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

Average Error: 6.1 → 0.4

Time: 2.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -4.057341715934731 \cdot 10^{-209}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 5.982970990941407 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 1.6037495896397977 \cdot 10^{+206}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

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) (- INFINITY))
     (* x (/ y z))
     (if (<= (* x y) -4.057341715934731e-209)
       t_0
       (if (<= (* x y) 5.982970990941407e-226)
         (/ x (/ z y))
         (if (<= (* x y) 1.6037495896397977e+206) 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) <= -((double) INFINITY)) {
		tmp = x * (y / z);
	} else if ((x * y) <= -4.057341715934731e-209) {
		tmp = t_0;
	} else if ((x * y) <= 5.982970990941407e-226) {
		tmp = x / (z / y);
	} else if ((x * y) <= 1.6037495896397977e+206) {
		tmp = t_0;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

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) <= -Double.POSITIVE_INFINITY) {
		tmp = x * (y / z);
	} else if ((x * y) <= -4.057341715934731e-209) {
		tmp = t_0;
	} else if ((x * y) <= 5.982970990941407e-226) {
		tmp = x / (z / y);
	} else if ((x * y) <= 1.6037495896397977e+206) {
		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) <= -math.inf:
		tmp = x * (y / z)
	elif (x * y) <= -4.057341715934731e-209:
		tmp = t_0
	elif (x * y) <= 5.982970990941407e-226:
		tmp = x / (z / y)
	elif (x * y) <= 1.6037495896397977e+206:
		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) <= Float64(-Inf))
		tmp = Float64(x * Float64(y / z));
	elseif (Float64(x * y) <= -4.057341715934731e-209)
		tmp = t_0;
	elseif (Float64(x * y) <= 5.982970990941407e-226)
		tmp = Float64(x / Float64(z / y));
	elseif (Float64(x * y) <= 1.6037495896397977e+206)
		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) <= -Inf)
		tmp = x * (y / z);
	elseif ((x * y) <= -4.057341715934731e-209)
		tmp = t_0;
	elseif ((x * y) <= 5.982970990941407e-226)
		tmp = x / (z / y);
	elseif ((x * y) <= 1.6037495896397977e+206)
		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], (-Infinity)], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -4.057341715934731e-209], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 5.982970990941407e-226], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.6037495896397977e+206], t$95$0, N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	6.4
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) < -inf.0

   1. Initial program 64.0

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

   2. Applied egg-rr 0.3

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

   3. Applied egg-rr 0.3

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

   4. Applied egg-rr 0.3

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

   if -inf.0 < (*.f64 x y) < -4.0573417159347308e-209 or 5.98297099094140689e-226 < (*.f64 x y) < 1.6037495896397977e206

   1. Initial program 0.2

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

   if -4.0573417159347308e-209 < (*.f64 x y) < 5.98297099094140689e-226

   1. Initial program 11.6

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

   2. Applied egg-rr 0.5

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

   3. Applied egg-rr 0.5

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

   if 1.6037495896397977e206 < (*.f64 x y)

   1. Initial program 25.7

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

   2. Applied egg-rr 1.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -4.057341715934731 \cdot 10^{-209}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5.982970990941407 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 1.6037495896397977 \cdot 10^{+206}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))