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

Average Error: 6.0 → 0.4

Time: 2.3s

Precision: binary64

Cost: 1360

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+222}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ y z))))
   (if (<= (* x y) (- INFINITY))
     t_1
     (if (<= (* x y) -2e-232)
       t_0
       (if (<= (* x y) 0.0) (* y (/ x z)) (if (<= (* x y) 5e+222) 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 * (y / z);
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((x * y) <= -2e-232) {
		tmp = t_0;
	} else if ((x * y) <= 0.0) {
		tmp = y * (x / z);
	} else if ((x * y) <= 5e+222) {
		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 * (y / z);
	double tmp;
	if ((x * y) <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if ((x * y) <= -2e-232) {
		tmp = t_0;
	} else if ((x * y) <= 0.0) {
		tmp = y * (x / z);
	} else if ((x * y) <= 5e+222) {
		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 * (y / z)
	tmp = 0
	if (x * y) <= -math.inf:
		tmp = t_1
	elif (x * y) <= -2e-232:
		tmp = t_0
	elif (x * y) <= 0.0:
		tmp = y * (x / z)
	elif (x * y) <= 5e+222:
		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(x * Float64(y / z))
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(x * y) <= -2e-232)
		tmp = t_0;
	elseif (Float64(x * y) <= 0.0)
		tmp = Float64(y * Float64(x / z));
	elseif (Float64(x * y) <= 5e+222)
		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 * (y / z);
	tmp = 0.0;
	if ((x * y) <= -Inf)
		tmp = t_1;
	elseif ((x * y) <= -2e-232)
		tmp = t_0;
	elseif ((x * y) <= 0.0)
		tmp = y * (x / z);
	elseif ((x * y) <= 5e+222)
		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[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -2e-232], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 0.0], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+222], t$95$0, t$95$1]]]]]]

Target

Original	6.0
Target	6.0
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 3 regimes

2. if (*.f64 x y) < -inf.0 or 5.00000000000000023e222 < (*.f64 x y)

   1. Initial program 38.1

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

   2. Simplified 0.8

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

   if -inf.0 < (*.f64 x y) < -2.00000000000000005e-232 or -0.0 < (*.f64 x y) < 5.00000000000000023e222

   1. Initial program 0.4

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

   if -2.00000000000000005e-232 < (*.f64 x y) < -0.0

   1. Initial program 15.1

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
      Proof
      (*.f64 (/.f64 x z) y): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x y) z)): 49 points increase in error, 51 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-232}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{-19}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Error	6.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Error	6.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Error	6.1
Cost	320

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

Reproduce

herbie shell --seed 2022331 
(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))