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

Average Error: 22.1 → 0.0

Time: 12.1s

Precision: binary64

Cost: 7940

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + -1}{y \cdot y}\\ \mathbf{if}\;y \leq -15000:\\ \;\;\;\;t_0 - \left(\frac{x + -1}{y} + \left(\frac{x + -1}{{y}^{3}} - x\right)\right)\\ \mathbf{elif}\;y \leq 400000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + t_0\right) + \frac{1 - x}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) (* y y))))
   (if (<= y -15000.0)
     (- t_0 (+ (/ (+ x -1.0) y) (- (/ (+ x -1.0) (pow y 3.0)) x)))
     (if (<= y 400000.0)
       (fma (/ y (+ y 1.0)) (+ x -1.0) 1.0)
       (+ (+ x t_0) (/ (- 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 t_0 = (x + -1.0) / (y * y);
	double tmp;
	if (y <= -15000.0) {
		tmp = t_0 - (((x + -1.0) / y) + (((x + -1.0) / pow(y, 3.0)) - x));
	} else if (y <= 400000.0) {
		tmp = fma((y / (y + 1.0)), (x + -1.0), 1.0);
	} else {
		tmp = (x + t_0) + ((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)
	t_0 = Float64(Float64(x + -1.0) / Float64(y * y))
	tmp = 0.0
	if (y <= -15000.0)
		tmp = Float64(t_0 - Float64(Float64(Float64(x + -1.0) / y) + Float64(Float64(Float64(x + -1.0) / (y ^ 3.0)) - x)));
	elseif (y <= 400000.0)
		tmp = fma(Float64(y / Float64(y + 1.0)), Float64(x + -1.0), 1.0);
	else
		tmp = Float64(Float64(x + t_0) + Float64(Float64(1.0 - x) / y));
	end
	return 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]}, If[LessEqual[y, -15000.0], N[(t$95$0 - N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] + N[(N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 400000.0], N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x + t$95$0), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	22.1
Target	0.2
Herbie	0.0

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

Derivation

1. Split input into 3 regimes

2. if y < -15000

   1. Initial program 44.7

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

   2. Taylor expanded in y around -inf 0.0

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

   3. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{x}{{y}^{2}} + \left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right)\right) - \frac{1}{{y}^{2}} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(\left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \frac{x}{{y}^{2}}\right)} - \frac{1}{{y}^{2}} \]
      associate--l+ [=>] 0.0	\[ \color{blue}{\left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)} \]
      associate-*r/ [=>] 0.0	\[ \left(\color{blue}{\frac{-1 \cdot \left(x - 1\right)}{y}} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      sub-neg [=>] 0.0	\[ \left(\frac{-1 \cdot \color{blue}{\left(x + \left(-1\right)\right)}}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      metadata-eval [=>] 0.0	\[ \left(\frac{-1 \cdot \left(x + \color{blue}{-1}\right)}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      distribute-lft-in [=>] 0.0	\[ \left(\frac{\color{blue}{-1 \cdot x + -1 \cdot -1}}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      metadata-eval [=>] 0.0	\[ \left(\frac{-1 \cdot x + \color{blue}{1}}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      +-commutative [<=] 0.0	\[ \left(\frac{\color{blue}{1 + -1 \cdot x}}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      mul-1-neg [=>] 0.0	\[ \left(\frac{1 + \color{blue}{\left(-x\right)}}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      sub-neg [<=] 0.0	\[ \left(\frac{\color{blue}{1 - x}}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      +-commutative [=>] 0.0	\[ \left(\frac{1 - x}{y} + \color{blue}{\left(x + -1 \cdot \frac{x - 1}{{y}^{3}}\right)}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      associate-*r/ [=>] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \color{blue}{\frac{-1 \cdot \left(x - 1\right)}{{y}^{3}}}\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      sub-neg [=>] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{-1 \cdot \color{blue}{\left(x + \left(-1\right)\right)}}{{y}^{3}}\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      metadata-eval [=>] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{-1 \cdot \left(x + \color{blue}{-1}\right)}{{y}^{3}}\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      distribute-lft-in [=>] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{\color{blue}{-1 \cdot x + -1 \cdot -1}}{{y}^{3}}\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      metadata-eval [=>] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{-1 \cdot x + \color{blue}{1}}{{y}^{3}}\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      +-commutative [<=] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{\color{blue}{1 + -1 \cdot x}}{{y}^{3}}\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      mul-1-neg [=>] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{1 + \color{blue}{\left(-x\right)}}{{y}^{3}}\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      sub-neg [<=] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{\color{blue}{1 - x}}{{y}^{3}}\right)\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) \]
      div-sub [<=] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{1 - x}{{y}^{3}}\right)\right) + \color{blue}{\frac{x - 1}{{y}^{2}}} \]
      unpow2 [=>] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{1 - x}{{y}^{3}}\right)\right) + \frac{x - 1}{\color{blue}{y \cdot y}} \]
      sub-neg [=>] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{1 - x}{{y}^{3}}\right)\right) + \frac{\color{blue}{x + \left(-1\right)}}{y \cdot y} \]
      metadata-eval [=>] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{1 - x}{{y}^{3}}\right)\right) + \frac{x + \color{blue}{-1}}{y \cdot y} \]
      +-commutative [=>] 0.0	\[ \left(\frac{1 - x}{y} + \left(x + \frac{1 - x}{{y}^{3}}\right)\right) + \frac{\color{blue}{-1 + x}}{y \cdot y} \]

