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

Average Error: 6.0 → 0.4

Time: 2.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+201}:\\ \;\;\;\;{\left(\frac{\frac{z}{x}}{y}\right)}^{-1}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-238}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 10^{+171}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

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) -4e+201)
     (pow (/ (/ z x) y) -1.0)
     (if (<= (* x y) -1e-252)
       t_0
       (if (<= (* x y) 5e-238)
         (* x (/ y z))
         (if (<= (* x y) 1e+171) 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) <= -4e+201) {
		tmp = pow(((z / x) / y), -1.0);
	} else if ((x * y) <= -1e-252) {
		tmp = t_0;
	} else if ((x * y) <= 5e-238) {
		tmp = x * (y / z);
	} else if ((x * y) <= 1e+171) {
		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) <= (-4d+201)) then
        tmp = ((z / x) / y) ** (-1.0d0)
    else if ((x * y) <= (-1d-252)) then
        tmp = t_0
    else if ((x * y) <= 5d-238) then
        tmp = x * (y / z)
    else if ((x * y) <= 1d+171) 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) <= -4e+201) {
		tmp = Math.pow(((z / x) / y), -1.0);
	} else if ((x * y) <= -1e-252) {
		tmp = t_0;
	} else if ((x * y) <= 5e-238) {
		tmp = x * (y / z);
	} else if ((x * y) <= 1e+171) {
		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) <= -4e+201:
		tmp = math.pow(((z / x) / y), -1.0)
	elif (x * y) <= -1e-252:
		tmp = t_0
	elif (x * y) <= 5e-238:
		tmp = x * (y / z)
	elif (x * y) <= 1e+171:
		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) <= -4e+201)
		tmp = Float64(Float64(z / x) / y) ^ -1.0;
	elseif (Float64(x * y) <= -1e-252)
		tmp = t_0;
	elseif (Float64(x * y) <= 5e-238)
		tmp = Float64(x * Float64(y / z));
	elseif (Float64(x * y) <= 1e+171)
		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) <= -4e+201)
		tmp = ((z / x) / y) ^ -1.0;
	elseif ((x * y) <= -1e-252)
		tmp = t_0;
	elseif ((x * y) <= 5e-238)
		tmp = x * (y / z);
	elseif ((x * y) <= 1e+171)
		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], -4e+201], N[Power[N[(N[(z / x), $MachinePrecision] / y), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-252], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 5e-238], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+171], t$95$0, N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	6.0
Target	6.1
Herbie	0.4

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -4.00000000000000015e201

   1. Initial program 25.9

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

   2. Simplified 0.8

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

   3. Applied egg-rr 26.0

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

   4. Applied egg-rr 1.4

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

   if -4.00000000000000015e201 < (*.f64 x y) < -9.99999999999999943e-253 or 5e-238 < (*.f64 x y) < 9.99999999999999954e170

   1. Initial program 0.2

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

   2. Simplified 9.2

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

   3. Applied egg-rr 9.2

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

   4. Taylor expanded in x around 0 0.2

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

   if -9.99999999999999943e-253 < (*.f64 x y) < 5e-238

   1. Initial program 13.5

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

   2. Simplified 0.3

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

   if 9.99999999999999954e170 < (*.f64 x y)

   1. Initial program 19.1

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

   2. Simplified 1.9

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

   3. Applied egg-rr 19.1

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

   4. Taylor expanded in z around 0 19.1

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

   5. Simplified 1.8

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+201}:\\ \;\;\;\;{\left(\frac{\frac{z}{x}}{y}\right)}^{-1}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-238}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 10^{+171}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(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))