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

Average Error: 6.1 → 0.7

Time: 2.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -2.5366365598671997 \cdot 10^{+214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -1.2470082916549296 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 4.074425885 \cdot 10^{-315}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 3.692596956485395 \cdot 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ y z))) (t_1 (/ (* x y) z)))
   (if (<= (* x y) -2.5366365598671997e+214)
     t_0
     (if (<= (* x y) -1.2470082916549296e-104)
       t_1
       (if (<= (* x y) 4.074425885e-315)
         t_0
         (if (<= (* x y) 3.692596956485395e+304) t_1 (* 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 t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -2.5366365598671997e+214) {
		tmp = t_0;
	} else if ((x * y) <= -1.2470082916549296e-104) {
		tmp = t_1;
	} else if ((x * y) <= 4.074425885e-315) {
		tmp = t_0;
	} else if ((x * y) <= 3.692596956485395e+304) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * (y / z)
    t_1 = (x * y) / z
    if ((x * y) <= (-2.5366365598671997d+214)) then
        tmp = t_0
    else if ((x * y) <= (-1.2470082916549296d-104)) then
        tmp = t_1
    else if ((x * y) <= 4.074425885d-315) then
        tmp = t_0
    else if ((x * y) <= 3.692596956485395d+304) then
        tmp = t_1
    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 t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -2.5366365598671997e+214) {
		tmp = t_0;
	} else if ((x * y) <= -1.2470082916549296e-104) {
		tmp = t_1;
	} else if ((x * y) <= 4.074425885e-315) {
		tmp = t_0;
	} else if ((x * y) <= 3.692596956485395e+304) {
		tmp = t_1;
	} 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)
	t_1 = (x * y) / z
	tmp = 0
	if (x * y) <= -2.5366365598671997e+214:
		tmp = t_0
	elif (x * y) <= -1.2470082916549296e-104:
		tmp = t_1
	elif (x * y) <= 4.074425885e-315:
		tmp = t_0
	elif (x * y) <= 3.692596956485395e+304:
		tmp = t_1
	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(x * Float64(y / z))
	t_1 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= -2.5366365598671997e+214)
		tmp = t_0;
	elseif (Float64(x * y) <= -1.2470082916549296e-104)
		tmp = t_1;
	elseif (Float64(x * y) <= 4.074425885e-315)
		tmp = t_0;
	elseif (Float64(x * y) <= 3.692596956485395e+304)
		tmp = t_1;
	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);
	t_1 = (x * y) / z;
	tmp = 0.0;
	if ((x * y) <= -2.5366365598671997e+214)
		tmp = t_0;
	elseif ((x * y) <= -1.2470082916549296e-104)
		tmp = t_1;
	elseif ((x * y) <= 4.074425885e-315)
		tmp = t_0;
	elseif ((x * y) <= 3.692596956485395e+304)
		tmp = t_1;
	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[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2.5366365598671997e+214], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], -1.2470082916549296e-104], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 4.074425885e-315], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 3.692596956485395e+304], t$95$1, N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	6.0
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.53663655986719974e214 or -1.24700829165492956e-104 < (*.f64 x y) < 4.074425885e-315

   1. Initial program 13.0

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

   2. Simplified 1.6

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

   3. Applied egg-rr 1.5

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

   4. Applied egg-rr 4.6

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2}}{z} \cdot \frac{\sqrt[3]{x}}{\frac{1}{y}}} \]

   5. Applied egg-rr 2.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{z}{y}}{x}}} \]

   6. Applied egg-rr 1.6

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

   if -2.53663655986719974e214 < (*.f64 x y) < -1.24700829165492956e-104 or 4.074425885e-315 < (*.f64 x y) < 3.69259695648539485e304

   1. Initial program 0.3

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

   if 3.69259695648539485e304 < (*.f64 x y)

   1. Initial program 62.8

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

   2. Simplified 0.3

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

   3. Taylor expanded in x around 0 62.8

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

   4. Simplified 0.3

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.5366365598671997 \cdot 10^{+214}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -1.2470082916549296 \cdot 10^{-104}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 4.074425885 \cdot 10^{-315}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 3.692596956485395 \cdot 10^{+304}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022151 
(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))