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

Average Error: 34.55% → 0.23%

Time: 10.3s

Precision: binary64

Cost: 1992

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + -1}{y \cdot y}\\ t_1 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -7800000000000:\\ \;\;\;\;\left(x + t_0\right) + t_1\\ \mathbf{elif}\;y \leq 11500:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \left(x + t_0 \cdot \frac{-1}{y}\right)\right) + t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) (* y y))) (t_1 (/ (- 1.0 x) y)))
   (if (<= y -7800000000000.0)
     (+ (+ x t_0) t_1)
     (if (<= y 11500.0)
       (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
       (+ (+ t_0 (+ x (* t_0 (/ -1.0 y)))) t_1)))))

C

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

↓

double code(double x, double y) {
	double t_0 = (x + -1.0) / (y * y);
	double t_1 = (1.0 - x) / y;
	double tmp;
	if (y <= -7800000000000.0) {
		tmp = (x + t_0) + t_1;
	} else if (y <= 11500.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = (t_0 + (x + (t_0 * (-1.0 / y)))) + t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + (-1.0d0)) / (y * y)
    t_1 = (1.0d0 - x) / y
    if (y <= (-7800000000000.0d0)) then
        tmp = (x + t_0) + t_1
    else if (y <= 11500.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = (t_0 + (x + (t_0 * ((-1.0d0) / y)))) + t_1
    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 t_0 = (x + -1.0) / (y * y);
	double t_1 = (1.0 - x) / y;
	double tmp;
	if (y <= -7800000000000.0) {
		tmp = (x + t_0) + t_1;
	} else if (y <= 11500.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = (t_0 + (x + (t_0 * (-1.0 / y)))) + t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = (x + -1.0) / (y * y)
	t_1 = (1.0 - x) / y
	tmp = 0
	if y <= -7800000000000.0:
		tmp = (x + t_0) + t_1
	elif y <= 11500.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = (t_0 + (x + (t_0 * (-1.0 / y)))) + t_1
	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)
	t_0 = Float64(Float64(x + -1.0) / Float64(y * y))
	t_1 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if (y <= -7800000000000.0)
		tmp = Float64(Float64(x + t_0) + t_1);
	elseif (y <= 11500.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(Float64(t_0 + Float64(x + Float64(t_0 * Float64(-1.0 / y)))) + t_1);
	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)
	t_0 = (x + -1.0) / (y * y);
	t_1 = (1.0 - x) / y;
	tmp = 0.0;
	if (y <= -7800000000000.0)
		tmp = (x + t_0) + t_1;
	elseif (y <= 11500.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = (t_0 + (x + (t_0 * (-1.0 / y)))) + t_1;
	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_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -7800000000000.0], N[(N[(x + t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[y, 11500.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(x + N[(t$95$0 * N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Target

Original	34.55%
Target	0.38%
Herbie	0.23%

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

Derivation

1. Split input into 3 regimes

2. if y < -7.8e12

   1. Initial program 70.21

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

   2. Simplified 46.1

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

      [Start] 70.21	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 70.21	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 70.21	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      neg-mul-1 [=>] 70.21	\[ \color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}} + 1 \]
      associate-*l/ [<=] 46.26	\[ -1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)} + 1 \]
      associate-*r* [=>] 46.26	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} + 1 \]
      fma-def [=>] 46.1	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{1 - x}{y + 1}, y, 1\right)} \]
      associate-*r/ [=>] 46.1	\[ \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \left(1 - x\right)}{y + 1}}, y, 1\right) \]
      neg-mul-1 [<=] 46.1	\[ \mathsf{fma}\left(\frac{\color{blue}{-\left(1 - x\right)}}{y + 1}, y, 1\right) \]
      neg-sub0 [=>] 46.1	\[ \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]
      associate--r- [=>] 46.1	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]
      metadata-eval [=>] 46.1	\[ \mathsf{fma}\left(\frac{\color{blue}{-1} + x}{y + 1}, y, 1\right) \]
      +-commutative [<=] 46.1	\[ \mathsf{fma}\left(\frac{\color{blue}{x + -1}}{y + 1}, y, 1\right) \]
      +-commutative [=>] 46.1	\[ \mathsf{fma}\left(\frac{x + -1}{\color{blue}{1 + y}}, y, 1\right) \]

