Average Error: 16.8 → 6.7
Time: 27.8s
Precision: binary64
Cost: 8388
\[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
\[\begin{array}{l} t_1 := \frac{y \cdot b}{t} + \left(a + 1\right)\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \left(a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)\right)}{z}}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-322}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{t_1}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+251}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* y b) t) (+ a 1.0))) (t_2 (/ (+ x (/ (* y z) t)) t_1)))
   (if (<= t_2 (- INFINITY))
     (/ y (/ (* t (+ a (fma (/ y t) b 1.0))) z))
     (if (<= t_2 -2e-322)
       (/ (+ x (* (* y z) (/ 1.0 t))) t_1)
       (if (<= t_2 0.0)
         (+ (/ z b) (* (/ t y) (/ x b)))
         (if (<= t_2 5e+251) t_2 (/ z b)))))))
double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * b) / t) + (a + 1.0);
	double t_2 = (x + ((y * z) / t)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y / ((t * (a + fma((y / t), b, 1.0))) / z);
	} else if (t_2 <= -2e-322) {
		tmp = (x + ((y * z) * (1.0 / t))) / t_1;
	} else if (t_2 <= 0.0) {
		tmp = (z / b) + ((t / y) * (x / b));
	} else if (t_2 <= 5e+251) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}
function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end
function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0))
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(y / Float64(Float64(t * Float64(a + fma(Float64(y / t), b, 1.0))) / z));
	elseif (t_2 <= -2e-322)
		tmp = Float64(Float64(x + Float64(Float64(y * z) * Float64(1.0 / t))) / t_1);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(t / y) * Float64(x / b)));
	elseif (t_2 <= 5e+251)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end
code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y / N[(N[(t * N[(a + N[(N[(y / t), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-322], N[(N[(x + N[(N[(y * z), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(t / y), $MachinePrecision] * N[(x / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+251], t$95$2, N[(z / b), $MachinePrecision]]]]]]]
\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}
\begin{array}{l}
t_1 := \frac{y \cdot b}{t} + \left(a + 1\right)\\
t_2 := \frac{x + \frac{y \cdot z}{t}}{t_1}\\
\mathbf{if}\;t_2 \leq -\infty:\\
\;\;\;\;\frac{y}{\frac{t \cdot \left(a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)\right)}{z}}\\

\mathbf{elif}\;t_2 \leq -2 \cdot 10^{-322}:\\
\;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{t_1}\\

\mathbf{elif}\;t_2 \leq 0:\\
\;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\

\mathbf{elif}\;t_2 \leq 5 \cdot 10^{+251}:\\
\;\;\;\;t_2\\

\mathbf{else}:\\
\;\;\;\;\frac{z}{b}\\


\end{array}

Error

Target

Original16.8
Target12.8
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \end{array} \]

Derivation

  1. Split input into 5 regimes
  2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -inf.0

    1. Initial program 64.0

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
    2. Taylor expanded in x around 0 41.9

      \[\leadsto \color{blue}{\frac{y \cdot z}{t \cdot \left(1 + \left(\frac{y \cdot b}{t} + a\right)\right)}} \]
    3. Simplified22.5

      \[\leadsto \color{blue}{\frac{y}{\frac{t \cdot \left(a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)\right)}{z}}} \]
      Proof
      (/.f64 y (/.f64 (*.f64 t (+.f64 a (fma.f64 (/.f64 y t) b 1))) z)): 0 points increase in error, 0 points decrease in error
      (/.f64 y (/.f64 (*.f64 t (Rewrite=> +-commutative_binary64 (+.f64 (fma.f64 (/.f64 y t) b 1) a))) z)): 0 points increase in error, 0 points decrease in error
      (/.f64 y (/.f64 (*.f64 t (+.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 (/.f64 y t) b) 1)) a)) z)): 0 points increase in error, 0 points decrease in error
      (/.f64 y (/.f64 (*.f64 t (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (/.f64 y t) b))) a)) z)): 0 points increase in error, 0 points decrease in error
      (/.f64 y (/.f64 (*.f64 t (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 (*.f64 (/.f64 y t) b) a)))) z)): 0 points increase in error, 0 points decrease in error
      (/.f64 y (/.f64 (*.f64 t (+.f64 1 (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y b) t)) a))) z)): 6 points increase in error, 12 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y z) (*.f64 t (+.f64 1 (+.f64 (/.f64 (*.f64 y b) t) a))))): 56 points increase in error, 31 points decrease in error

