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

Average Error: 6.1 → 0.8

Time: 2.3s

Precision: binary64

Cost: 1360

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ t_1 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-283}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (/ z y))) (t_1 (/ (* x y) z)))
   (if (<= (* x y) -1e+169)
     t_0
     (if (<= (* x y) -5e-164)
       t_1
       (if (<= (* x y) 5e-283)
         (* y (/ x z))
         (if (<= (* x y) 2e+121) t_1 t_0))))))

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 / (z / y);
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -1e+169) {
		tmp = t_0;
	} else if ((x * y) <= -5e-164) {
		tmp = t_1;
	} else if ((x * y) <= 5e-283) {
		tmp = y * (x / z);
	} else if ((x * y) <= 2e+121) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 / (z / y)
    t_1 = (x * y) / z
    if ((x * y) <= (-1d+169)) then
        tmp = t_0
    else if ((x * y) <= (-5d-164)) then
        tmp = t_1
    else if ((x * y) <= 5d-283) then
        tmp = y * (x / z)
    else if ((x * y) <= 2d+121) then
        tmp = t_1
    else
        tmp = t_0
    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 / (z / y);
	double t_1 = (x * y) / z;
	double tmp;
	if ((x * y) <= -1e+169) {
		tmp = t_0;
	} else if ((x * y) <= -5e-164) {
		tmp = t_1;
	} else if ((x * y) <= 5e-283) {
		tmp = y * (x / z);
	} else if ((x * y) <= 2e+121) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = x / (z / y)
	t_1 = (x * y) / z
	tmp = 0
	if (x * y) <= -1e+169:
		tmp = t_0
	elif (x * y) <= -5e-164:
		tmp = t_1
	elif (x * y) <= 5e-283:
		tmp = y * (x / z)
	elif (x * y) <= 2e+121:
		tmp = t_1
	else:
		tmp = t_0
	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(z / y))
	t_1 = Float64(Float64(x * y) / z)
	tmp = 0.0
	if (Float64(x * y) <= -1e+169)
		tmp = t_0;
	elseif (Float64(x * y) <= -5e-164)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e-283)
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(x * y) <= 2e+121)
		tmp = t_1;
	else
		tmp = t_0;
	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 / (z / y);
	t_1 = (x * y) / z;
	tmp = 0.0;
	if ((x * y) <= -1e+169)
		tmp = t_0;
	elseif ((x * y) <= -5e-164)
		tmp = t_1;
	elseif ((x * y) <= 5e-283)
		tmp = y * (x / z);
	elseif ((x * y) <= 2e+121)
		tmp = t_1;
	else
		tmp = t_0;
	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[(z / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e+169], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], -5e-164], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e-283], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+121], t$95$1, t$95$0]]]]]]

Target

Original	6.1
Target	5.9
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 3 regimes

2. if (*.f64 x y) < -9.99999999999999934e168 or 2.00000000000000007e121 < (*.f64 x y)

   1. Initial program 18.3

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

   2. Simplified 2.8

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
      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)): 48 points increase in error, 57 points decrease in error

   3. Applied egg-rr 3.0

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

   if -9.99999999999999934e168 < (*.f64 x y) < -4.99999999999999962e-164 or 5.0000000000000001e-283 < (*.f64 x y) < 2.00000000000000007e121

   1. Initial program 0.2

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

   if -4.99999999999999962e-164 < (*.f64 x y) < 5.0000000000000001e-283

   1. Initial program 10.8

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

   2. Taylor expanded in x around 0 10.8

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

   3. Simplified 0.6

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]
      Proof
      (*.f64 y (/.f64 x z)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y x) z)): 55 points increase in error, 54 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+169}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-164}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-283}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.4
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ t_1 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -5.682759260278493 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.0954529066542809 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	6.2
Cost	848

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ t_1 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -5.682759260278493 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.0954529066542809 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	6.2
Cost	320

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

Alternative 4
Error	6.3
Cost	320

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

Reproduce

herbie shell --seed 2022300 
(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))