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

Average Error: 6.0 → 0.5

Time: 2.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ t_1 := \frac{\frac{x}{z}}{\frac{1}{y}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x \cdot \left(y \cdot {z}^{-1}\right)\\ \mathbf{elif}\;t_0 \leq -8.65 \cdot 10^{-322}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 6.967586597423559 \cdot 10^{+277}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)) (t_1 (/ (/ x z) (/ 1.0 y))))
   (if (<= t_0 (- INFINITY))
     (* x (* y (pow z -1.0)))
     (if (<= t_0 -8.65e-322)
       t_0
       (if (<= t_0 0.0) t_1 (if (<= t_0 6.967586597423559e+277) t_0 t_1))))))

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 t_1 = (x / z) / (1.0 / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x * (y * pow(z, -1.0));
	} else if (t_0 <= -8.65e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 6.967586597423559e+277) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 t_1 = (x / z) / (1.0 / y);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = x * (y * Math.pow(z, -1.0));
	} else if (t_0 <= -8.65e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 6.967586597423559e+277) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x * y) / z
	t_1 = (x / z) / (1.0 / y)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = x * (y * math.pow(z, -1.0))
	elif t_0 <= -8.65e-322:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 6.967586597423559e+277:
		tmp = t_0
	else:
		tmp = t_1
	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)
	t_1 = Float64(Float64(x / z) / Float64(1.0 / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x * Float64(y * (z ^ -1.0)));
	elseif (t_0 <= -8.65e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 6.967586597423559e+277)
		tmp = t_0;
	else
		tmp = t_1;
	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;
	t_1 = (x / z) / (1.0 / y);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = x * (y * (z ^ -1.0));
	elseif (t_0 <= -8.65e-322)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 6.967586597423559e+277)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] / N[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x * N[(y * N[Power[z, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -8.65e-322], t$95$0, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 6.967586597423559e+277], t$95$0, t$95$1]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	6.2
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x y) z) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.3

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

   3. Applied egg-rr 1.3

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{\frac{z}{y}}}\right)}^{3}} \]

   4. Applied egg-rr 0.3

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

   5. Applied egg-rr 0.3

      \[\leadsto \color{blue}{x \cdot \left({z}^{-1} \cdot y\right)} \]

   if -inf.0 < (/.f64 (*.f64 x y) z) < -8.64615e-322 or -0.0 < (/.f64 (*.f64 x y) z) < 6.9675865974235591e277

   1. Initial program 0.5

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

   if -8.64615e-322 < (/.f64 (*.f64 x y) z) < -0.0 or 6.9675865974235591e277 < (/.f64 (*.f64 x y) z)

   1. Initial program 15.7

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

   2. Simplified 1.4

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

   3. Applied egg-rr 1.6

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{\frac{z}{y}}}\right)}^{3}} \]

   4. Applied egg-rr 0.7

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{1}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -\infty:\\ \;\;\;\;x \cdot \left(y \cdot {z}^{-1}\right)\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq -8.65 \cdot 10^{-322}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 0:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 6.967586597423559 \cdot 10^{+277}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))