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

Average Error: 6.5 → 0.8

Time: 2.7s

Precision: binary64

Cost: 1360

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+187}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-294}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+161}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= (* x y) -4e+187)
     (* x (/ y z))
     (if (<= (* x y) -1e-144)
       t_0
       (if (<= (* x y) 5e-294)
         (* y (/ x z))
         (if (<= (* x y) 2e+161) 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 = (x * y) / z;
	double tmp;
	if ((x * y) <= -4e+187) {
		tmp = x * (y / z);
	} else if ((x * y) <= -1e-144) {
		tmp = t_0;
	} else if ((x * y) <= 5e-294) {
		tmp = y * (x / z);
	} else if ((x * y) <= 2e+161) {
		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 = (x * y) / z
    if ((x * y) <= (-4d+187)) then
        tmp = x * (y / z)
    else if ((x * y) <= (-1d-144)) then
        tmp = t_0
    else if ((x * y) <= 5d-294) then
        tmp = y * (x / z)
    else if ((x * y) <= 2d+161) 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 = (x * y) / z;
	double tmp;
	if ((x * y) <= -4e+187) {
		tmp = x * (y / z);
	} else if ((x * y) <= -1e-144) {
		tmp = t_0;
	} else if ((x * y) <= 5e-294) {
		tmp = y * (x / z);
	} else if ((x * y) <= 2e+161) {
		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 = (x * y) / z
	tmp = 0
	if (x * y) <= -4e+187:
		tmp = x * (y / z)
	elif (x * y) <= -1e-144:
		tmp = t_0
	elif (x * y) <= 5e-294:
		tmp = y * (x / z)
	elif (x * y) <= 2e+161:
		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(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= -4e+187)
		tmp = Float64(x * Float64(y / z));
	elseif (Float64(x * y) <= -1e-144)
		tmp = t_0;
	elseif (Float64(x * y) <= 5e-294)
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(x * y) <= 2e+161)
		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 = (x * y) / z;
	tmp = 0.0;
	if ((x * y) <= -4e+187)
		tmp = x * (y / z);
	elseif ((x * y) <= -1e-144)
		tmp = t_0;
	elseif ((x * y) <= 5e-294)
		tmp = y * (x / z);
	elseif ((x * y) <= 2e+161)
		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[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -4e+187], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-144], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 5e-294], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+161], t$95$0, N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	6.5
Target	6.5
Herbie	0.8

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -3.99999999999999963e187

   1. Initial program 25.1

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

   2. Simplified 1.6

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      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)): 0 points increase in error, 1 points decrease in error

   if -3.99999999999999963e187 < (*.f64 x y) < -9.9999999999999995e-145 or 5.0000000000000003e-294 < (*.f64 x y) < 2.0000000000000001e161

   1. Initial program 0.2

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

   if -9.9999999999999995e-145 < (*.f64 x y) < 5.0000000000000003e-294

   1. Initial program 11.4

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

   2. Simplified 1.4

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
      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)): 0 points increase in error, 1 points decrease in error

   if 2.0000000000000001e161 < (*.f64 x y)

   1. Initial program 21.3

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

   2. Simplified 2.1

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
      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)): 0 points increase in error, 1 points decrease in error

   3. Applied egg-rr 2.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+187}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-144}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-294}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.5
Cost	320

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

Alternative 2
Error	6.3
Cost	320

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

Reproduce

herbie shell --seed 2022343 
(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))