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

Average Error: 6.3 → 0.5

Time: 1.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+181}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 10^{-197}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+192}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= (* x y) -2e+181)
     (* y (* x (/ 1.0 z)))
     (if (<= (* x y) -1e-165)
       t_0
       (if (<= (* x y) 1e-197)
         (/ x (/ z y))
         (if (<= (* x y) 5e+192) t_0 (* 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 t_0 = (x * y) / z;
	double tmp;
	if ((x * y) <= -2e+181) {
		tmp = y * (x * (1.0 / z));
	} else if ((x * y) <= -1e-165) {
		tmp = t_0;
	} else if ((x * y) <= 1e-197) {
		tmp = x / (z / y);
	} else if ((x * y) <= 5e+192) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x * y) / z
    if ((x * y) <= (-2d+181)) then
        tmp = y * (x * (1.0d0 / z))
    else if ((x * y) <= (-1d-165)) then
        tmp = t_0
    else if ((x * y) <= 1d-197) then
        tmp = x / (z / y)
    else if ((x * y) <= 5d+192) then
        tmp = t_0
    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 t_0 = (x * y) / z;
	double tmp;
	if ((x * y) <= -2e+181) {
		tmp = y * (x * (1.0 / z));
	} else if ((x * y) <= -1e-165) {
		tmp = t_0;
	} else if ((x * y) <= 1e-197) {
		tmp = x / (z / y);
	} else if ((x * y) <= 5e+192) {
		tmp = t_0;
	} else {
		tmp = y * (x / 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 (x * y) <= -2e+181:
		tmp = y * (x * (1.0 / z))
	elif (x * y) <= -1e-165:
		tmp = t_0
	elif (x * y) <= 1e-197:
		tmp = x / (z / y)
	elif (x * y) <= 5e+192:
		tmp = t_0
	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)
	t_0 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= -2e+181)
		tmp = Float64(y * Float64(x * Float64(1.0 / z)));
	elseif (Float64(x * y) <= -1e-165)
		tmp = t_0;
	elseif (Float64(x * y) <= 1e-197)
		tmp = Float64(x / Float64(z / y));
	elseif (Float64(x * y) <= 5e+192)
		tmp = t_0;
	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)
	t_0 = (x * y) / z;
	tmp = 0.0;
	if ((x * y) <= -2e+181)
		tmp = y * (x * (1.0 / z));
	elseif ((x * y) <= -1e-165)
		tmp = t_0;
	elseif ((x * y) <= 1e-197)
		tmp = x / (z / y);
	elseif ((x * y) <= 5e+192)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+181], N[(y * N[(x * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-165], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 1e-197], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+192], t$95$0, N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	6.3
Herbie	0.5

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1.9999999999999998e181

   1. Initial program 23.0

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

   2. Applied egg-rr 2.2

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{1}{z}\right)} \]

   if -1.9999999999999998e181 < (*.f64 x y) < -1e-165 or 9.9999999999999999e-198 < (*.f64 x y) < 5.00000000000000033e192

   1. Initial program 0.2

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

   if -1e-165 < (*.f64 x y) < 9.9999999999999999e-198

   1. Initial program 9.8

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

   2. Applied egg-rr 9.9

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}} \]

   3. Applied egg-rr 0.6

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

   if 5.00000000000000033e192 < (*.f64 x y)

   1. Initial program 24.7

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

   2. Applied egg-rr 24.8

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}} \]

   3. Applied egg-rr 0.7

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

   4. Applied egg-rr 0.9

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+181}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-165}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 10^{-197}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+192}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))