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

Average Error: 8.4 → 0.0

Time: 3.3s

Precision: binary64

Cost: 712

Math

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

↓

\[\begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -33000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -33000000.0)
     t_0
     (if (<= y 155000000.0) (/ (* x y) (+ y 1.0)) t_0))))

C

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

↓

double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -33000000.0) {
		tmp = t_0;
	} else if (y <= 155000000.0) {
		tmp = (x * y) / (y + 1.0);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = x - (x / y)
    if (y <= (-33000000.0d0)) then
        tmp = t_0
    else if (y <= 155000000.0d0) then
        tmp = (x * y) / (y + 1.0d0)
    else
        tmp = t_0
    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 t_0 = x - (x / y);
	double tmp;
	if (y <= -33000000.0) {
		tmp = t_0;
	} else if (y <= 155000000.0) {
		tmp = (x * y) / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = x - (x / y)
	tmp = 0
	if y <= -33000000.0:
		tmp = t_0
	elif y <= 155000000.0:
		tmp = (x * y) / (y + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = Float64(x - Float64(x / y))
	tmp = 0.0
	if (y <= -33000000.0)
		tmp = t_0;
	elseif (y <= 155000000.0)
		tmp = Float64(Float64(x * y) / Float64(y + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = x - (x / y);
	tmp = 0.0;
	if (y <= -33000000.0)
		tmp = t_0;
	elseif (y <= 155000000.0)
		tmp = (x * y) / (y + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -33000000.0], t$95$0, If[LessEqual[y, 155000000.0], N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	8.4
Target	0.0
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if y < -3.3e7 or 1.55e8 < y

   1. Initial program 17.2

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

   2. Taylor expanded in y around inf 0.0

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

   3. Simplified 0.0

      \[\leadsto \color{blue}{x + \left(-\frac{x}{y}\right)} \]
      Proof

      [Start] 0.0	\[ -1 \cdot \frac{x}{y} + x \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.0	\[ \color{blue}{x + -1 \cdot \frac{x}{y}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ x + \color{blue}{\frac{x}{y} \cdot -1} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 0.0	\[ x + \color{blue}{\left(-\frac{x}{y}\right)} \]

   4. Taylor expanded in y around 0 0.0

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

   5. Simplified 0.0

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

      [Start] 0.0	\[ x + -1 \cdot \frac{x}{y} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ x + \color{blue}{\frac{x}{y} \cdot -1} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 0.0	\[ x + \color{blue}{\left(-\frac{x}{y}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 0.0	\[ x + \color{blue}{\left(0 - \frac{x}{y}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 0.0	\[ \color{blue}{\left(0 + x\right) - \frac{x}{y}} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.0	\[ \color{blue}{\left(x + 0\right)} - \frac{x}{y} \]
      rational_best_oopsla_all_46_json_45_simplify-38 [<=] 0.0	\[ \left(x + \color{blue}{\frac{x - \frac{x}{y}}{x - \frac{x}{y}} \cdot 0}\right) - \frac{x}{y} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.0	\[ \left(x + \color{blue}{0 \cdot \frac{x - \frac{x}{y}}{x - \frac{x}{y}}}\right) - \frac{x}{y} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [<=] 0.0	\[ \color{blue}{0 \cdot \frac{x - \frac{x}{y}}{x - \frac{x}{y}} + \left(x - \frac{x}{y}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.0	\[ \color{blue}{\left(x - \frac{x}{y}\right) + 0 \cdot \frac{x - \frac{x}{y}}{x - \frac{x}{y}}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [<=] 0.0	\[ \color{blue}{\left(x - \frac{x}{y}\right) \cdot 1} + 0 \cdot \frac{x - \frac{x}{y}}{x - \frac{x}{y}} \]
      rational_best_oopsla_all_46_json_45_simplify-89 [=>] 0.0	\[ \color{blue}{\left(x - \frac{x}{y}\right) \cdot \left(1 \cdot \frac{x - \frac{x}{y}}{x - \frac{x}{y}}\right)} + 0 \cdot \frac{x - \frac{x}{y}}{x - \frac{x}{y}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ \left(x - \frac{x}{y}\right) \cdot \color{blue}{\left(\frac{x - \frac{x}{y}}{x - \frac{x}{y}} \cdot 1\right)} + 0 \cdot \frac{x - \frac{x}{y}}{x - \frac{x}{y}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 0.0	\[ \left(x - \frac{x}{y}\right) \cdot \color{blue}{\frac{x - \frac{x}{y}}{x - \frac{x}{y}}} + 0 \cdot \frac{x - \frac{x}{y}}{x - \frac{x}{y}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ \color{blue}{\frac{x - \frac{x}{y}}{x - \frac{x}{y}} \cdot \left(x - \frac{x}{y}\right)} + 0 \cdot \frac{x - \frac{x}{y}}{x - \frac{x}{y}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ \frac{x - \frac{x}{y}}{x - \frac{x}{y}} \cdot \left(x - \frac{x}{y}\right) + \color{blue}{\frac{x - \frac{x}{y}}{x - \frac{x}{y}} \cdot 0} \]
      rational_best_oopsla_all_46_json_45_simplify-38 [=>] 0.0	\[ \frac{x - \frac{x}{y}}{x - \frac{x}{y}} \cdot \left(x - \frac{x}{y}\right) + \color{blue}{0} \]
      rational_best_oopsla_all_46_json_45_simplify-38 [<=] 0.0	\[ \frac{x - \frac{x}{y}}{x - \frac{x}{y}} \cdot \left(x - \frac{x}{y}\right) + \color{blue}{\left(x - \frac{x}{y}\right) \cdot 0} \]
      rational_best_oopsla_all_46_json_45_simplify-37 [<=] 0.0	\[ \color{blue}{\left(x - \frac{x}{y}\right) \cdot \left(\frac{x - \frac{x}{y}}{x - \frac{x}{y}} + 0\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 0.0	\[ \left(x - \frac{x}{y}\right) \cdot \color{blue}{\frac{x - \frac{x}{y}}{x - \frac{x}{y}}} \]

   if -3.3e7 < y < 1.55e8

   1. Initial program 0.0

      \[\frac{x \cdot y}{y + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -33000000:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	584

\[\begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.3
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 3
Error	31.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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)))