    if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < -1.97626e-322

    1. Initial program 0.5

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
    2. Applied egg-rr0.5

      \[\leadsto \frac{x + \color{blue}{\frac{1}{t} \cdot \left(y \cdot z\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

    if -1.97626e-322 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 0.0

    1. Initial program 31.8

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
    2. Simplified25.8

      \[\leadsto \color{blue}{\frac{x + \frac{y}{t} \cdot z}{1 + \left(a + \frac{y}{t} \cdot b\right)}} \]
      Proof
      (/.f64 (+.f64 x (*.f64 (/.f64 y t) z)) (+.f64 1 (+.f64 a (*.f64 (/.f64 y t) b)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y z) t))) (+.f64 1 (+.f64 a (*.f64 (/.f64 y t) b)))): 18 points increase in error, 20 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 1 (+.f64 a (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y b) t))))): 12 points increase in error, 14 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 a) (/.f64 (*.f64 y b) t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 a 1)) (/.f64 (*.f64 y b) t))): 0 points increase in error, 0 points decrease in error
    3. Taylor expanded in b around inf 33.1

      \[\leadsto \color{blue}{\frac{t \cdot \left(\frac{y \cdot z}{t} + x\right)}{y \cdot b}} \]
    4. Simplified34.5

      \[\leadsto \color{blue}{\frac{t}{b} \cdot \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{y}} \]
      Proof
      (*.f64 (/.f64 t b) (/.f64 (fma.f64 z (/.f64 y t) x) y)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 t b) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (/.f64 y t)) x)) y)): 2 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 t b) (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 y t) z)) x) y)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 t b) (/.f64 (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y z) t)) x) y)): 12 points increase in error, 21 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t (+.f64 (/.f64 (*.f64 y z) t) x)) (*.f64 b y))): 33 points increase in error, 63 points decrease in error
      (/.f64 (*.f64 t (+.f64 (/.f64 (*.f64 y z) t) x)) (Rewrite<= *-commutative_binary64 (*.f64 y b))): 0 points increase in error, 0 points decrease in error
    5. Taylor expanded in t around 0 25.8

      \[\leadsto \color{blue}{\frac{t \cdot x}{y \cdot b} + \frac{z}{b}} \]
    6. Simplified17.5

      \[\leadsto \color{blue}{\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}} \]
      Proof
      (+.f64 (/.f64 z b) (*.f64 (/.f64 t y) (/.f64 x b))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 z b) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t x) (*.f64 y b)))): 21 points increase in error, 32 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 t x) (*.f64 y b)) (/.f64 z b))): 0 points increase in error, 0 points decrease in error

    if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t))) < 5.0000000000000005e251

    1. Initial program 0.6

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

    if 5.0000000000000005e251 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a 1) (/.f64 (*.f64 y b) t)))

    1. Initial program 58.1

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
    2. Simplified46.9

      \[\leadsto \color{blue}{\frac{x + \frac{y}{t} \cdot z}{1 + \left(a + \frac{y}{t} \cdot b\right)}} \]
      Proof
      (/.f64 (+.f64 x (*.f64 (/.f64 y t) z)) (+.f64 1 (+.f64 a (*.f64 (/.f64 y t) b)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y z) t))) (+.f64 1 (+.f64 a (*.f64 (/.f64 y t) b)))): 18 points increase in error, 20 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 1 (+.f64 a (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y b) t))))): 12 points increase in error, 14 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 a) (/.f64 (*.f64 y b) t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 a 1)) (/.f64 (*.f64 y b) t))): 0 points increase in error, 0 points decrease in error
    3. Taylor expanded in y around inf 17.1

