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

Average Error: 22.4 → 0.1

Time: 9.1s

Precision: binary64

Cost: 1225

Math

\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -280000 \lor \neg \left(y \leq 360000\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -280000.0) (not (<= y 360000.0)))
   (+ (+ x (/ (+ x -1.0) (* y y))) (/ (- 1.0 x) y))
   (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if ((y <= -280000.0) || !(y <= 360000.0)) {
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - (((1.0d0 - 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 <= (-280000.0d0)) .or. (.not. (y <= 360000.0d0))) then
        tmp = (x + ((x + (-1.0d0)) / (y * y))) + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y) {
	double tmp;
	if ((y <= -280000.0) || !(y <= 360000.0)) {
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if (y <= -280000.0) or not (y <= 360000.0):
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y)
	else:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	return tmp

Julia

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

↓

function code(x, y)
	tmp = 0.0
	if ((y <= -280000.0) || !(y <= 360000.0))
		tmp = Float64(Float64(x + Float64(Float64(x + -1.0) / Float64(y * y))) + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -280000.0) || ~((y <= 360000.0)))
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y);
	else
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := If[Or[LessEqual[y, -280000.0], N[Not[LessEqual[y, 360000.0]], $MachinePrecision]], N[(N[(x + N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	22.4
Target	0.2
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if y < -2.8e5 or 3.6e5 < y

   1. Initial program 45.4

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Simplified 29.1

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

      [Start] 45.4	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      remove-double-neg [<=] 45.4	\[ 1 - \color{blue}{\left(-\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)\right)} \]
      distribute-neg-frac [=>] 45.4	\[ 1 - \left(-\color{blue}{\frac{-\left(1 - x\right) \cdot y}{y + 1}}\right) \]
      distribute-rgt-neg-in [=>] 45.4	\[ 1 - \left(-\frac{\color{blue}{\left(1 - x\right) \cdot \left(-y\right)}}{y + 1}\right) \]
      associate-/l* [=>] 29.1	\[ 1 - \left(-\color{blue}{\frac{1 - x}{\frac{y + 1}{-y}}}\right) \]
      distribute-frac-neg [<=] 29.1	\[ 1 - \color{blue}{\frac{-\left(1 - x\right)}{\frac{y + 1}{-y}}} \]
      /-rgt-identity [<=] 29.1	\[ 1 - \frac{\color{blue}{\frac{-\left(1 - x\right)}{1}}}{\frac{y + 1}{-y}} \]
      neg-mul-1 [=>] 29.1	\[ 1 - \frac{\frac{\color{blue}{-1 \cdot \left(1 - x\right)}}{1}}{\frac{y + 1}{-y}} \]
      *-commutative [=>] 29.1	\[ 1 - \frac{\frac{\color{blue}{\left(1 - x\right) \cdot -1}}{1}}{\frac{y + 1}{-y}} \]
      associate-/l* [=>] 29.1	\[ 1 - \frac{\color{blue}{\frac{1 - x}{\frac{1}{-1}}}}{\frac{y + 1}{-y}} \]
      metadata-eval [=>] 29.1	\[ 1 - \frac{\frac{1 - x}{\color{blue}{-1}}}{\frac{y + 1}{-y}} \]
      associate-/r* [<=] 29.1	\[ 1 - \color{blue}{\frac{1 - x}{-1 \cdot \frac{y + 1}{-y}}} \]
      mul-1-neg [=>] 29.1	\[ 1 - \frac{1 - x}{\color{blue}{-\frac{y + 1}{-y}}} \]
      *-rgt-identity [<=] 29.1	\[ 1 - \frac{1 - x}{-\color{blue}{\frac{y + 1}{-y} \cdot 1}} \]
      mul-1-neg [<=] 29.1	\[ 1 - \frac{1 - x}{-\frac{y + 1}{\color{blue}{-1 \cdot y}} \cdot 1} \]
      associate-*l/ [=>] 29.1	\[ 1 - \frac{1 - x}{-\color{blue}{\frac{\left(y + 1\right) \cdot 1}{-1 \cdot y}}} \]
      *-commutative [=>] 29.1	\[ 1 - \frac{1 - x}{-\frac{\color{blue}{1 \cdot \left(y + 1\right)}}{-1 \cdot y}} \]
      times-frac [=>] 29.1	\[ 1 - \frac{1 - x}{-\color{blue}{\frac{1}{-1} \cdot \frac{y + 1}{y}}} \]
      metadata-eval [=>] 29.1	\[ 1 - \frac{1 - x}{-\color{blue}{-1} \cdot \frac{y + 1}{y}} \]
      mul-1-neg [=>] 29.1	\[ 1 - \frac{1 - x}{-\color{blue}{\left(-\frac{y + 1}{y}\right)}} \]
      remove-double-neg [=>] 29.1	\[ 1 - \frac{1 - x}{\color{blue}{\frac{y + 1}{y}}} \]

   3. Taylor expanded in y around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{y} + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right)\right) - \frac{x}{y}} \]

   4. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{y} + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right)\right) - \frac{x}{y} \]
      associate--l+ [=>] 0.0	\[ \color{blue}{\frac{1}{y} + \left(\left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right) - \frac{x}{y}\right)} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(\left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right) - \frac{x}{y}\right) + \frac{1}{y}} \]
      associate-+l- [=>] 0.0	\[ \color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right) - \left(\frac{x}{y} - \frac{1}{y}\right)} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(x + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}}\right)} - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      mul-1-neg [=>] 0.0	\[ \left(x + \color{blue}{\left(-\frac{1 + -1 \cdot x}{{y}^{2}}\right)}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      unsub-neg [=>] 0.0	\[ \color{blue}{\left(x - \frac{1 + -1 \cdot x}{{y}^{2}}\right)} - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      mul-1-neg [=>] 0.0	\[ \left(x - \frac{1 + \color{blue}{\left(-x\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      sub-neg [<=] 0.0	\[ \left(x - \frac{\color{blue}{1 - x}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      unpow2 [=>] 0.0	\[ \left(x - \frac{1 - x}{\color{blue}{y \cdot y}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      div-sub [<=] 0.0	\[ \left(x - \frac{1 - x}{y \cdot y}\right) - \color{blue}{\frac{x - 1}{y}} \]
      sub-neg [=>] 0.0	\[ \left(x - \frac{1 - x}{y \cdot y}\right) - \frac{\color{blue}{x + \left(-1\right)}}{y} \]
      metadata-eval [=>] 0.0	\[ \left(x - \frac{1 - x}{y \cdot y}\right) - \frac{x + \color{blue}{-1}}{y} \]

   if -2.8e5 < y < 3.6e5

   1. Initial program 0.1

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -280000 \lor \neg \left(y \leq 360000\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -98000000 \lor \neg \left(y \leq 120000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -130000000 \lor \neg \left(y \leq 165000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternative 3
Error	1.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.78\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \end{array} \]

Alternative 4
Error	0.9
Cost	713

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

Alternative 5
Error	9.1
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.048\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 6
Error	1.3
Cost	585

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

Alternative 7
Error	16.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	16.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.7:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	39.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023040 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

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

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