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

Average Error: 6.5 → 0.9

Time: 2.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y \cdot \left(-x\right)}{-z}\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+181}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-229}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 10^{+158}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* y (- x)) (- z))))
   (if (<= (* x y) -1e+181)
     (* x (/ y z))
     (if (<= (* x y) -5e-104)
       t_0
       (if (<= (* x y) 2e-229)
         (* y (/ x z))
         (if (<= (* x y) 1e+158) t_0 (/ y (/ z x))))))))

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 tmp;
	if ((x * y) <= -1e+181) {
		tmp = x * (y / z);
	} else if ((x * y) <= -5e-104) {
		tmp = t_0;
	} else if ((x * y) <= 2e-229) {
		tmp = y * (x / z);
	} else if ((x * y) <= 1e+158) {
		tmp = t_0;
	} else {
		tmp = y / (z / x);
	}
	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 = (y * -x) / -z
    if ((x * y) <= (-1d+181)) then
        tmp = x * (y / z)
    else if ((x * y) <= (-5d-104)) then
        tmp = t_0
    else if ((x * y) <= 2d-229) then
        tmp = y * (x / z)
    else if ((x * y) <= 1d+158) then
        tmp = t_0
    else
        tmp = y / (z / x)
    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 tmp;
	if ((x * y) <= -1e+181) {
		tmp = x * (y / z);
	} else if ((x * y) <= -5e-104) {
		tmp = t_0;
	} else if ((x * y) <= 2e-229) {
		tmp = y * (x / z);
	} else if ((x * y) <= 1e+158) {
		tmp = t_0;
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (y * -x) / -z
	tmp = 0
	if (x * y) <= -1e+181:
		tmp = x * (y / z)
	elif (x * y) <= -5e-104:
		tmp = t_0
	elif (x * y) <= 2e-229:
		tmp = y * (x / z)
	elif (x * y) <= 1e+158:
		tmp = t_0
	else:
		tmp = y / (z / x)
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(-x)) / Float64(-z))
	tmp = 0.0
	if (Float64(x * y) <= -1e+181)
		tmp = Float64(x * Float64(y / z));
	elseif (Float64(x * y) <= -5e-104)
		tmp = t_0;
	elseif (Float64(x * y) <= 2e-229)
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(x * y) <= 1e+158)
		tmp = t_0;
	else
		tmp = Float64(y / Float64(z / x));
	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;
	tmp = 0.0;
	if ((x * y) <= -1e+181)
		tmp = x * (y / z);
	elseif ((x * y) <= -5e-104)
		tmp = t_0;
	elseif ((x * y) <= 2e-229)
		tmp = y * (x / z);
	elseif ((x * y) <= 1e+158)
		tmp = t_0;
	else
		tmp = y / (z / x);
	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[(y * (-x)), $MachinePrecision] / (-z)), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+181], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-104], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 2e-229], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+158], t$95$0, N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.5
Target	6.1
Herbie	0.9

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -9.9999999999999992e180

   1. Initial program 23.0

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

   2. Simplified 2.1

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

   if -9.9999999999999992e180 < (*.f64 x y) < -4.99999999999999979e-104 or 2.00000000000000014e-229 < (*.f64 x y) < 9.99999999999999953e157

   1. Initial program 0.2

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

   2. Simplified 10.9

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

   3. Taylor expanded in x around 0 0.2

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

   4. Simplified 10.7

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

   5. Applied egg-rr 0.2

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

   if -4.99999999999999979e-104 < (*.f64 x y) < 2.00000000000000014e-229

   1. Initial program 9.5

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

   2. Simplified 1.6

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

   3. Taylor expanded in x around 0 9.5

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

   4. Simplified 1.5

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

   5. Applied egg-rr 1.5

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

   6. Applied egg-rr 1.5

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

   if 9.99999999999999953e157 < (*.f64 x y)

   1. Initial program 19.3

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

   2. Simplified 2.8

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

   3. Taylor expanded in x around 0 19.3

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

   4. Simplified 1.6

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

   5. Applied egg-rr 2.0

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+181}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{-z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-229}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 10^{+158}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022159 
(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))