Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, J

Average Error: 21.0 → 7.5

Time: 26.4s

Precision: binary64

Cost: 6352

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

↓

\[\begin{array}{l} t_1 := x \cdot \left(9 \cdot y\right)\\ t_2 := \frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + t_1}{z}}{c}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-283}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 10^{-166}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b + \left(t_1 + \left(t \cdot a\right) \cdot \left(z \cdot -4\right)\right)}{c}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

↓

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* x (* 9.0 y)))
        (t_2 (/ (+ (+ (* (* x 9.0) y) (* a (* t (* z -4.0)))) b) (* z c))))
   (if (<= t_2 (- INFINITY))
     (/ (+ (* a (* t -4.0)) (/ (+ b t_1) z)) c)
     (if (<= t_2 -2e-283)
       t_2
       (if (<= t_2 1e-166)
         (* (/ 1.0 z) (/ (+ b (+ t_1 (* (* t a) (* z -4.0)))) c))
         (if (<= t_2 2e+292) t_2 (* t (/ (* a -4.0) c))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = x * (9.0 * y);
	double t_2 = ((((x * 9.0) * y) + (a * (t * (z * -4.0)))) + b) / (z * c);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((a * (t * -4.0)) + ((b + t_1) / z)) / c;
	} else if (t_2 <= -2e-283) {
		tmp = t_2;
	} else if (t_2 <= 1e-166) {
		tmp = (1.0 / z) * ((b + (t_1 + ((t * a) * (z * -4.0)))) / c);
	} else if (t_2 <= 2e+292) {
		tmp = t_2;
	} else {
		tmp = t * ((a * -4.0) / c);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

↓

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = x * (9.0 * y);
	double t_2 = ((((x * 9.0) * y) + (a * (t * (z * -4.0)))) + b) / (z * c);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = ((a * (t * -4.0)) + ((b + t_1) / z)) / c;
	} else if (t_2 <= -2e-283) {
		tmp = t_2;
	} else if (t_2 <= 1e-166) {
		tmp = (1.0 / z) * ((b + (t_1 + ((t * a) * (z * -4.0)))) / c);
	} else if (t_2 <= 2e+292) {
		tmp = t_2;
	} else {
		tmp = t * ((a * -4.0) / c);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)

↓

def code(x, y, z, t, a, b, c):
	t_1 = x * (9.0 * y)
	t_2 = ((((x * 9.0) * y) + (a * (t * (z * -4.0)))) + b) / (z * c)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = ((a * (t * -4.0)) + ((b + t_1) / z)) / c
	elif t_2 <= -2e-283:
		tmp = t_2
	elif t_2 <= 1e-166:
		tmp = (1.0 / z) * ((b + (t_1 + ((t * a) * (z * -4.0)))) / c)
	elif t_2 <= 2e+292:
		tmp = t_2
	else:
		tmp = t * ((a * -4.0) / c)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
end

↓

function code(x, y, z, t, a, b, c)
	t_1 = Float64(x * Float64(9.0 * y))
	t_2 = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) + Float64(a * Float64(t * Float64(z * -4.0)))) + b) / Float64(z * c))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(a * Float64(t * -4.0)) + Float64(Float64(b + t_1) / z)) / c);
	elseif (t_2 <= -2e-283)
		tmp = t_2;
	elseif (t_2 <= 1e-166)
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(b + Float64(t_1 + Float64(Float64(t * a) * Float64(z * -4.0)))) / c));
	elseif (t_2 <= 2e+292)
		tmp = t_2;
	else
		tmp = Float64(t * Float64(Float64(a * -4.0) / c));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
end

↓

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = x * (9.0 * y);
	t_2 = ((((x * 9.0) * y) + (a * (t * (z * -4.0)))) + b) / (z * c);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = ((a * (t * -4.0)) + ((b + t_1) / z)) / c;
	elseif (t_2 <= -2e-283)
		tmp = t_2;
	elseif (t_2 <= 1e-166)
		tmp = (1.0 / z) * ((b + (t_1 + ((t * a) * (z * -4.0)))) / c);
	elseif (t_2 <= 2e+292)
		tmp = t_2;
	else
		tmp = t * ((a * -4.0) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(x * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] + N[(a * N[(t * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(a * N[(t * -4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b + t$95$1), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$2, -2e-283], t$95$2, If[LessEqual[t$95$2, 1e-166], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(b + N[(t$95$1 + N[(N[(t * a), $MachinePrecision] * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+292], t$95$2, N[(t * N[(N[(a * -4.0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	21.0
Target	14.7
Herbie	7.5

