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

Average Error: 35.49% → 0.42%

Time: 9.9s

Precision: binary64

Cost: 969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -4600000000 \lor \neg \left(y \leq 52000000000\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4600000000.0) (not (<= y 52000000000.0)))
   (+ x (/ 1.0 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 <= -4600000000.0) || !(y <= 52000000000.0)) {
		tmp = x + (1.0 / 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 <= (-4600000000.0d0)) .or. (.not. (y <= 52000000000.0d0))) then
        tmp = x + (1.0d0 / 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 <= -4600000000.0) || !(y <= 52000000000.0)) {
		tmp = x + (1.0 / 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 <= -4600000000.0) or not (y <= 52000000000.0):
		tmp = x + (1.0 / 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 <= -4600000000.0) || !(y <= 52000000000.0))
		tmp = Float64(x + Float64(1.0 / 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 <= -4600000000.0) || ~((y <= 52000000000.0)))
		tmp = x + (1.0 / 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, -4600000000.0], N[Not[LessEqual[y, 52000000000.0]], $MachinePrecision]], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	35.49%
Target	0.44%
Herbie	0.42%

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

Derivation

1. Split input into 2 regimes

2. if y < -4.6e9 or 5.2e10 < y

   1. Initial program 72.09

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

   2. Simplified 45.29

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

      [Start] 72.09	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      remove-double-neg [<=] 72.09	\[ 1 - \color{blue}{\left(-\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)\right)} \]
      neg-mul-1 [=>] 72.09	\[ 1 - \left(-\color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}}\right) \]
      associate-*l/ [<=] 45.29	\[ 1 - \left(--1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)}\right) \]
      associate-*r* [=>] 45.29	\[ 1 - \left(-\color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y}\right) \]
      distribute-lft-neg-in [=>] 45.29	\[ 1 - \color{blue}{\left(--1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} \]
      distribute-lft-neg-in [=>] 45.29	\[ 1 - \color{blue}{\left(\left(--1\right) \cdot \frac{1 - x}{y + 1}\right)} \cdot y \]
      metadata-eval [=>] 45.29	\[ 1 - \left(\color{blue}{1} \cdot \frac{1 - x}{y + 1}\right) \cdot y \]
      *-lft-identity [=>] 45.29	\[ 1 - \color{blue}{\frac{1 - x}{y + 1}} \cdot y \]
      +-commutative [=>] 45.29	\[ 1 - \frac{1 - x}{\color{blue}{1 + y}} \cdot y \]

   3. Taylor expanded in y around inf 0.14

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

   4. Simplified 0.14

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

      [Start] 0.14	\[ \left(\frac{1}{y} + x\right) - \frac{x}{y} \]
      +-commutative [=>] 0.14	\[ \color{blue}{\left(x + \frac{1}{y}\right)} - \frac{x}{y} \]
      associate--l+ [=>] 0.14	\[ \color{blue}{x + \left(\frac{1}{y} - \frac{x}{y}\right)} \]
      div-sub [<=] 0.14	\[ x + \color{blue}{\frac{1 - x}{y}} \]
      sub-neg [=>] 0.14	\[ x + \frac{\color{blue}{1 + \left(-x\right)}}{y} \]
      mul-1-neg [<=] 0.14	\[ x + \frac{1 + \color{blue}{-1 \cdot x}}{y} \]
      +-commutative [=>] 0.14	\[ x + \frac{\color{blue}{-1 \cdot x + 1}}{y} \]
      metadata-eval [<=] 0.14	\[ x + \frac{-1 \cdot x + \color{blue}{-1 \cdot -1}}{y} \]
      distribute-lft-in [<=] 0.14	\[ x + \frac{\color{blue}{-1 \cdot \left(x + -1\right)}}{y} \]
      metadata-eval [<=] 0.14	\[ x + \frac{-1 \cdot \left(x + \color{blue}{\left(-1\right)}\right)}{y} \]
      sub-neg [<=] 0.14	\[ x + \frac{-1 \cdot \color{blue}{\left(x - 1\right)}}{y} \]
      associate-*r/ [<=] 0.14	\[ x + \color{blue}{-1 \cdot \frac{x - 1}{y}} \]
      mul-1-neg [=>] 0.14	\[ x + \color{blue}{\left(-\frac{x - 1}{y}\right)} \]
      unsub-neg [=>] 0.14	\[ \color{blue}{x - \frac{x - 1}{y}} \]
      sub-neg [=>] 0.14	\[ x - \frac{\color{blue}{x + \left(-1\right)}}{y} \]
      metadata-eval [=>] 0.14	\[ x - \frac{x + \color{blue}{-1}}{y} \]

   5. Taylor expanded in x around 0 0.28

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

   if -4.6e9 < y < 5.2e10

   1. Initial program 0.55

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

4. Final simplification 0.42

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4600000000 \lor \neg \left(y \leq 52000000000\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -215000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 410000000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 2
Error	1.78%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.8\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 3
Error	1.64%
Cost	712

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

Alternative 4
Error	1.64%
Cost	712

\[\begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 0.82:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	13.73%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.11\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 6
Error	2.02%
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	25.95%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.95:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	26.19%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.485:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	61.59%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023115 
(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))))