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

Average Error: 22.4 → 0.2

Time: 10.9s

Precision: binary64

Cost: 8264

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ t_1 := \mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)\\ \mathbf{if}\;t_0 \leq 0.005:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;\left(x - \frac{x + -1}{y}\right) + \frac{x + -1}{y \cdot y} \cdot \left(1 + \frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y)))
        (t_1 (fma (- 1.0 x) (/ y (- -1.0 y)) 1.0)))
   (if (<= t_0 0.005)
     t_1
     (if (<= t_0 2.0)
       (+ (- x (/ (+ x -1.0) y)) (* (/ (+ x -1.0) (* y y)) (+ 1.0 (/ -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 = ((1.0 - x) * y) / (1.0 + y);
	double t_1 = fma((1.0 - x), (y / (-1.0 - y)), 1.0);
	double tmp;
	if (t_0 <= 0.005) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = (x - ((x + -1.0) / y)) + (((x + -1.0) / (y * y)) * (1.0 + (-1.0 / y)));
	} else {
		tmp = 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(Float64(1.0 - x) * y) / Float64(1.0 + y))
	t_1 = fma(Float64(1.0 - x), Float64(y / Float64(-1.0 - y)), 1.0)
	tmp = 0.0
	if (t_0 <= 0.005)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(x - Float64(Float64(x + -1.0) / y)) + Float64(Float64(Float64(x + -1.0) / Float64(y * y)) * Float64(1.0 + Float64(-1.0 / y))));
	else
		tmp = t_1;
	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[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.005], t$95$1, If[LessEqual[t$95$0, 2.0], N[(N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	22.4
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 2 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 0.0050000000000000001 or 2 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 10.7

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)} \]
      Proof
      (fma.f64 (-.f64 1 x) (/.f64 y (-.f64 -1 y)) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (Rewrite=> sub-neg_binary64 (+.f64 -1 (neg.f64 y)))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (+.f64 (Rewrite<= metadata-eval (*.f64 -1 1)) (neg.f64 y))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (+.f64 (*.f64 -1 1) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 y)))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (Rewrite<= distribute-lft-in_binary64 (*.f64 -1 (+.f64 1 y)))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (*.f64 -1 (Rewrite<= +-commutative_binary64 (+.f64 y 1)))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (*.f64 (Rewrite<= metadata-eval (/.f64 1 -1)) (+.f64 y 1))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (Rewrite<= associate-/r/_binary64 (/.f64 1 (/.f64 -1 (+.f64 y 1))))) 1): 17 points increase in error, 6 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1 (+.f64 y 1)) -1))) 1): 6 points increase in error, 17 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 y 1) 1)) -1)) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (Rewrite=> associate-/l*_binary64 (/.f64 (+.f64 y 1) (/.f64 -1 1)))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (/.f64 (+.f64 y 1) (Rewrite=> metadata-eval -1))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 y (+.f64 y 1)) -1)) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (Rewrite=> *-commutative_binary64 (*.f64 -1 (/.f64 y (+.f64 y 1)))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 y (+.f64 y 1)))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (neg.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 y)) (+.f64 y 1))) 1): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 1 x) (neg.f64 (Rewrite=> associate-/l*_binary64 (/.f64 1 (/.f64 (+.f64 y 1) y)))) 1): 4 points increase in error, 1 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 1 x) (neg.f64 (/.f64 1 (/.f64 (+.f64 y 1) y)))) 1)): 3 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (-.f64 1 x) (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1 y) (+.f64 y 1))))) 1): 1 points increase in error, 4 points decrease in error
      (+.f64 (*.f64 (-.f64 1 x) (neg.f64 (/.f64 (Rewrite=> *-lft-identity_binary64 y) (+.f64 y 1)))) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (-.f64 1 x) (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 y) (+.f64 y 1)))) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 1 x) (neg.f64 y)) (+.f64 y 1))) 1): 40 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (-.f64 1 x) y))) (+.f64 y 1)) 1): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)))) 1): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 1 (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)))): 0 points increase in error, 0 points decrease in error