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -inf.0

   1. Initial program 64.0

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Simplified 25.4

      \[\leadsto \color{blue}{\frac{a \cdot \left(t \cdot -4\right) + \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}}{c}} \]
      Proof
      (/.f64 (+.f64 (*.f64 a (*.f64 t -4)) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 a (*.f64 t (Rewrite<= metadata-eval (neg.f64 4)))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 1 points increase in error, 12 points decrease in error
      (/.f64 (+.f64 (*.f64 a (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 t 4)))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 4 points increase in error, 6 points decrease in error
      (/.f64 (+.f64 (*.f64 a (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 t)))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 0 points increase in error, 4 points decrease in error
      (/.f64 (+.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 a 1)) (neg.f64 (*.f64 4 t))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 (/.f64 a (Rewrite<= *-inverses_binary64 (/.f64 z z))) (neg.f64 (*.f64 4 t))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 22 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a z) z)) (neg.f64 (*.f64 4 t))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 6 points increase in error, 1 points decrease in error
      (/.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (*.f64 a z) z) (*.f64 4 t)))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 1 points increase in error, 21 points decrease in error
      (/.f64 (+.f64 (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 a z) (/.f64 z (*.f64 4 t))))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 22 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 a z) (*.f64 4 t)) z))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 0 points increase in error, 22 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 a (*.f64 z (*.f64 4 t)))) z)) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 10 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (*.f64 a (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z 4) t))) z)) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 5 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 z 4) t) a)) z)) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 0 points increase in error, 15 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (*.f64 (*.f64 (*.f64 z 4) t) a) z)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (*.f64 9 y)) b)) z)) c): 1 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (*.f64 (*.f64 (*.f64 z 4) t) a) z)) (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 9) y)) b) z)) c): 13 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (*.f64 (*.f64 (*.f64 z 4) t) a) z)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 b (*.f64 (*.f64 x 9) y))) z)) c): 9 points increase in error, 13 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (neg.f64 (/.f64 (*.f64 (*.f64 (*.f64 z 4) t) a) z)))) c): 21 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (/.f64 (*.f64 (*.f64 (*.f64 z 4) t) a) z))) c): 0 points increase in error, 21 points decrease in error
      (/.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) (*.f64 (*.f64 (*.f64 z 4) t) a)) z)) c): 22 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= associate-+r-_binary64 (+.f64 b (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)))) z) c): 5 points increase in error, 17 points decrease in error
      (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b)) z) c): 7 points increase in error, 5 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))): 22 points increase in error, 0 points decrease in error

   3. Applied egg-rr 25.4

      \[\leadsto \frac{a \cdot \left(t \cdot -4\right) + \frac{\color{blue}{x \cdot \left(9 \cdot y\right) + b}}{z}}{c} \]

   if -inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -1.99999999999999989e-283 or 1.00000000000000004e-166 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 2e292

   1. Initial program 0.7

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   if -1.99999999999999989e-283 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 1.00000000000000004e-166

   1. Initial program 32.0

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Applied egg-rr 2.5

      \[\leadsto \color{blue}{\frac{1}{z} \cdot \frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{c}} \]

   if 2e292 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))

   1. Initial program 60.8

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Simplified 53.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(t, \left(z \cdot a\right) \cdot -4, b\right)\right)}{z \cdot c}} \]
      Proof
      (/.f64 (+.f64 (*.f64 a (*.f64 t -4)) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 a (*.f64 t (Rewrite<= metadata-eval (neg.f64 4)))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 1 points increase in error, 12 points decrease in error
      (/.f64 (+.f64 (*.f64 a (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 t 4)))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 4 points increase in error, 6 points decrease in error
      (/.f64 (+.f64 (*.f64 a (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 t)))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 0 points increase in error, 4 points decrease in error
      (/.f64 (+.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 a 1)) (neg.f64 (*.f64 4 t))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 (/.f64 a (Rewrite<= *-inverses_binary64 (/.f64 z z))) (neg.f64 (*.f64 4 t))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 22 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a z) z)) (neg.f64 (*.f64 4 t))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 6 points increase in error, 1 points decrease in error
      (/.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (*.f64 a z) z) (*.f64 4 t)))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 1 points increase in error, 21 points decrease in error
      (/.f64 (+.f64 (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 a z) (/.f64 z (*.f64 4 t))))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 22 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 a z) (*.f64 4 t)) z))) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 0 points increase in error, 22 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 a (*.f64 z (*.f64 4 t)))) z)) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 10 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (*.f64 a (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z 4) t))) z)) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 5 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 z 4) t) a)) z)) (/.f64 (fma.f64 x (*.f64 9 y) b) z)) c): 0 points increase in error, 15 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (*.f64 (*.f64 (*.f64 z 4) t) a) z)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (*.f64 9 y)) b)) z)) c): 1 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (*.f64 (*.f64 (*.f64 z 4) t) a) z)) (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 9) y)) b) z)) c): 13 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 (/.f64 (*.f64 (*.f64 (*.f64 z 4) t) a) z)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 b (*.f64 (*.f64 x 9) y))) z)) c): 9 points increase in error, 13 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (neg.f64 (/.f64 (*.f64 (*.f64 (*.f64 z 4) t) a) z)))) c): 21 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (/.f64 (*.f64 (*.f64 (*.f64 z 4) t) a) z))) c): 0 points increase in error, 21 points decrease in error
      (/.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) (*.f64 (*.f64 (*.f64 z 4) t) a)) z)) c): 22 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= associate-+r-_binary64 (+.f64 b (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)))) z) c): 5 points increase in error, 17 points decrease in error
      (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b)) z) c): 7 points increase in error, 5 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))): 22 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around inf 32.2

