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

Average Error: 6.1 → 2.0

Time: 2.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -1.761720498107075 \cdot 10^{-142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 3.8895485895542 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.4951491249076507 \cdot 10^{+235}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= (* x y) -1.761720498107075e-142)
     t_0
     (if (<= (* x y) 3.8895485895542e-310)
       (* x (/ y z))
       (if (<= (* x y) 1.4951491249076507e+235) t_0 (/ x (/ z y)))))))

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 ((x * y) <= -1.761720498107075e-142) {
		tmp = t_0;
	} else if ((x * y) <= 3.8895485895542e-310) {
		tmp = x * (y / z);
	} else if ((x * y) <= 1.4951491249076507e+235) {
		tmp = t_0;
	} else {
		tmp = x / (z / y);
	}
	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 ((x * y) <= (-1.761720498107075d-142)) then
        tmp = t_0
    else if ((x * y) <= 3.8895485895542d-310) then
        tmp = x * (y / z)
    else if ((x * y) <= 1.4951491249076507d+235) then
        tmp = t_0
    else
        tmp = x / (z / y)
    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 ((x * y) <= -1.761720498107075e-142) {
		tmp = t_0;
	} else if ((x * y) <= 3.8895485895542e-310) {
		tmp = x * (y / z);
	} else if ((x * y) <= 1.4951491249076507e+235) {
		tmp = t_0;
	} else {
		tmp = x / (z / y);
	}
	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 (x * y) <= -1.761720498107075e-142:
		tmp = t_0
	elif (x * y) <= 3.8895485895542e-310:
		tmp = x * (y / z)
	elif (x * y) <= 1.4951491249076507e+235:
		tmp = t_0
	else:
		tmp = x / (z / y)
	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 (Float64(x * y) <= -1.761720498107075e-142)
		tmp = t_0;
	elseif (Float64(x * y) <= 3.8895485895542e-310)
		tmp = Float64(x * Float64(y / z));
	elseif (Float64(x * y) <= 1.4951491249076507e+235)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(z / y));
	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 ((x * y) <= -1.761720498107075e-142)
		tmp = t_0;
	elseif ((x * y) <= 3.8895485895542e-310)
		tmp = x * (y / z);
	elseif ((x * y) <= 1.4951491249076507e+235)
		tmp = t_0;
	else
		tmp = x / (z / y);
	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[N[(x * y), $MachinePrecision], -1.761720498107075e-142], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 3.8895485895542e-310], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.4951491249076507e+235], t$95$0, N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	6.6
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 x y) < -1.7617204981070749e-142 or 3.88954858955419e-310 < (*.f64 x y) < 1.4951491249076507e235

   1. Initial program 2.5

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

   if -1.7617204981070749e-142 < (*.f64 x y) < 3.88954858955419e-310

   1. Initial program 11.1

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

   2. Simplified 1.0

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

   if 1.4951491249076507e235 < (*.f64 x y)

   1. Initial program 35.8

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

   2. Applied egg-rr 1.3

      \[\leadsto \color{blue}{\frac{x}{{\left(\sqrt[3]{z}\right)}^{2}} \cdot \frac{y}{\sqrt[3]{z}}} \]

   3. Applied egg-rr 0.5

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.761720498107075 \cdot 10^{-142}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 3.8895485895542 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.4951491249076507 \cdot 10^{+235}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))