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

Average Error: 6.3 → 2.3

Time: 4.3s

Precision: binary64

Cost: 1100

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

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) -1e-190)
     t_0
     (if (<= (* x y) 0.0)
       (/ x (/ z y))
       (if (<= (* x y) 4e+123) 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) <= -1e-190) {
		tmp = t_0;
	} else if ((x * y) <= 0.0) {
		tmp = x / (z / y);
	} else if ((x * y) <= 4e+123) {
		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) <= (-1d-190)) then
        tmp = t_0
    else if ((x * y) <= 0.0d0) then
        tmp = x / (z / y)
    else if ((x * y) <= 4d+123) 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) <= -1e-190) {
		tmp = t_0;
	} else if ((x * y) <= 0.0) {
		tmp = x / (z / y);
	} else if ((x * y) <= 4e+123) {
		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) <= -1e-190:
		tmp = t_0
	elif (x * y) <= 0.0:
		tmp = x / (z / y)
	elif (x * y) <= 4e+123:
		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) <= -1e-190)
		tmp = t_0;
	elseif (Float64(x * y) <= 0.0)
		tmp = Float64(x / Float64(z / y));
	elseif (Float64(x * y) <= 4e+123)
		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) <= -1e-190)
		tmp = t_0;
	elseif ((x * y) <= 0.0)
		tmp = x / (z / y);
	elseif ((x * y) <= 4e+123)
		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], -1e-190], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 0.0], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+123], t$95$0, N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	6.3
Target	6.1
Herbie	2.3

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e-190 or -0.0 < (*.f64 x y) < 3.99999999999999991e123

   1. Initial program 2.6

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

   if -1e-190 < (*.f64 x y) < -0.0

   1. Initial program 13.9

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

   2. Simplified 0.6

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

      [Start] 13.9	\[ \frac{x \cdot y}{z} \]
      rational.json-simplify-2 [=>] 13.9	\[ \frac{\color{blue}{y \cdot x}}{z} \]
      rational.json-simplify-49 [=>] 0.6	\[ \color{blue}{x \cdot \frac{y}{z}} \]

   3. Applied egg-rr 0.7

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

   if 3.99999999999999991e123 < (*.f64 x y)

   1. Initial program 14.7

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

   2. Simplified 3.6

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

      [Start] 14.7	\[ \frac{x \cdot y}{z} \]
      rational.json-simplify-2 [=>] 14.7	\[ \frac{\color{blue}{y \cdot x}}{z} \]
      rational.json-simplify-49 [=>] 3.6	\[ \color{blue}{x \cdot \frac{y}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-190}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+123}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-153}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Error	6.5
Cost	320

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

Reproduce

herbie shell --seed 2023075 
(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))