\[\frac{x + y}{1 - \frac{y}{z}} \]

↓

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ t_1 := \mathsf{fma}\left(x, z, z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-274} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-z}{\frac{y}{x}} - \left(\frac{t_1}{\frac{{y}^{3}}{z \cdot z}} + \left(z + \frac{t_1}{y} \cdot \frac{z}{y}\right)\right)\right) - \frac{z}{\frac{y}{z}}\\ \end{array} \]

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))) (t_1 (fma x z (* z z))))
   (if (or (<= t_0 -5e-274) (not (<= t_0 0.0)))
     t_0
     (-
      (-
       (/ (- z) (/ y x))
       (+ (/ t_1 (/ (pow y 3.0) (* z z))) (+ z (* (/ t_1 y) (/ z y)))))
      (/ z (/ y z))))))

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

↓

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double t_1 = fma(x, z, (z * z));
	double tmp;
	if ((t_0 <= -5e-274) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = ((-z / (y / x)) - ((t_1 / (pow(y, 3.0) / (z * z))) + (z + ((t_1 / y) * (z / y))))) - (z / (y / z));
	}
	return tmp;
}

function code(x, y, z)
	return Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	t_1 = fma(x, z, Float64(z * z))
	tmp = 0.0
	if ((t_0 <= -5e-274) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(-z) / Float64(y / x)) - Float64(Float64(t_1 / Float64((y ^ 3.0) / Float64(z * z))) + Float64(z + Float64(Float64(t_1 / y) * Float64(z / y))))) - Float64(z / Float64(y / z)));
	end
	return tmp
end

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * z + N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-274], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(N[((-z) / N[(y / x), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 / N[(N[Power[y, 3.0], $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z + N[(N[(t$95$1 / y), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

\frac{x + y}{1 - \frac{y}{z}}

↓

\begin{array}{l}
t_0 := \frac{x + y}{1 - \frac{y}{z}}\\
t_1 := \mathsf{fma}\left(x, z, z \cdot z\right)\\
\mathbf{if}\;t_0 \leq -5 \cdot 10^{-274} \lor \neg \left(t_0 \leq 0\right):\\
\;\;\;\;t_0\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{-z}{\frac{y}{x}} - \left(\frac{t_1}{\frac{{y}^{3}}{z \cdot z}} + \left(z + \frac{t_1}{y} \cdot \frac{z}{y}\right)\right)\right) - \frac{z}{\frac{y}{z}}\\