   if -15000 < y < 4e5

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 0.1	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 0.1	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      *-lft-identity [<=] 0.1	\[ \color{blue}{1 \cdot \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} + 1 \]
      associate-/l* [=>] 0.1	\[ 1 \cdot \left(-\color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\right) + 1 \]
      distribute-neg-frac [=>] 0.1	\[ 1 \cdot \color{blue}{\frac{-\left(1 - x\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*r/ [=>] 0.1	\[ \color{blue}{\frac{1 \cdot \left(-\left(1 - x\right)\right)}{\frac{y + 1}{y}}} + 1 \]
      associate-*l/ [<=] 0.1	\[ \color{blue}{\frac{1}{\frac{y + 1}{y}} \cdot \left(-\left(1 - x\right)\right)} + 1 \]
      fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(\frac{1}{\frac{y + 1}{y}}, -\left(1 - x\right), 1\right)} \]
      associate-/l* [<=] 0.1	\[ \mathsf{fma}\left(\color{blue}{\frac{1 \cdot y}{y + 1}}, -\left(1 - x\right), 1\right) \]
      *-lft-identity [=>] 0.1	\[ \mathsf{fma}\left(\frac{\color{blue}{y}}{y + 1}, -\left(1 - x\right), 1\right) \]
      +-commutative [=>] 0.1	\[ \mathsf{fma}\left(\frac{y}{\color{blue}{1 + y}}, -\left(1 - x\right), 1\right) \]
      neg-sub0 [=>] 0.1	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{0 - \left(1 - x\right)}, 1\right) \]
      associate--r- [=>] 0.1	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{\left(0 - 1\right) + x}, 1\right) \]
      metadata-eval [=>] 0.1	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{-1} + x, 1\right) \]
      +-commutative [<=] 0.1	\[ \mathsf{fma}\left(\frac{y}{1 + y}, \color{blue}{x + -1}, 1\right) \]

   if 4e5 < y

   1. Initial program 45.5

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

   2. Simplified 29.2

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

      [Start] 45.5	\[ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
      sub-neg [=>] 45.5	\[ \color{blue}{1 + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)} \]
      +-commutative [=>] 45.5	\[ \color{blue}{\left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) + 1} \]
      neg-mul-1 [=>] 45.5	\[ \color{blue}{-1 \cdot \frac{\left(1 - x\right) \cdot y}{y + 1}} + 1 \]
      associate-*l/ [<=] 29.1	\[ -1 \cdot \color{blue}{\left(\frac{1 - x}{y + 1} \cdot y\right)} + 1 \]
      associate-*r* [=>] 29.1	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{y + 1}\right) \cdot y} + 1 \]
      fma-def [=>] 29.2	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{1 - x}{y + 1}, y, 1\right)} \]
      associate-*r/ [=>] 29.2	\[ \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \left(1 - x\right)}{y + 1}}, y, 1\right) \]
      neg-mul-1 [<=] 29.2	\[ \mathsf{fma}\left(\frac{\color{blue}{-\left(1 - x\right)}}{y + 1}, y, 1\right) \]
      neg-sub0 [=>] 29.2	\[ \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]
      associate--r- [=>] 29.2	\[ \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]
      metadata-eval [=>] 29.2	\[ \mathsf{fma}\left(\frac{\color{blue}{-1} + x}{y + 1}, y, 1\right) \]
      +-commutative [<=] 29.2	\[ \mathsf{fma}\left(\frac{\color{blue}{x + -1}}{y + 1}, y, 1\right) \]
      +-commutative [=>] 29.2	\[ \mathsf{fma}\left(\frac{x + -1}{\color{blue}{1 + y}}, y, 1\right) \]

