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

Average Error: 22.6 → 0.2

Time: 16.5s

Precision: binary64

Cost: 1224

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+15}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 170000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -1.95e+15)
   (+ x (/ 1.0 y))
   (if (<= y 170000.0)
     (- 1.0 (/ (* y (- 1.0 x)) (+ y 1.0)))
     (+ (+ x (/ (+ x -1.0) (* y y))) (/ (- 1.0 x) y)))))

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 <= -1.95e+15) {
		tmp = x + (1.0 / y);
	} else if (y <= 170000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y);
	}
	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 <= (-1.95d+15)) then
        tmp = x + (1.0d0 / y)
    else if (y <= 170000.0d0) then
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (y + 1.0d0))
    else
        tmp = (x + ((x + (-1.0d0)) / (y * y))) + ((1.0d0 - x) / y)
    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 <= -1.95e+15) {
		tmp = x + (1.0 / y);
	} else if (y <= 170000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y);
	}
	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 <= -1.95e+15:
		tmp = x + (1.0 / y)
	elif y <= 170000.0:
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0))
	else:
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y)
	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 <= -1.95e+15)
		tmp = Float64(x + Float64(1.0 / y));
	elseif (y <= 170000.0)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x + Float64(Float64(x + -1.0) / Float64(y * y))) + Float64(Float64(1.0 - x) / y));
	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 <= -1.95e+15)
		tmp = x + (1.0 / y);
	elseif (y <= 170000.0)
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	else
		tmp = (x + ((x + -1.0) / (y * y))) + ((1.0 - x) / y);
	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[LessEqual[y, -1.95e+15], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 170000.0], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $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]]]

Target

Original	22.6
Target	0.2
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if y < -1.95e15

   1. Initial program 46.1

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

   2. Simplified 29.3

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

      [Start] 46.1	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      remove-double-neg [<=] 46.1	\[ 1 - \color{blue}{\left(-\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)\right)} \]
      neg-mul-1 [=>] 46.1	\[ 1 - \left(-\color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}}\right) \]
      associate-*l/ [<=] 29.3	\[ 1 - \left(--1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)}\right) \]
      associate-*r* [=>] 29.3	\[ 1 - \left(-\color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y}\right) \]
      distribute-lft-neg-in [=>] 29.3	\[ 1 - \color{blue}{\left(--1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} \]
      distribute-lft-neg-in [=>] 29.3	\[ 1 - \color{blue}{\left(\left(--1\right) \cdot \frac{1 - x}{y + 1}\right)} \cdot y \]
      metadata-eval [=>] 29.3	\[ 1 - \left(\color{blue}{1} \cdot \frac{1 - x}{y + 1}\right) \cdot y \]
      *-lft-identity [=>] 29.3	\[ 1 - \color{blue}{\frac{1 - x}{y + 1}} \cdot y \]
      +-commutative [=>] 29.3	\[ 1 - \frac{1 - x}{\color{blue}{1 + y}} \cdot y \]

   3. Taylor expanded in y around inf 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{y} + x\right) - \frac{x}{y} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(x + \frac{1}{y}\right)} - \frac{x}{y} \]
      associate--l+ [=>] 0.0	\[ \color{blue}{x + \left(\frac{1}{y} - \frac{x}{y}\right)} \]
      div-sub [<=] 0.0	\[ x + \color{blue}{\frac{1 - x}{y}} \]
      sub-neg [=>] 0.0	\[ x + \frac{\color{blue}{1 + \left(-x\right)}}{y} \]
      mul-1-neg [<=] 0.0	\[ x + \frac{1 + \color{blue}{-1 \cdot x}}{y} \]
      +-commutative [=>] 0.0	\[ x + \frac{\color{blue}{-1 \cdot x + 1}}{y} \]
      metadata-eval [<=] 0.0	\[ x + \frac{-1 \cdot x + \color{blue}{-1 \cdot -1}}{y} \]
      distribute-lft-in [<=] 0.0	\[ x + \frac{\color{blue}{-1 \cdot \left(x + -1\right)}}{y} \]
      metadata-eval [<=] 0.0	\[ x + \frac{-1 \cdot \left(x + \color{blue}{\left(-1\right)}\right)}{y} \]
      sub-neg [<=] 0.0	\[ x + \frac{-1 \cdot \color{blue}{\left(x - 1\right)}}{y} \]
      associate-*r/ [<=] 0.0	\[ x + \color{blue}{-1 \cdot \frac{x - 1}{y}} \]
      mul-1-neg [=>] 0.0	\[ x + \color{blue}{\left(-\frac{x - 1}{y}\right)} \]
      unsub-neg [=>] 0.0	\[ \color{blue}{x - \frac{x - 1}{y}} \]
      sub-neg [=>] 0.0	\[ x - \frac{\color{blue}{x + \left(-1\right)}}{y} \]
      metadata-eval [=>] 0.0	\[ x - \frac{x + \color{blue}{-1}}{y} \]

