Average Error: 21.9 → 0.2
Time: 10.5s
Precision: binary64
Cost: 1608
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -2.271805485292431 \cdot 10^{+19}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 47.11716370500957:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) + \frac{x + -1}{y \cdot y} \cdot \left(1 + \frac{-1}{y}\right)\\ \end{array} \]
(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= y -2.271805485292431e+19)
   (+ x (/ 1.0 y))
   (if (<= y 47.11716370500957)
     (- 1.0 (/ (* y (- 1.0 x)) (+ y 1.0)))
     (+ (+ x (/ (- 1.0 x) y)) (* (/ (+ x -1.0) (* y y)) (+ 1.0 (/ -1.0 y)))))))
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 <= -2.271805485292431e+19) {
		tmp = x + (1.0 / y);
	} else if (y <= 47.11716370500957) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = (x + ((1.0 - x) / y)) + (((x + -1.0) / (y * y)) * (1.0 + (-1.0 / y)));
	}
	return tmp;
}

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 <= (-2.271805485292431d+19)) then
        tmp = x + (1.0d0 / y)
    else if (y <= 47.11716370500957d0) then
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (y + 1.0d0))
    else
        tmp = (x + ((1.0d0 - x) / y)) + (((x + (-1.0d0)) / (y * y)) * (1.0d0 + ((-1.0d0) / y)))
    end if
    code = tmp
end function

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 <= -2.271805485292431e+19) {
		tmp = x + (1.0 / y);
	} else if (y <= 47.11716370500957) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = (x + ((1.0 - x) / y)) + (((x + -1.0) / (y * y)) * (1.0 + (-1.0 / y)));
	}
	return tmp;
}

def code(x, y):
	return 1.0 - (((1.0 - x) * y) / (y + 1.0))
↓
def code(x, y):
	tmp = 0
	if y <= -2.271805485292431e+19:
		tmp = x + (1.0 / y)
	elif y <= 47.11716370500957:
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0))
	else:
		tmp = (x + ((1.0 - x) / y)) + (((x + -1.0) / (y * y)) * (1.0 + (-1.0 / y)))
	return tmp

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 <= -2.271805485292431e+19)
		tmp = Float64(x + Float64(1.0 / y));
	elseif (y <= 47.11716370500957)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x + Float64(Float64(1.0 - x) / y)) + Float64(Float64(Float64(x + -1.0) / Float64(y * y)) * Float64(1.0 + Float64(-1.0 / y))));
	end
	return tmp
end

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 <= -2.271805485292431e+19)
		tmp = x + (1.0 / y);
	elseif (y <= 47.11716370500957)
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	else
		tmp = (x + ((1.0 - x) / y)) + (((x + -1.0) / (y * y)) * (1.0 + (-1.0 / y)));
	end
	tmp_2 = tmp;
end

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, -2.271805485292431e+19], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 47.11716370500957], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(1.0 - x), $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]]]

1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -2.271805485292431 \cdot 10^{+19}:\\
\;\;\;\;x + \frac{1}{y}\\

\mathbf{elif}\;y \leq 47.11716370500957:\\
\;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\

\mathbf{else}:\\
\;\;\;\;\left(x + \frac{1 - x}{y}\right) + \frac{x + -1}{y \cdot y} \cdot \left(1 + \frac{-1}{y}\right)\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.9
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if y < -22718054852924310000

    1. Initial program 46.9

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

    2. Simplified29.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, \frac{y}{-1 - y}, 1\right)} \]
      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): 25 points increase in error, 3 points decrease in error
      (fma.f64 (-.f64 1 x) (/.f64 y (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1 (+.f64 y 1)) -1))) 1): 3 points increase in error, 25 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): 8 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 1 x) (neg.f64 (/.f64 1 (/.f64 (+.f64 y 1) y)))) 1)): 0 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): 0 points increase in error, 8 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): 46 points increase in error, 4 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

    3. Taylor expanded in y around inf 15.3

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

    4. Simplified0.0

      \[\leadsto \color{blue}{x + \frac{1 - x}{y}} \]
      Proof
      (+.f64 x (/.f64 (-.f64 1 x) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-lft-identity_binary64 (+.f64 0 x)) (/.f64 (-.f64 1 x) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (Rewrite<= metadata-eval (-.f64 1 1)) x) (/.f64 (-.f64 1 x) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate--r-_binary64 (-.f64 1 (-.f64 1 x))) (/.f64 (-.f64 1 x) y)): 38 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 (-.f64 1 x)))) (/.f64 (-.f64 1 x) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (-.f64 1 x)))) (/.f64 (-.f64 1 x) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 1 (*.f64 -1 (-.f64 1 x))) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 1 y) (/.f64 x y)))): 2 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (+.f64 1 (*.f64 -1 (-.f64 1 x))) (/.f64 1 y)) (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 y) (+.f64 1 (*.f64 -1 (-.f64 1 x))))) (/.f64 x y)): 0 points increase in error, 0 points decrease in error

    5. Taylor expanded in x around 0 0.0

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

    if -22718054852924310000 < y < 47.1171637050095669

    1. Initial program 0.4

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

    if 47.1171637050095669 < y

    1. Initial program 44.0

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

    2. Taylor expanded in y around -inf 0.1

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

    3. Simplified0.1

      \[\leadsto \color{blue}{\left(x + \frac{1 - x}{y}\right) + \frac{-1 + x}{y \cdot y} \cdot \left(\frac{-1}{y} + 1\right)} \]
      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, 2 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))))): 8 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))))): 5 points increase in error, 5 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)))): 2 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, 0 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 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.271805485292431 \cdot 10^{+19}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 47.11716370500957:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) + \frac{x + -1}{y \cdot y} \cdot \left(1 + \frac{-1}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error0.2
Cost1096
\[\begin{array}{l} \mathbf{if}\;y \leq -2.271805485292431 \cdot 10^{+19}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 144992.14485709165:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + -1}{y} \cdot \left(\frac{1}{y} + -1\right)\\ \end{array} \]
Alternative 2
Error0.3
Cost968
\[\begin{array}{l} \mathbf{if}\;y \leq -2.271805485292431 \cdot 10^{+19}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 411729.85051168053:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
Alternative 3
Error9.3
Cost712
\[\begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -11.62750023525451:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.1926242047658488 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error9.0
Cost712
\[\begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -11.62750023525451:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 411729.85051168053:\\ \;\;\;\;1 - \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error1.1
Cost712
\[\begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -11.62750023525451:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.1926242047658488 \cdot 10^{-6}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error9.5
Cost584
\[\begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -11.62750023525451:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.8289001453256795 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error17.2
Cost328
\[\begin{array}{l} \mathbf{if}\;y \leq -11.62750023525451:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 411729.85051168053:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 8
Error39.5
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022308 
(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))))