   3. Taylor expanded in y around -inf 0.02

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

   4. Simplified 0.02

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

      [Start] 0.02	\[ \left(\frac{1}{y} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \frac{x}{y} \]
      associate--l+ [=>] 0.02	\[ \color{blue}{\frac{1}{y} + \left(\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \frac{x}{y}\right)} \]
      +-commutative [=>] 0.02	\[ \color{blue}{\left(\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \frac{x}{y}\right) + \frac{1}{y}} \]
      associate-+l- [=>] 0.02	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \left(\frac{x}{y} - \frac{1}{y}\right)} \]
      +-commutative [=>] 0.02	\[ \color{blue}{\left(x + -1 \cdot \frac{1 - x}{{y}^{2}}\right)} - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      associate-*r/ [=>] 0.02	\[ \left(x + \color{blue}{\frac{-1 \cdot \left(1 - x\right)}{{y}^{2}}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      sub-neg [=>] 0.02	\[ \left(x + \frac{-1 \cdot \color{blue}{\left(1 + \left(-x\right)\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-lft-in [=>] 0.02	\[ \left(x + \frac{\color{blue}{-1 \cdot 1 + -1 \cdot \left(-x\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      *-commutative [<=] 0.02	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{\left(-x\right) \cdot -1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-lft-neg-in [<=] 0.02	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{\left(-x \cdot -1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-rgt-neg-in [=>] 0.02	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{x \cdot \left(--1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      metadata-eval [=>] 0.02	\[ \left(x + \frac{-1 \cdot 1 + x \cdot \color{blue}{1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-rgt-in [<=] 0.02	\[ \left(x + \frac{\color{blue}{1 \cdot \left(-1 + x\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      +-commutative [<=] 0.02	\[ \left(x + \frac{1 \cdot \color{blue}{\left(x + -1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      *-lft-identity [=>] 0.02	\[ \left(x + \frac{\color{blue}{x + -1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      +-commutative [=>] 0.02	\[ \left(x + \frac{\color{blue}{-1 + x}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      unpow2 [=>] 0.02	\[ \left(x + \frac{-1 + x}{\color{blue}{y \cdot y}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      div-sub [<=] 0.02	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \color{blue}{\frac{x - 1}{y}} \]
      sub-neg [=>] 0.02	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{\color{blue}{x + \left(-1\right)}}{y} \]
      metadata-eval [=>] 0.02	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{x + \color{blue}{-1}}{y} \]
      +-commutative [=>] 0.02	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{\color{blue}{-1 + x}}{y} \]

   if -7.8e12 < y < 11500

   1. Initial program 0.41

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

   2. Simplified 0.42

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

      [Start] 0.41	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      remove-double-neg [<=] 0.41	\[ 1 - \color{blue}{\left(-\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)\right)} \]
      neg-mul-1 [=>] 0.41	\[ 1 - \left(-\color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}}\right) \]
      associate-*l/ [<=] 0.42	\[ 1 - \left(--1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)}\right) \]
      associate-*r* [=>] 0.42	\[ 1 - \left(-\color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y}\right) \]
      distribute-lft-neg-in [=>] 0.42	\[ 1 - \color{blue}{\left(--1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} \]
      distribute-lft-neg-in [=>] 0.42	\[ 1 - \color{blue}{\left(\left(--1\right) \cdot \frac{1 - x}{y + 1}\right)} \cdot y \]
      metadata-eval [=>] 0.42	\[ 1 - \left(\color{blue}{1} \cdot \frac{1 - x}{y + 1}\right) \cdot y \]
      *-lft-identity [=>] 0.42	\[ 1 - \color{blue}{\frac{1 - x}{y + 1}} \cdot y \]
      +-commutative [=>] 0.42	\[ 1 - \frac{1 - x}{\color{blue}{1 + y}} \cdot y \]

