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

Average Error: 6.2 → 1.4

Time: 4.2s

Precision: binary64

Cost: 1360

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 10^{+168}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

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) -5e+85)
     (/ 1.0 (* (/ 1.0 x) (/ z y)))
     (if (<= (* x y) -5e-196)
       t_0
       (if (<= (* x y) 2e-143)
         (/ x (/ z y))
         (if (<= (* x y) 1e+168) t_0 (* x (/ y 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) <= -5e+85) {
		tmp = 1.0 / ((1.0 / x) * (z / y));
	} else if ((x * y) <= -5e-196) {
		tmp = t_0;
	} else if ((x * y) <= 2e-143) {
		tmp = x / (z / y);
	} else if ((x * y) <= 1e+168) {
		tmp = t_0;
	} else {
		tmp = x * (y / 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) <= (-5d+85)) then
        tmp = 1.0d0 / ((1.0d0 / x) * (z / y))
    else if ((x * y) <= (-5d-196)) then
        tmp = t_0
    else if ((x * y) <= 2d-143) then
        tmp = x / (z / y)
    else if ((x * y) <= 1d+168) then
        tmp = t_0
    else
        tmp = x * (y / 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) <= -5e+85) {
		tmp = 1.0 / ((1.0 / x) * (z / y));
	} else if ((x * y) <= -5e-196) {
		tmp = t_0;
	} else if ((x * y) <= 2e-143) {
		tmp = x / (z / y);
	} else if ((x * y) <= 1e+168) {
		tmp = t_0;
	} else {
		tmp = x * (y / 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) <= -5e+85:
		tmp = 1.0 / ((1.0 / x) * (z / y))
	elif (x * y) <= -5e-196:
		tmp = t_0
	elif (x * y) <= 2e-143:
		tmp = x / (z / y)
	elif (x * y) <= 1e+168:
		tmp = t_0
	else:
		tmp = x * (y / 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) <= -5e+85)
		tmp = Float64(1.0 / Float64(Float64(1.0 / x) * Float64(z / y)));
	elseif (Float64(x * y) <= -5e-196)
		tmp = t_0;
	elseif (Float64(x * y) <= 2e-143)
		tmp = Float64(x / Float64(z / y));
	elseif (Float64(x * y) <= 1e+168)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(y / 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) <= -5e+85)
		tmp = 1.0 / ((1.0 / x) * (z / y));
	elseif ((x * y) <= -5e-196)
		tmp = t_0;
	elseif ((x * y) <= 2e-143)
		tmp = x / (z / y);
	elseif ((x * y) <= 1e+168)
		tmp = t_0;
	else
		tmp = x * (y / 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], -5e+85], N[(1.0 / N[(N[(1.0 / x), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-196], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 2e-143], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+168], t$95$0, N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	6.2
Target	6.0
Herbie	1.4

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -5.0000000000000001e85

   1. Initial program 13.0

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

   2. Simplified 4.9

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      Proof
      (*.f64 x (/.f64 y z)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x y) z)): 53 points increase in error, 46 points decrease in error

   3. Applied egg-rr 13.0

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

   4. Applied egg-rr 5.3

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

   if -5.0000000000000001e85 < (*.f64 x y) < -5.0000000000000005e-196 or 1.9999999999999999e-143 < (*.f64 x y) < 9.9999999999999993e167

   1. Initial program 0.2

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

   if -5.0000000000000005e-196 < (*.f64 x y) < 1.9999999999999999e-143

   1. Initial program 9.3

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

   2. Simplified 1.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      Proof
      (*.f64 x (/.f64 y z)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x y) z)): 53 points increase in error, 46 points decrease in error

   3. Applied egg-rr 1.2

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

   if 9.9999999999999993e167 < (*.f64 x y)

   1. Initial program 19.9

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

   2. Simplified 3.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      Proof
      (*.f64 x (/.f64 y z)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x y) z)): 53 points increase in error, 46 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 10^{+168}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	1360

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 10^{+168}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Error	6.4
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;z \leq 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.0828879542302007 \cdot 10^{+299}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	6.2
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;z \leq 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.0828879542302007 \cdot 10^{+299}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	6.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq 10^{-166}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 1.0828879542302007 \cdot 10^{+299}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 5
Error	6.5
Cost	320

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

Reproduce

herbie shell --seed 2022291 
(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))