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

Average Error: 22.7 → 0.1

Time: 15.4s

Precision: binary64

Cost: 1476

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -6238223.946121859:\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) + \frac{1 - x}{y \cdot y} \cdot \left(-1 + \frac{1}{y}\right)\\ \mathbf{elif}\;y \leq 31835.723135063974:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{y} + \frac{\frac{-1}{y}}{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -6238223.946121859)
   (+ (+ x (/ (- 1.0 x) y)) (* (/ (- 1.0 x) (* y y)) (+ -1.0 (/ 1.0 y))))
   (if (<= y 31835.723135063974)
     (- 1.0 (/ (* y (- 1.0 x)) (+ y 1.0)))
     (+ x (+ (/ 1.0 y) (/ (/ -1.0 y) 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 <= -6238223.946121859) {
		tmp = (x + ((1.0 - x) / y)) + (((1.0 - x) / (y * y)) * (-1.0 + (1.0 / y)));
	} else if (y <= 31835.723135063974) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = x + ((1.0 / y) + ((-1.0 / y) / 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 <= (-6238223.946121859d0)) then
        tmp = (x + ((1.0d0 - x) / y)) + (((1.0d0 - x) / (y * y)) * ((-1.0d0) + (1.0d0 / y)))
    else if (y <= 31835.723135063974d0) then
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (y + 1.0d0))
    else
        tmp = x + ((1.0d0 / y) + (((-1.0d0) / y) / 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 <= -6238223.946121859) {
		tmp = (x + ((1.0 - x) / y)) + (((1.0 - x) / (y * y)) * (-1.0 + (1.0 / y)));
	} else if (y <= 31835.723135063974) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = x + ((1.0 / y) + ((-1.0 / y) / 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 <= -6238223.946121859:
		tmp = (x + ((1.0 - x) / y)) + (((1.0 - x) / (y * y)) * (-1.0 + (1.0 / y)))
	elif y <= 31835.723135063974:
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0))
	else:
		tmp = x + ((1.0 / y) + ((-1.0 / y) / 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 <= -6238223.946121859)
		tmp = Float64(Float64(x + Float64(Float64(1.0 - x) / y)) + Float64(Float64(Float64(1.0 - x) / Float64(y * y)) * Float64(-1.0 + Float64(1.0 / y))));
	elseif (y <= 31835.723135063974)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(1.0 / y) + Float64(Float64(-1.0 / y) / 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 <= -6238223.946121859)
		tmp = (x + ((1.0 - x) / y)) + (((1.0 - x) / (y * y)) * (-1.0 + (1.0 / y)));
	elseif (y <= 31835.723135063974)
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	else
		tmp = x + ((1.0 / y) + ((-1.0 / y) / 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, -6238223.946121859], N[(N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 - x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 31835.723135063974], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 / y), $MachinePrecision] + N[(N[(-1.0 / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	22.7
Target	0.2
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if y < -6238223.94612185936

   1. Initial program 46.8

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

   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

      \[\leadsto \color{blue}{\left(x + \frac{1 - x}{y}\right) + \frac{-1 + x}{y \cdot y} \cdot \left(\frac{-1}{y} + 1\right)} \]
      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)))))): 1 points increase in error, 0 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))))): 9 points increase in error, 7 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))))): 2 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)))): 1 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))): 1 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

