118 < (*.f64 x y) < 3.99999999999999978e194

   1. Initial program 0.2

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

   if -9.99999999999999925e-208 < (*.f64 x y) < 1.99999999999999997e-118 or 3.99999999999999978e194 < (*.f64 x y)

   1. Initial program 11.0

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

   2. Applied egg-rr 1.1

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

Alternatives

Alternative 1
Error	6.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 1.08 \cdot 10^{-268}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]

Alternative 2
Error	6.0
Cost	320

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

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))