   3. Taylor expanded in y around -inf 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{y} + \left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right)\right) - \frac{x}{y} \]
      associate--l+ [=>] 0.0	\[ \color{blue}{\frac{1}{y} + \left(\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \frac{x}{y}\right)} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \frac{x}{y}\right) + \frac{1}{y}} \]
      associate-+l- [=>] 0.0	\[ \color{blue}{\left(-1 \cdot \frac{1 - x}{{y}^{2}} + x\right) - \left(\frac{x}{y} - \frac{1}{y}\right)} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(x + -1 \cdot \frac{1 - x}{{y}^{2}}\right)} - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      associate-*r/ [=>] 0.0	\[ \left(x + \color{blue}{\frac{-1 \cdot \left(1 - x\right)}{{y}^{2}}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      sub-neg [=>] 0.0	\[ \left(x + \frac{-1 \cdot \color{blue}{\left(1 + \left(-x\right)\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-lft-in [=>] 0.0	\[ \left(x + \frac{\color{blue}{-1 \cdot 1 + -1 \cdot \left(-x\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      *-commutative [<=] 0.0	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{\left(-x\right) \cdot -1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-lft-neg-in [<=] 0.0	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{\left(-x \cdot -1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-rgt-neg-in [=>] 0.0	\[ \left(x + \frac{-1 \cdot 1 + \color{blue}{x \cdot \left(--1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      metadata-eval [=>] 0.0	\[ \left(x + \frac{-1 \cdot 1 + x \cdot \color{blue}{1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      distribute-rgt-in [<=] 0.0	\[ \left(x + \frac{\color{blue}{1 \cdot \left(-1 + x\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      +-commutative [<=] 0.0	\[ \left(x + \frac{1 \cdot \color{blue}{\left(x + -1\right)}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      *-lft-identity [=>] 0.0	\[ \left(x + \frac{\color{blue}{x + -1}}{{y}^{2}}\right) - \left(\frac{x}{y} - \frac{1}{y}\right) \]
      +-commutative [=>] 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	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{x + \color{blue}{-1}}{y} \]
      +-commutative [=>] 0.0	\[ \left(x + \frac{-1 + x}{y \cdot y}\right) - \frac{\color{blue}{-1 + x}}{y} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15000:\\ \;\;\;\;\frac{x + -1}{y \cdot y} - \left(\frac{x + -1}{y} + \left(\frac{x + -1}{{y}^{3}} - x\right)\right)\\ \mathbf{elif}\;y \leq 400000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	7240

\[\begin{array}{l} t_0 := \frac{x + -1}{y \cdot y}\\ \mathbf{if}\;y \leq -15000:\\ \;\;\;\;x + \left(t_0 + \frac{x + -1}{y} \cdot \left(-1 + \frac{-1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;y \leq 400000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + 1}, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + t_0\right) + \frac{1 - x}{y}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	1604

\[\begin{array}{l} t_0 := \frac{x + -1}{y \cdot y}\\ \mathbf{if}\;y \leq -15000:\\ \;\;\;\;x + \left(t_0 + \frac{x + -1}{y} \cdot \left(-1 + \frac{-1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;y \leq 110000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + t_0\right) + \frac{1 - x}{y}\\ \end{array} \]

Alternative 3
Error	0.1
Cost	1476

\[\begin{array}{l} t_0 := \frac{-1}{y \cdot y}\\ \mathbf{if}\;y \leq -63000:\\ \;\;\;\;x + \left(t_0 + \frac{x + -1}{y} \cdot \left(-1 + t_0\right)\right)\\ \mathbf{elif}\;y \leq 400000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	1225

\[\begin{array}{l} \mathbf{if}\;y \leq -31000000000000 \lor \neg \left(y \leq 190000\right):\\ \;\;\;\;\left(x + \frac{x + -1}{y \cdot y}\right) + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternative 5
Error	0.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -31000000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 280000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 6
Error	0.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -31000000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 53000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 7
Error	9.3
Cost	848

\[\begin{array}{l} t_0 := x - \frac{-1}{y}\\ t_1 := \frac{1}{y + 1}\\ \mathbf{if}\;y \leq -37000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-60}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 220000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	9.9
Cost	716

\[\begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-110}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 0.06:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	1.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.84\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \end{array} \]

Alternative 10
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 11
Error	0.9
Cost	712

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

Alternative 12
Error	17.4
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	17.4
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-112}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	1.2
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 15
Error	16.4
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	38.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022364 
(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))))