   if -6238223.94612185936 < y < 31835.723135063974

   1. Initial program 0.1

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

   if 31835.723135063974 < y

   1. Initial program 45.4

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

   2. Taylor expanded in y around inf 0.1

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{x + \frac{-1 + x}{y} \cdot \left(\frac{1}{y} + -1\right)} \]
      Proof
      (+.f64 x (*.f64 (/.f64 (+.f64 -1 x) y) (+.f64 (/.f64 1 y) -1))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1)) y) (+.f64 (/.f64 1 y) -1))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (*.f64 (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) y) (+.f64 (/.f64 1 y) -1))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (*.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) y) (+.f64 (/.f64 1 y) -1))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (/.f64 1 y) (/.f64 (-.f64 x 1) y)) (*.f64 -1 (/.f64 (-.f64 x 1) y))))): 2 points increase in error, 2 points decrease in error
      (+.f64 x (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 1 (-.f64 x 1)) (*.f64 y y))) (*.f64 -1 (/.f64 (-.f64 x 1) y)))): 14 points increase in error, 13 points decrease in error
      (+.f64 x (+.f64 (/.f64 (*.f64 1 (-.f64 x 1)) (Rewrite<= unpow2_binary64 (pow.f64 y 2))) (*.f64 -1 (/.f64 (-.f64 x 1) y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (+.f64 (/.f64 (*.f64 1 (-.f64 x 1)) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (pow.f64 y 2)))) (*.f64 -1 (/.f64 (-.f64 x 1) y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (+.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 1 1) (/.f64 (-.f64 x 1) (pow.f64 y 2)))) (*.f64 -1 (/.f64 (-.f64 x 1) y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (+.f64 (*.f64 (Rewrite=> metadata-eval 1) (/.f64 (-.f64 x 1) (pow.f64 y 2))) (*.f64 -1 (/.f64 (-.f64 x 1) y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (+.f64 (Rewrite=> *-lft-identity_binary64 (/.f64 (-.f64 x 1) (pow.f64 y 2))) (*.f64 -1 (/.f64 (-.f64 x 1) y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 x (/.f64 (-.f64 x 1) (pow.f64 y 2))) (*.f64 -1 (/.f64 (-.f64 x 1) y)))): 0 points increase in error, 2 points decrease in error
      (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 (-.f64 x 1) (pow.f64 y 2)) x)) (*.f64 -1 (/.f64 (-.f64 x 1) y))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-+l+_binary64 (+.f64 (/.f64 (-.f64 x 1) (pow.f64 y 2)) (+.f64 x (*.f64 -1 (/.f64 (-.f64 x 1) y))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (-.f64 x 1) (pow.f64 y 2)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) 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)) x) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 x (pow.f64 y 2)) (/.f64 1 (pow.f64 y 2))))): 2 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 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)) x))) (/.f64 1 (pow.f64 y 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate--l+_binary64 (+.f64 (/.f64 x (pow.f64 y 2)) (-.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (/.f64 1 (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (-.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) x) (/.f64 1 (pow.f64 y 2))) (/.f64 x (pow.f64 y 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate--l+_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (-.f64 x (/.f64 1 (pow.f64 y 2))))) (/.f64 x (pow.f64 y 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-+l+_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (+.f64 (-.f64 x (/.f64 1 (pow.f64 y 2))) (/.f64 x (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (Rewrite<= associate--r-_binary64 (-.f64 x (-.f64 (/.f64 1 (pow.f64 y 2)) (/.f64 x (pow.f64 y 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (-.f64 x (Rewrite<= div-sub_binary64 (/.f64 (-.f64 1 x) (pow.f64 y 2))))): 0 points increase in error, 2 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (-.f64 x (/.f64 (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 x))) (pow.f64 y 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (-.f64 x (/.f64 (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))) (pow.f64 y 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (Rewrite<= unsub-neg_binary64 (+.f64 x (neg.f64 (/.f64 (+.f64 1 (*.f64 -1 x)) (pow.f64 y 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (+.f64 x (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 1 (*.f64 -1 x)) (pow.f64 y 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (-.f64 x 1) y)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 1 (*.f64 -1 x)) (pow.f64 y 2))) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 1 (*.f64 -1 x)) (pow.f64 y 2))) x) (*.f64 -1 (/.f64 (-.f64 x 1) y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 1 (*.f64 -1 x)) (pow.f64 y 2))) x) (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 (-.f64 x 1) y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> unsub-neg_binary64 (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 1 (*.f64 -1 x)) (pow.f64 y 2))) x) (/.f64 (-.f64 x 1) y))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 1 (*.f64 -1 x)) (pow.f64 y 2))) x) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 x y) (/.f64 1 y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 1 (*.f64 -1 x)) (pow.f64 y 2))) x) (/.f64 x y)) (/.f64 1 y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 y) (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 1 (*.f64 -1 x)) (pow.f64 y 2))) x) (/.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 1 y) (+.f64 (*.f64 -1 (/.f64 (+.f64 1 (*.f64 -1 x)) (pow.f64 y 2))) x)) (/.f64 x y))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in x around 0 0.4