   5. Taylor expanded in x around 0 0.0

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

   if -1.95e15 < y < 1.7e5

   1. Initial program 0.3

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

   if 1.7e5 < y

   1. Initial program 46.1

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

   2. Simplified 29.1

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

      [Start] 46.1	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 46.1	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 46.1	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      neg-mul-1 [=>] 46.1	\[ \color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}} + 1 \]
      associate-*l/ [<=] 29.0	\[ -1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)} + 1 \]
      associate-*r* [=>] 29.0	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} + 1 \]
      fma-def [=>] 29.1	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{1 - x}{y + 1}, y, 1\right)} \]
      associate-*r/ [=>] 29.1	\[ \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \left(1 - x\right)}{y + 1}}, y, 1\right) \]
      neg-mul-1 [<=] 29.1	\[ \mathsf{fma}\left(\frac{\color{blue}{-\left(1 - x\right)}}{y + 1}, y, 1\right) \]
      neg-sub0 [=>] 29.1	\[ \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]
      associate--r- [=>] 29.1	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]
      metadata-eval [=>] 29.1	\[ \mathsf{fma}\left(\frac{\color{blue}{-1} + x}{y + 1}, y, 1\right) \]
      +-commutative [<=] 29.1	\[ \mathsf{fma}\left(\frac{\color{blue}{x + -1}}{y + 1}, y, 1\right) \]
      +-commutative [=>] 29.1	\[ \mathsf{fma}\left(\frac{x + -1}{\color{blue}{1 + y}}, y, 1\right) \]

   3. Taylor expanded in y around -inf 0.1

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

   4. Simplified 0.1

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

      [Start] 0.1	\[ \left(\frac{1}{y} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \frac{x}{y} \]
      associate--l+ [=>] 0.1	\[ \color{blue}{\frac{1}{y} + \left(\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \frac{x}{y}\right)} \]
      +-commutative [=>] 0.1	\[ \color{blue}{\left(\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \frac{x}{y}\right) + \frac{1}{y}} \]
      associate-+l- [=>] 0.1	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \left(\frac{x}{y} - \frac{1}{y}\right)} \]
      +-commutative [=>] 0.1	\[ \color{blue}{\left(x + -1 \cdot \frac{1 - x}{{y}^{2}}\right)} - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      associate-*r/ [=>] 0.1	\[ \left(x + \color{blue}{\frac{-1 \cdot \left(1 - x\right)}{{y}^{2}}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      sub-neg [=>] 0.1	\[ \left(x + \frac{-1 \cdot \color{blue}{\left(1 + \left(-x\right)\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-lft-in [=>] 0.1	\[ \left(x + \frac{\color{blue}{-1 \cdot 1 + -1 \cdot \left(-x\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      *-commutative [<=] 0.1	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{\left(-x\right) \cdot -1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-lft-neg-in [<=] 0.1	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{\left(-x \cdot -1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-rgt-neg-in [=>] 0.1	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{x \cdot \left(--1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      metadata-eval [=>] 0.1	\[ \left(x + \frac{-1 \cdot 1 + x \cdot \color{blue}{1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-rgt-in [<=] 0.1	\[ \left(x + \frac{\color{blue}{1 \cdot \left(-1 + x\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      +-commutative [<=] 0.1	\[ \left(x + \frac{1 \cdot \color{blue}{\left(x + -1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      *-lft-identity [=>] 0.1	\[ \left(x + \frac{\color{blue}{x + -1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      +-commutative [=>] 0.1	\[ \left(x + \frac{\color{blue}{-1 + x}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      unpow2 [=>] 0.1	\[ \left(x + \frac{-1 + x}{\color{blue}{y \cdot y}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      div-sub [<=] 0.1	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \color{blue}{\frac{x - 1}{y}} \]
      sub-neg [=>] 0.1	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{\color{blue}{x + \left(-1\right)}}{y} \]
      metadata-eval [=>] 0.1	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{x + \color{blue}{-1}}{y} \]
      +-commutative [=>] 0.1	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{\color{blue}{-1 + x}}{y} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+15}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 170000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -280000000 \lor \neg \left(y \leq 170000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+15}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 170000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 3
Error	9.6
Cost	848

\[\begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-107}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-54}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.018:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	16.9
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-108}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-54}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.015:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	1.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.79\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \end{array} \]

Alternative 6
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 7
Error	17.1
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-55}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.02:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
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 9
Error	16.2
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.022:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	39.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023046 
(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))))