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

Average Accuracy: 90.8% → 98.9%

Time: 3.2s

Precision: binary64

Cost: 1360.00

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

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-137}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

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))
     t_0
     (if (<= (* x y) -5e-137)
       (* (* x y) (/ 1.0 z))
       (if (<= (* x y) 0.0)
         t_0
         (if (<= (* x y) 2e+188) (/ (* x y) z) (* 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 = t_0;
	} else if ((x * y) <= -5e-137) {
		tmp = (x * y) * (1.0 / z);
	} else if ((x * y) <= 0.0) {
		tmp = t_0;
	} else if ((x * y) <= 2e+188) {
		tmp = (x * y) / z;
	} 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 = t_0;
	} else if ((x * y) <= -5e-137) {
		tmp = (x * y) * (1.0 / z);
	} else if ((x * y) <= 0.0) {
		tmp = t_0;
	} else if ((x * y) <= 2e+188) {
		tmp = (x * y) / z;
	} 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 = t_0
	elif (x * y) <= -5e-137:
		tmp = (x * y) * (1.0 / z)
	elif (x * y) <= 0.0:
		tmp = t_0
	elif (x * y) <= 2e+188:
		tmp = (x * y) / z
	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(x * Float64(y / z))
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(x * y) <= -5e-137)
		tmp = Float64(Float64(x * y) * Float64(1.0 / z));
	elseif (Float64(x * y) <= 0.0)
		tmp = t_0;
	elseif (Float64(x * y) <= 2e+188)
		tmp = Float64(Float64(x * y) / z);
	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 = t_0;
	elseif ((x * y) <= -5e-137)
		tmp = (x * y) * (1.0 / z);
	elseif ((x * y) <= 0.0)
		tmp = t_0;
	elseif ((x * y) <= 2e+188)
		tmp = (x * y) / z;
	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[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], -5e-137], N[(N[(x * y), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 2e+188], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	90.8%
Target	89.8%
Herbie	98.9%

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -inf.0 or -5.00000000000000001e-137 < (*.f64 x y) < 0.0

   1. Initial program 76.5%

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

   2. Simplified 98.0%

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

      [Start] 76.5	\[ \frac{x \cdot y}{z} \]
      associate-*r/ [<=] 98.0	\[ \color{blue}{x \cdot \frac{y}{z}} \]

   if -inf.0 < (*.f64 x y) < -5.00000000000000001e-137

   1. Initial program 99.6%

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

   2. Applied egg-rr 99.5%

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

   if 0.0 < (*.f64 x y) < 2e188

   1. Initial program 99.3%

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

   if 2e188 < (*.f64 x y)

   1. Initial program 65.3%

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

   2. Simplified 97.7%

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

      [Start] 65.3	\[ \frac{x \cdot y}{z} \]
      associate-*l/ [<=] 97.7	\[ \color{blue}{\frac{x}{z} \cdot y} \]

3. Recombined 4 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-137}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.7%
Cost	1360.00

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+188}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	90.4%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-209}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Accuracy	90.1%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-229}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Accuracy	90.2%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 5
Accuracy	89.9%
Cost	320.00

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

Reproduce

herbie shell --seed 2023096 
(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))