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

   5. Simplified 0.4

      \[\leadsto x + \color{blue}{\frac{1 + \frac{-1}{y}}{y}} \]
      Proof
      (/.f64 (+.f64 1 (/.f64 -1 y)) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) y)) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 y)))) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (/.f64 1 y)) 1)) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (/.f64 1 y))) 1) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 (/.f64 1 y) 1))) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 (/.f64 1 y) 1))) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (-.f64 (/.f64 1 y) 1))) y): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (-.f64 (/.f64 1 y) 1) y))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in y around 0 0.4

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

   7. Simplified 0.4

      \[\leadsto x + \color{blue}{\left(\frac{1}{y} + \frac{\frac{-1}{y}}{y}\right)} \]
      Proof
      (+.f64 (/.f64 1 y) (/.f64 (/.f64 -1 y) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 y) (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 y y)))): 13 points increase in error, 12 points decrease in error
      (+.f64 (/.f64 1 y) (/.f64 -1 (Rewrite<= unpow2_binary64 (pow.f64 y 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 y) (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) (pow.f64 y 2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 y) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 y) (/.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 simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6238223.946121859:\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) + \frac{1 - x}{y \cdot y} \cdot \left(-1 + \frac{1}{y}\right)\\ \mathbf{elif}\;y \leq 31835.723135063974:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{y} + \frac{\frac{-1}{y}}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	3280

\[\begin{array}{l} t_0 := \frac{y \cdot \left(1 - x\right)}{y + 1}\\ t_1 := 1 - t_0\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;t_0 \leq 0.98:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 1.002:\\ \;\;\;\;x + \left(\frac{1}{y} + \frac{\frac{-1}{y}}{y}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	1.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;x + \frac{1 + \frac{-1}{y}}{y}\\ \mathbf{elif}\;y \leq 1.607625927258275 \cdot 10^{-8}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{y} + \frac{\frac{-1}{y}}{y}\right)\\ \end{array} \]

Alternative 3
Error	0.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -6238223.946121859:\\ \;\;\;\;x + \frac{1 - x}{y} \cdot \left(1 + \frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 31835.723135063974:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{y} + \frac{\frac{-1}{y}}{y}\right)\\ \end{array} \]

Alternative 4
Error	16.7
Cost	852

\[\begin{array}{l} \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.8628430812644452 \cdot 10^{-37}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 2.250932980465915 \cdot 10^{-12}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.607625927258275 \cdot 10^{-8}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 5.405557843775767 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	9.0
Cost	848

\[\begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.8628430812644452 \cdot 10^{-37}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 2.250932980465915 \cdot 10^{-12}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.607625927258275 \cdot 10^{-8}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.2
Cost	840

\[\begin{array}{l} t_0 := x + \frac{1 + \frac{-1}{y}}{y}\\ \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.607625927258275 \cdot 10^{-8}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.3
Cost	712

\[\begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.607625927258275 \cdot 10^{-8}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.2
Cost	712

\[\begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.607625927258275 \cdot 10^{-8}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	16.7
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.607625927258275 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.405557843775767 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	16.6
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.607625927258275 \cdot 10^{-8}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 5.405557843775767 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	1.4
Cost	584

\[\begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.607625927258275 \cdot 10^{-8}:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	16.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -218.53919179681262:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 25.213549419601623:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	39.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022291 
(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))))