\end{array}

Original	7.6
Target	4.2
Herbie	0.3

Derivation

Split input into 2 regimes
if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5e-274 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))
1. Initial program 0.1
  \[\frac{x + y}{1 - \frac{y}{z}} \]
if -5e-274 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0
1. Initial program 57.6
  \[\frac{x + y}{1 - \frac{y}{z}} \]
2. Simplified57.6
  \[\leadsto \color{blue}{\frac{y + x}{1 - \frac{y}{z}}} \]
  Proof
  (/.f64 (+.f64 y x) (-.f64 1 (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x y)) (-.f64 1 (/.f64 y z))): 0 points increase in error, 0 points decrease in error
3. Taylor expanded in y around inf 1.6
  \[\leadsto \color{blue}{\left(-1 \cdot z + \left(-1 \cdot \frac{z \cdot x}{y} + \left(\frac{\left(-1 \cdot \left(z \cdot x\right) - {z}^{2}\right) \cdot {z}^{2}}{{y}^{3}} + \frac{\left(-1 \cdot \left(z \cdot x\right) - {z}^{2}\right) \cdot z}{{y}^{2}}\right)\right)\right) - \frac{{z}^{2}}{y}} \]
4. Simplified1.6
  \[\leadsto \color{blue}{\left(\frac{-z}{\frac{y}{x}} + \left(\left(\left(-z\right) - \frac{\mathsf{fma}\left(x, z, z \cdot z\right)}{y} \cdot \frac{z}{y}\right) - \frac{\mathsf{fma}\left(x, z, z \cdot z\right)}{\frac{{y}^{3}}{z \cdot z}}\right)\right) - \frac{z}{\frac{y}{z}}} \]
  Proof
  (-.f64 (+.f64 (/.f64 (neg.f64 z) (/.f64 y x)) (-.f64 (-.f64 (neg.f64 z) (*.f64 (/.f64 (fma.f64 x z (*.f64 z z)) y) (/.f64 z y))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z)) (/.f64 y x)) (-.f64 (-.f64 (neg.f64 z) (*.f64 (/.f64 (fma.f64 x z (*.f64 z z)) y) (/.f64 z y))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 z (/.f64 y x)))) (-.f64 (-.f64 (neg.f64 z) (*.f64 (/.f64 (fma.f64 x z (*.f64 z z)) y) (/.f64 z y))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z x) y))) (-.f64 (-.f64 (neg.f64 z) (*.f64 (/.f64 (fma.f64 x z (*.f64 z z)) y) (/.f64 z y))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z)) (*.f64 (/.f64 (fma.f64 x z (*.f64 z z)) y) (/.f64 z y))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 36 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (-.f64 (*.f64 -1 z) (*.f64 (/.f64 (fma.f64 x z (Rewrite<= unpow2_binary64 (pow.f64 z 2))) y) (/.f64 z y))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 10 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (-.f64 (*.f64 -1 z) (*.f64 (/.f64 (fma.f64 x z (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (pow.f64 z 2))))) y) (/.f64 z y))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (-.f64 (*.f64 -1 z) (*.f64 (/.f64 (fma.f64 x z (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 z 2))))) y) (/.f64 z y))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 46 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (-.f64 (*.f64 -1 z) (*.f64 (/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x z) (*.f64 -1 (pow.f64 z 2)))) y) (/.f64 z y))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (-.f64 (*.f64 -1 z) (*.f64 (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 z x)) (*.f64 -1 (pow.f64 z 2))) y) (/.f64 z y))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (-.f64 (*.f64 -1 z) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (*.f64 y y)))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (-.f64 (*.f64 -1 z) (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (Rewrite<= unpow2_binary64 (pow.f64 y 2)))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 z) (neg.f64 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2))))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 46 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (+.f64 (*.f64 -1 z) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2))))) (/.f64 (fma.f64 x z (*.f64 z z)) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (+.f64 (*.f64 -1 z) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2)))) (/.f64 (fma.f64 x z (Rewrite<= unpow2_binary64 (pow.f64 z 2))) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (+.f64 (*.f64 -1 z) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2)))) (/.f64 (fma.f64 x z (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (pow.f64 z 2))))) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 20 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (+.f64 (*.f64 -1 z) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2)))) (/.f64 (fma.f64 x z (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 z 2))))) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (+.f64 (*.f64 -1 z) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2)))) (/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x z) (*.f64 -1 (pow.f64 z 2)))) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 46 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (+.f64 (*.f64 -1 z) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2)))) (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 z x)) (*.f64 -1 (pow.f64 z 2))) (/.f64 (pow.f64 y 3) (*.f64 z z))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (+.f64 (*.f64 -1 z) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2)))) (/.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (/.f64 (pow.f64 y 3) (Rewrite<= unpow2_binary64 (pow.f64 z 2)))))) (/.f64 z (/.f64 y z))): 46 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (-.f64 (+.f64 (*.f64 -1 z) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2)))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (*.f64 -1 z) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2)))) (neg.f64 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3)))))) (/.f64 z (/.f64 y z))): 46 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (+.f64 (*.f64 -1 z) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2)))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3)))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (Rewrite=> associate-+l+_binary64 (+.f64 (*.f64 -1 z) (+.f64 (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z) (pow.f64 y 2))) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3))))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) z)) (pow.f64 y 2))) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3)))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 -1 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2)))) z)) (pow.f64 y 2)) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3)))))) (/.f64 z (/.f64 y z))): 46 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (*.f64 -1 (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (*.f64 z x) (*.f64 (neg.f64 -1) (pow.f64 z 2))))) z) (pow.f64 y 2)) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3)))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 -1 (*.f64 z x)) (*.f64 -1 (*.f64 (neg.f64 -1) (pow.f64 z 2))))) z) (pow.f64 y 2)) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3)))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (+.f64 (*.f64 -1 (*.f64 z x)) (*.f64 -1 (*.f64 (Rewrite=> metadata-eval 1) (pow.f64 z 2)))) z) (pow.f64 y 2)) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3)))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (+.f64 (*.f64 -1 (*.f64 z x)) (*.f64 -1 (Rewrite=> *-lft-identity_binary64 (pow.f64 z 2)))) z) (pow.f64 y 2)) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3)))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (+.f64 (*.f64 -1 (*.f64 z x)) (Rewrite=> mul-1-neg_binary64 (neg.f64 (pow.f64 z 2)))) z) (pow.f64 y 2)) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3)))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2))) z) (pow.f64 y 2)) (*.f64 -1 (/.f64 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3)))))) (/.f64 z (/.f64 y z))): 46 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2)) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1 (*.f64 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2))) (pow.f64 z 2))) (pow.f64 y 3)))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2)) (/.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 -1 (-.f64 (*.f64 z x) (*.f64 -1 (pow.f64 z 2)))) (pow.f64 z 2))) (pow.f64 y 3))))) (/.f64 z (/.f64 y z))): 46 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2)) (/.f64 (*.f64 (*.f64 -1 (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (*.f64 z x) (*.f64 (neg.f64 -1) (pow.f64 z 2))))) (pow.f64 z 2)) (pow.f64 y 3))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2)) (/.f64 (*.f64 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 -1 (*.f64 z x)) (*.f64 -1 (*.f64 (neg.f64 -1) (pow.f64 z 2))))) (pow.f64 z 2)) (pow.f64 y 3))))) (/.f64 z (/.f64 y z))): 46 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2)) (/.f64 (*.f64 (+.f64 (*.f64 -1 (*.f64 z x)) (*.f64 -1 (*.f64 (Rewrite=> metadata-eval 1) (pow.f64 z 2)))) (pow.f64 z 2)) (pow.f64 y 3))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2)) (/.f64 (*.f64 (+.f64 (*.f64 -1 (*.f64 z x)) (*.f64 -1 (Rewrite=> *-lft-identity_binary64 (pow.f64 z 2)))) (pow.f64 z 2)) (pow.f64 y 3))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2)) (/.f64 (*.f64 (+.f64 (*.f64 -1 (*.f64 z x)) (Rewrite=> mul-1-neg_binary64 (neg.f64 (pow.f64 z 2)))) (pow.f64 z 2)) (pow.f64 y 3))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2)) (/.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2))) (pow.f64 z 2)) (pow.f64 y 3))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (*.f64 -1 z) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) (pow.f64 z 2)) (pow.f64 y 3)) (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2)))))) (/.f64 z (/.f64 y z))): 43 points increase in error, 0 points decrease in error
  (-.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (*.f64 -1 z)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) (pow.f64 z 2)) (pow.f64 y 3)) (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 43 points decrease in error
  (-.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 z) (*.f64 -1 (/.f64 (*.f64 z x) y)))) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) (pow.f64 z 2)) (pow.f64 y 3)) (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2)))) (/.f64 z (/.f64 y z))): 46 points increase in error, 0 points decrease in error
  (-.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -1 z) (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) (pow.f64 z 2)) (pow.f64 y 3)) (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2)))))) (/.f64 z (/.f64 y z))): 0 points increase in error, 0 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 z) (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) (pow.f64 z 2)) (pow.f64 y 3)) (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2))))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z z) y))): 0 points increase in error, 46 points decrease in error
  (-.f64 (+.f64 (*.f64 -1 z) (+.f64 (*.f64 -1 (/.f64 (*.f64 z x) y)) (+.f64 (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) (pow.f64 z 2)) (pow.f64 y 3)) (/.f64 (*.f64 (-.f64 (*.f64 -1 (*.f64 z x)) (pow.f64 z 2)) z) (pow.f64 y 2))))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) y)): 0 points increase in error, 0 points decrease in error
Recombined 2 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-274} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-z}{\frac{y}{x}} - \left(\frac{\mathsf{fma}\left(x, z, z \cdot z\right)}{\frac{{y}^{3}}{z \cdot z}} + \left(z + \frac{\mathsf{fma}\left(x, z, z \cdot z\right)}{y} \cdot \frac{z}{y}\right)\right)\right) - \frac{z}{\frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1865