      \[\leadsto \color{blue}{\frac{z}{b}} \]
  3. Recombined 5 regimes into one program.
  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\frac{t \cdot \left(a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)\right)}{z}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -2 \cdot 10^{-322}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error6.4
Cost8388
\[\begin{array}{l} t_1 := \frac{y \cdot b}{t} + \left(a + 1\right)\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{a + \mathsf{fma}\left(\frac{y}{t}, b, 1\right)}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-322}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{t_1}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+251}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
Alternative 2
Error6.5
Cost5712
\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-322}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+251}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
Alternative 3
Error6.5
Cost5712
\[\begin{array}{l} t_1 := \frac{y \cdot b}{t} + \left(a + 1\right)\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{t} \cdot \frac{z}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-322}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{t_1}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+251}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
Alternative 4
Error30.3
Cost2024
\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t_2}{a}\\ \mathbf{if}\;a \leq -4.9 \cdot 10^{+182}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -9.5 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.45 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{elif}\;a \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.26 \cdot 10^{-106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -5.6 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-225}:\\ \;\;\;\;\frac{x}{1 + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;a \leq 1.46 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{-34}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 5
Error30.2
Cost1896
\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t_2}{a}\\ t_4 := \frac{x}{1 + \left(a + \frac{y}{\frac{t}{b}}\right)}\\ \mathbf{if}\;a \leq -8.8 \cdot 10^{+181}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.42 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -59000000000000:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{elif}\;a \leq -5.9 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7.8 \cdot 10^{-111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-237}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 100000:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 6
Error30.2
Cost1896
\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t_2}{a}\\ t_4 := \frac{x}{1 + \left(a + \frac{y \cdot b}{t}\right)}\\ \mathbf{if}\;a \leq -8.8 \cdot 10^{+181}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1750000000:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.95 \cdot 10^{-106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.3 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.42 \cdot 10^{-233}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 2.55 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3000000:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 7
Error24.4
Cost1752
\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t_2}{1 + \frac{y \cdot b}{t}}\\ \mathbf{if}\;a \leq -8.8 \cdot 10^{+181}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{a}\\ \mathbf{elif}\;a \leq -1.2 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -47000000:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-214}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 5.7 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 0.00048:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{a + 1}\\ \end{array} \]
Alternative 8
Error30.3
Cost1500
\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ t_2 := \frac{t_1}{a}\\ \mathbf{if}\;a \leq -5.8 \cdot 10^{+199}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{+156}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -60000000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-225}:\\ \;\;\;\;\frac{x}{1 + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 9
Error29.7
Cost1500
\[\begin{array}{l} t_1 := \frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{if}\;a \leq -1.8 \cdot 10^{+199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.7 \cdot 10^{+156}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -32000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 10^{-223}:\\ \;\;\;\;\frac{x}{1 + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error30.1
Cost1500
\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{a}\\ \mathbf{if}\;a \leq -1.8 \cdot 10^{+199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -5.5 \cdot 10^{+156}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -29000:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a}\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-225}:\\ \;\;\;\;\frac{x}{1 + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-59}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 11
Error22.3
Cost1496
\[\begin{array}{l} t_1 := \frac{x + z \cdot \frac{y}{t}}{a + 1}\\ t_2 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6600:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 12
Error23.5
Cost1496
\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{+30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7.8 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 13
Error23.6
Cost1496
\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{+30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-80}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 14
Error23.5
Cost1496
\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{+30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{a + 1} + \frac{y \cdot z}{t \cdot \left(a + 1\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{x + \left(y \cdot z\right) \cdot \frac{1}{t}}{a + 1}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 15
Error11.7
Cost1484
\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{+195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-121}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t}}{\frac{y}{\frac{t}{b}} + \left(a + 1\right)}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+222}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 16
Error11.5
Cost1484
\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;y \leq -1.75 \cdot 10^{+196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-97}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\frac{y}{\frac{t}{b}} + \left(a + 1\right)}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+222}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 17
Error11.5
Cost1484
\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\frac{y}{\frac{t}{b}} + \left(a + 1\right)}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+222}:\\ \;\;\;\;\frac{x + \frac{z}{\frac{t}{y}}}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 18
Error11.8
Cost1352
\[\begin{array}{l} t_1 := \frac{z}{b} + \frac{t}{y} \cdot \frac{x}{b}\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+222}:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{1 + \left(a + b \cdot \frac{y}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 19
Error31.6
Cost1244
\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;y \leq -3.75 \cdot 10^{+195}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{1 + y \cdot \frac{b}{t}}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+49}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
Alternative 20
Error37.7
Cost984
\[\begin{array}{l} \mathbf{if}\;a \leq -5.1 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq -18000000:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-229}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]
Alternative 21
Error30.8
Cost980
\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;y \leq -7.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
Alternative 22
Error28.8
Cost584
\[\begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -0.118:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 23
Error36.6
Cost456
\[\begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]
Alternative 24
Error51.3
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022338 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))