   if 11500 < y

   1. Initial program 71.23

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

   2. Simplified 46.2

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

      [Start] 71.23	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 71.23	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 71.23	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      neg-mul-1 [=>] 71.23	\[ \color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}} + 1 \]
      associate-*l/ [<=] 46.16	\[ -1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)} + 1 \]
      associate-*r* [=>] 46.16	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} + 1 \]
      fma-def [=>] 46.2	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{1 - x}{y + 1}, y, 1\right)} \]
      associate-*r/ [=>] 46.2	\[ \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \left(1 - x\right)}{y + 1}}, y, 1\right) \]
      neg-mul-1 [<=] 46.2	\[ \mathsf{fma}\left(\frac{\color{blue}{-\left(1 - x\right)}}{y + 1}, y, 1\right) \]
      neg-sub0 [=>] 46.2	\[ \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]
      associate--r- [=>] 46.2	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]
      metadata-eval [=>] 46.2	\[ \mathsf{fma}\left(\frac{\color{blue}{-1} + x}{y + 1}, y, 1\right) \]
      +-commutative [<=] 46.2	\[ \mathsf{fma}\left(\frac{\color{blue}{x + -1}}{y + 1}, y, 1\right) \]
      +-commutative [=>] 46.2	\[ \mathsf{fma}\left(\frac{x + -1}{\color{blue}{1 + y}}, y, 1\right) \]

   3. Taylor expanded in y around -inf 0.03

      \[\leadsto \color{blue}{\left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \frac{x}{y}\right)} \]

   4. Simplified 0.03

      \[\leadsto \color{blue}{\left(\frac{-1 + x}{y \cdot y} + \left(x + \frac{1 - x}{{y}^{3}}\right)\right) - \frac{-1 + x}{y}} \]
      Proof

      [Start] 0.03	\[ \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \frac{x}{y}\right) \]
      associate--l+ [=>] 0.03	\[ \color{blue}{\frac{1}{y} + \left(\left(\frac{1}{{y}^{3}} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \left(\frac{x}{{y}^{3}} + \frac{x}{y}\right)\right)} \]
      +-commutative [=>] 0.03	\[ \color{blue}{\left(\left(\frac{1}{{y}^{3}} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \left(\frac{x}{{y}^{3}} + \frac{x}{y}\right)\right) + \frac{1}{y}} \]
      associate--r+ [=>] 0.03	\[ \color{blue}{\left(\left(\left(\frac{1}{{y}^{3}} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \frac{x}{{y}^{3}}\right) - \frac{x}{y}\right)} + \frac{1}{y} \]
      associate-+l- [=>] 0.03	\[ \color{blue}{\left(\left(\frac{1}{{y}^{3}} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \frac{x}{{y}^{3}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right)} \]

   5. Applied egg-rr 0.03

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

4. Final simplification 0.23

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7800000000000:\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 11500:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x + -1}{y \cdot y} + \left(x + \frac{x + -1}{y \cdot y} \cdot \frac{-1}{y}\right)\right) + \frac{1 - x}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.25%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;y \leq -7800000000000 \lor \neg \left(y \leq 380000\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 2
Error	0.36%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -7800000000000:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 180000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 3
Error	14.57%
Cost	848

\[\begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-143}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-125}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.038:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.64%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -7800000000000:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 95000:\\ \;\;\;\;1 + y \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 5
Error	26.37%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-143}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-125}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.65:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	1.91%
Cost	713

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

Alternative 7
Error	1.67%
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 8
Error	26.8%
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-125}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 340000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	2.26%
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 10
Error	26.11%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 340000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	61.13%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023089 
(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))))