   if 0.0050000000000000001 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 2

   1. Initial program 57.5

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

   2. Taylor expanded in y around -inf 0.9

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

   3. Simplified 0.9

      \[\leadsto \color{blue}{\left(x + \frac{1 - x}{y}\right) + \frac{-1 + x}{y \cdot y} \cdot \left(\frac{-1}{y} + 1\right)} \]
      Proof
      (+.f64 (+.f64 x (/.f64 (-.f64 1 x) y)) (*.f64 (/.f64 (+.f64 -1 x) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 x (Rewrite=> div-sub_binary64 (-.f64 (/.f64 1 y) (/.f64 x y)))) (*.f64 (/.f64 (+.f64 -1 x) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 x (/.f64 1 y)) (/.f64 x y))) (*.f64 (/.f64 (+.f64 -1 x) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 y) x)) (/.f64 x y)) (*.f64 (/.f64 (+.f64 -1 x) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate--l+_binary64 (+.f64 (/.f64 1 y) (-.f64 x (/.f64 x y)))) (*.f64 (/.f64 (+.f64 -1 x) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (-.f64 x (/.f64 x y)) (/.f64 1 y))) (*.f64 (/.f64 (+.f64 -1 x) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate--r-_binary64 (-.f64 x (-.f64 (/.f64 x y) (/.f64 1 y)))) (*.f64 (/.f64 (+.f64 -1 x) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 x (Rewrite<= div-sub_binary64 (/.f64 (-.f64 x 1) y))) (*.f64 (/.f64 (+.f64 -1 x) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 x (neg.f64 (/.f64 (-.f64 x 1) y)))) (*.f64 (/.f64 (+.f64 -1 x) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 x (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 x 1) y)))) (*.f64 (/.f64 (+.f64 -1 x) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x)) (*.f64 (/.f64 (+.f64 -1 x) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1)) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (*.f64 (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (*.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (*.f64 y y)) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (*.f64 (/.f64 (-.f64 x 1) (Rewrite<= unpow2_binary64 (pow.f64 y 2))) (+.f64 (/.f64 -1 y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (*.f64 (/.f64 (-.f64 x 1) (pow.f64 y 2)) (+.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) y) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (*.f64 (/.f64 (-.f64 x 1) (pow.f64 y 2)) (+.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 y))) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (neg.f64 (/.f64 1 y)) (/.f64 (-.f64 x 1) (pow.f64 y 2))) (*.f64 1 (/.f64 (-.f64 x 1) (pow.f64 y 2)))))): 0 points increase in error, 1 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (+.f64 (*.f64 (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 1) y)) (/.f64 (-.f64 x 1) (pow.f64 y 2))) (*.f64 1 (/.f64 (-.f64 x 1) (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (+.f64 (*.f64 (/.f64 (Rewrite=> metadata-eval -1) y) (/.f64 (-.f64 x 1) (pow.f64 y 2))) (*.f64 1 (/.f64 (-.f64 x 1) (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 x 1)) (*.f64 y (pow.f64 y 2)))) (*.f64 1 (/.f64 (-.f64 x 1) (pow.f64 y 2))))): 5 points increase in error, 9 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (+.f64 (/.f64 (*.f64 -1 (-.f64 x 1)) (*.f64 y (Rewrite=> unpow2_binary64 (*.f64 y y)))) (*.f64 1 (/.f64 (-.f64 x 1) (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (+.f64 (/.f64 (*.f64 -1 (-.f64 x 1)) (Rewrite<= cube-mult_binary64 (pow.f64 y 3))) (*.f64 1 (/.f64 (-.f64 x 1) (pow.f64 y 2))))): 3 points increase in error, 2 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (-.f64 x 1) (pow.f64 y 3)))) (*.f64 1 (/.f64 (-.f64 x 1) (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) (pow.f64 y 3))) (Rewrite=> *-lft-identity_binary64 (/.f64 (-.f64 x 1) (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (*.f64 -1 (/.f64 (-.f64 x 1) (pow.f64 y 3)))) (/.f64 (-.f64 x 1) (pow.f64 y 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (+.f64 x (*.f64 -1 (/.f64 (-.f64 x 1) (pow.f64 y 3)))))) (/.f64 (-.f64 x 1) (pow.f64 y 2))): 0 points increase in error, 1 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) (pow.f64 y 3))) x))) (/.f64 (-.f64 x 1) (pow.f64 y 2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) (pow.f64 y 3))) x)) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 x (pow.f64 y 2)) (/.f64 1 (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) (pow.f64 y 3))) x)) (/.f64 x (pow.f64 y 2))) (/.f64 1 (pow.f64 y 2)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 x (pow.f64 y 2)) (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) (pow.f64 y 3))) x)))) (/.f64 1 (pow.f64 y 2))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 2:\\ \;\;\;\;\left(x - \frac{x + -1}{y}\right) + \frac{x + -1}{y \cdot y} \cdot \left(1 + \frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := 1 + \frac{-1}{y}\\ \mathbf{if}\;y \leq -142508742465221.1:\\ \;\;\;\;\left(x - \frac{x + -1}{y}\right) + \frac{x + -1}{y \cdot y} \cdot t_0\\ \mathbf{elif}\;y \leq 8.254783054693725:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y} \cdot t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1096

\[\begin{array}{l} t_0 := x + \frac{1 - x}{y} \cdot \left(1 + \frac{-1}{y}\right)\\ \mathbf{if}\;y \leq -142508742465221.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.254783054693725:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	17.7
Cost	984

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2645642163707999 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -21929357.71019714:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -9.732671360181056 \cdot 10^{-72}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 6.847924196923214 \cdot 10^{-92}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.996062656067024 \cdot 10^{-57}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 7.44095667816971 \cdot 10^{-8}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	1.1
Cost	968

\[\begin{array}{l} t_0 := x - \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -21929357.71019714:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7596658380727143 \cdot 10^{-5}:\\ \;\;\;\;1 + y \cdot \left(\left(1 - x\right) \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.4
Cost	968

\[\begin{array}{l} t_0 := x - \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -142508742465221.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.254783054693725:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	17.7
Cost	856

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2645642163707999 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -21929357.71019714:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -9.732671360181056 \cdot 10^{-72}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 6.847924196923214 \cdot 10^{-92}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.996062656067024 \cdot 10^{-57}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 7.44095667816971 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	8.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2645642163707999 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -21929357.71019714:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 1.7596658380727143 \cdot 10^{-5}:\\ \;\;\;\;1 + \left(x \cdot y - y\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \]

Alternative 8
Error	17.7
Cost	724

\[\begin{array}{l} \mathbf{if}\;y \leq -21929357.71019714:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -9.732671360181056 \cdot 10^{-72}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 6.847924196923214 \cdot 10^{-92}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.996062656067024 \cdot 10^{-57}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 7.44095667816971 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	9.4
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2645642163707999 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -21929357.71019714:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 10029534862266856:\\ \;\;\;\;1 + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	9.1
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2645642163707999 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -21929357.71019714:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 1.7596658380727143 \cdot 10^{-5}:\\ \;\;\;\;1 + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \]

Alternative 11
Error	1.3
Cost	712

\[\begin{array}{l} t_0 := x - \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -21929357.71019714:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7596658380727143 \cdot 10^{-5}:\\ \;\;\;\;1 + \left(x \cdot y - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	16.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -21929357.71019714:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.44095667816971 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	39.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022316 
(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))))