Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, B

Average Accuracy: 87.4% → 100.0%

Time: 3.8s

Precision: binary64

Cost: 712

Math

\[\frac{x \cdot y}{y + 1} \]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -125000000:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* x y) (+ y 1.0)))

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -125000000.0)
   (- x (/ x y))
   (if (<= y 5.5e+15) (* y (/ x (+ y 1.0))) x)))

C

double code(double x, double y) {
	return (x * y) / (y + 1.0);
}

↓

double code(double x, double y) {
	double tmp;
	if (y <= -125000000.0) {
		tmp = x - (x / y);
	} else if (y <= 5.5e+15) {
		tmp = y * (x / (y + 1.0));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (y + 1.0d0)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-125000000.0d0)) then
        tmp = x - (x / y)
    else if (y <= 5.5d+15) then
        tmp = y * (x / (y + 1.0d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return (x * y) / (y + 1.0);
}

↓

public static double code(double x, double y) {
	double tmp;
	if (y <= -125000000.0) {
		tmp = x - (x / y);
	} else if (y <= 5.5e+15) {
		tmp = y * (x / (y + 1.0));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	return (x * y) / (y + 1.0)

↓

def code(x, y):
	tmp = 0
	if y <= -125000000.0:
		tmp = x - (x / y)
	elif y <= 5.5e+15:
		tmp = y * (x / (y + 1.0))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(y + 1.0))
end

↓

function code(x, y)
	tmp = 0.0
	if (y <= -125000000.0)
		tmp = Float64(x - Float64(x / y));
	elseif (y <= 5.5e+15)
		tmp = Float64(y * Float64(x / Float64(y + 1.0)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (y + 1.0);
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -125000000.0)
		tmp = x - (x / y);
	elseif (y <= 5.5e+15)
		tmp = y * (x / (y + 1.0));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := If[LessEqual[y, -125000000.0], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e+15], N[(y * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Target

Original	87.4%
Target	100.0%
Herbie	100.0%

\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if y < -1.25e8

   1. Initial program 74.4%

      \[\frac{x \cdot y}{y + 1} \]

   2. Simplified 100.0%

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

      [Start] 74.4	\[ \frac{x \cdot y}{y + 1} \]
      associate-*r/ [<=] 100.0	\[ \color{blue}{x \cdot \frac{y}{y + 1}} \]
      *-commutative [<=] 100.0	\[ \color{blue}{\frac{y}{y + 1} \cdot x} \]

   3. Taylor expanded in y around inf 100.0%

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

   4. Simplified 100.0%

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

      [Start] 100.0	\[ -1 \cdot \frac{x}{y} + x \]
      +-commutative [=>] 100.0	\[ \color{blue}{x + -1 \cdot \frac{x}{y}} \]
      mul-1-neg [=>] 100.0	\[ x + \color{blue}{\left(-\frac{x}{y}\right)} \]
      unsub-neg [=>] 100.0	\[ \color{blue}{x - \frac{x}{y}} \]

   if -1.25e8 < y < 5.5e15

   1. Initial program 99.9%

      \[\frac{x \cdot y}{y + 1} \]

   2. Simplified 99.6%

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

      [Start] 99.9	\[ \frac{x \cdot y}{y + 1} \]
      *-commutative [=>] 99.9	\[ \frac{\color{blue}{y \cdot x}}{y + 1} \]
      associate-/l* [=>] 99.6	\[ \color{blue}{\frac{y}{\frac{y + 1}{x}}} \]

   3. Applied egg-rr 100.0%

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

      [Start] 99.6	\[ \frac{y}{\frac{y + 1}{x}} \]
      clear-num [=>] 98.5	\[ \color{blue}{\frac{1}{\frac{\frac{y + 1}{x}}{y}}} \]
      associate-/r/ [=>] 99.8	\[ \color{blue}{\frac{1}{\frac{y + 1}{x}} \cdot y} \]
      clear-num [<=] 100.0	\[ \color{blue}{\frac{x}{y + 1}} \cdot y \]

   if 5.5e15 < y

   1. Initial program 73.3%

      \[\frac{x \cdot y}{y + 1} \]

   2. Simplified 100.0%

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

      [Start] 73.3	\[ \frac{x \cdot y}{y + 1} \]
      associate-*r/ [<=] 100.0	\[ \color{blue}{x \cdot \frac{y}{y + 1}} \]
      *-commutative [<=] 100.0	\[ \color{blue}{\frac{y}{y + 1} \cdot x} \]

   3. Taylor expanded in y around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -125000000:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	98.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.3\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Accuracy	98.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	100.0%
Cost	448

\[\frac{y}{y + 1} \cdot x \]

Alternative 5
Accuracy	50.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023153 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 6799310503.41891) (/ (* x y) (+ y 1.0)) (- (/ x (* y y)) (- (/ x y) x))))

  (/ (* x y) (+ y 1.0)))