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

Average Error: 6.0 → 2.2

Time: 1.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))) (t_1 (/ (* x y) z)))
   (if (<= (* x y) -2e+104)
     t_0
     (if (<= (* x y) -1e-247) t_1 (if (<= (* x y) 1e-291) t_0 t_1)))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -2e+104) {
		tmp = t_0;
	} else if ((x * y) <= -1e-247) {
		tmp = t_1;
	} else if ((x * y) <= 1e-291) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = y * (x / z)
    t_1 = (x * y) / z
    if ((x * y) <= (-2d+104)) then
        tmp = t_0
    else if ((x * y) <= (-1d-247)) then
        tmp = t_1
    else if ((x * y) <= 1d-291) then
        tmp = t_0
    else
        tmp = t_1
    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 = y * (x / z);
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -2e+104) {
		tmp = t_0;
	} else if ((x * y) <= -1e-247) {
		tmp = t_1;
	} else if ((x * y) <= 1e-291) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = y * (x / z)
	t_1 = (x * y) / z
	tmp = 0
	if (x * y) <= -2e+104:
		tmp = t_0
	elif (x * y) <= -1e-247:
		tmp = t_1
	elif (x * y) <= 1e-291:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	t_1 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= -2e+104)
		tmp = t_0;
	elseif (Float64(x * y) <= -1e-247)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e-291)
		tmp = t_0;
	else
		tmp = t_1;
	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 = y * (x / z);
	t_1 = (x * y) / z;
	tmp = 0.0;
	if ((x * y) <= -2e+104)
		tmp = t_0;
	elseif ((x * y) <= -1e-247)
		tmp = t_1;
	elseif ((x * y) <= 1e-291)
		tmp = t_0;
	else
		tmp = t_1;
	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[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+104], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], -1e-247], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e-291], t$95$0, t$95$1]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	6.6
Herbie	2.2

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2e104 or -1e-247 < (*.f64 x y) < 9.99999999999999962e-292

   1. Initial program 13.9

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

   2. Simplified 1.5

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

   3. Taylor expanded in x around 0 13.9

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

   4. Simplified 1.3

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

   if -2e104 < (*.f64 x y) < -1e-247 or 9.99999999999999962e-292 < (*.f64 x y)

   1. Initial program 2.6

      \[\frac{x \cdot y}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-247}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 10^{-291}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022169 
(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))