\[\begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-274} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternative 2
Error	18.2
Cost	1372

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.62 \cdot 10^{+81}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.52 \cdot 10^{-52}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-169}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	18.1
Cost	1372

\[\begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.08 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.62 \cdot 10^{+81}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-52}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-84}:\\ \;\;\;\;\frac{\left(-y\right) - x}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-169}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	18.2
Cost	1372

\[\begin{array}{l} t_0 := \left(1 + \frac{y}{z}\right) \cdot \left(x + y\right)\\ t_1 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.62 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.52 \cdot 10^{-52}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{\left(-y\right) - x}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	22.8
Cost	1308

\[\begin{array}{l} t_0 := x \cdot \frac{-z}{y}\\ \mathbf{if}\;y \leq -1.12 \cdot 10^{+145}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+73}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -19000000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.52 \cdot 10^{-52}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-82}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	22.8
Cost	1308

\[\begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+145}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{+74}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -700000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.52 \cdot 10^{-52}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{-x}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-82}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+15}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	18.7
Cost	1243

\[\begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+145} \lor \neg \left(y \leq -1.62 \cdot 10^{+81}\right) \land \left(y \leq -3 \cdot 10^{-13} \lor \neg \left(y \leq -1.52 \cdot 10^{-52} \lor \neg \left(y \leq -4.6 \cdot 10^{-84}\right) \land y \leq 4.8 \cdot 10^{-109}\right)\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 8
Error	22.4
Cost	1044

\[\begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+145}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{+74}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -950000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-82}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 115000000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Error	21.2
Cost	721

\[\begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+145}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+72} \lor \neg \left(y \leq -4100000000000\right) \land y \leq 4 \cdot 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10
Error	26.3
Cost	392

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-16}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 11
Error	41.4
Cost	64

\[x \]

Error

Target

Derivation

`if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5e-274 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))`

`if -5e-274 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0`

Alternatives

Error

Reproduce