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

   4. Simplified 25.5

      \[\leadsto \color{blue}{\frac{-4 \cdot a}{c} \cdot t} \]
      Proof
      (*.f64 (/.f64 (*.f64 -4 a) c) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 a -4)) c) t): 0 points increase in error, 6 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 a -4) (/.f64 c t))): 6 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 -4 a)) (/.f64 c t)): 0 points increase in error, 6 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -4 (/.f64 a (/.f64 c t)))): 6 points increase in error, 0 points decrease in error
      (*.f64 -4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a t) c))): 0 points increase in error, 6 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq -\infty:\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq -2 \cdot 10^{-283}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq 10^{-166}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b + \left(x \cdot \left(9 \cdot y\right) + \left(t \cdot a\right) \cdot \left(z \cdot -4\right)\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.2
Cost	6352

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ t_2 := \frac{a \cdot \left(t \cdot -4\right) + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \end{array} \]

Alternative 2
Error	37.1
Cost	1504

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+256}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{+160}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-60}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;t \leq -1.42 \cdot 10^{-81}:\\ \;\;\;\;9 \cdot \frac{y}{z \cdot \frac{c}{x}}\\ \mathbf{elif}\;t \leq -1.12 \cdot 10^{-100}:\\ \;\;\;\;\frac{a}{c \cdot \frac{-0.25}{t}}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-204}:\\ \;\;\;\;9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right)\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \end{array} \]

Alternative 3
Error	37.0
Cost	1504

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+256}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \mathbf{elif}\;t \leq -6 \cdot 10^{+161}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-59}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-81}:\\ \;\;\;\;9 \cdot \frac{y}{z \cdot \frac{c}{x}}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{a}{c \cdot \frac{-0.25}{t}}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-203}:\\ \;\;\;\;\left(x \cdot \frac{y}{c}\right) \cdot \frac{9}{z}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-172}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \end{array} \]

Alternative 4
Error	23.6
Cost	1496

\[\begin{array}{l} t_1 := b + 9 \cdot \left(x \cdot y\right)\\ t_2 := \frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{-58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{b}{z} + 9 \cdot \frac{y}{\frac{z}{x}}}{c}\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7.6 \cdot 10^{-181}:\\ \;\;\;\;\frac{t_1}{z \cdot c}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-206}:\\ \;\;\;\;\frac{t_1}{z} \cdot \frac{1}{c}\\ \mathbf{elif}\;t \leq 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 5
Error	23.6
Cost	1496

\[\begin{array}{l} t_1 := \frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{b}{z} + 9 \cdot \frac{y}{\frac{z}{x}}}{c}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{-181}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{b}{z} + 9 \cdot \frac{x \cdot y}{z}}{c}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 6
Error	23.7
Cost	1496

\[\begin{array}{l} t_1 := \frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{b}{z} + 9 \cdot \frac{y}{\frac{z}{x}}}{c}\\ \mathbf{elif}\;t \leq -1.86 \cdot 10^{-97}:\\ \;\;\;\;\frac{b - 4 \cdot \left(a \cdot \left(z \cdot t\right)\right)}{z \cdot c}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{b}{z} + 9 \cdot \frac{x \cdot y}{z}}{c}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 7
Error	10.2
Cost	1481

\[\begin{array}{l} t_1 := x \cdot \left(9 \cdot y\right)\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-27} \lor \neg \left(z \leq 2.5 \cdot 10^{-155}\right):\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + t_1}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(t_1 + \left(t \cdot a\right) \cdot \left(z \cdot -4\right)\right)}{z \cdot c}\\ \end{array} \]

Alternative 8
Error	12.5
Cost	1353

\[\begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{-25} \lor \neg \left(z \leq 1.2 \cdot 10^{-239}\right):\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \end{array} \]

Alternative 9
Error	35.1
Cost	1240

\[\begin{array}{l} t_1 := t \cdot \frac{a \cdot -4}{c}\\ \mathbf{if}\;b \leq -25000000:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b}{c}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-284}:\\ \;\;\;\;9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right)\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+131}:\\ \;\;\;\;b \cdot \frac{1}{z \cdot c}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+202}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 10
Error	30.3
Cost	1232

\[\begin{array}{l} t_1 := \frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+257}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-58}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \end{array} \]

Alternative 11
Error	23.6
Cost	1232

\[\begin{array}{l} t_1 := b + 9 \cdot \left(x \cdot y\right)\\ t_2 := \frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-179}:\\ \;\;\;\;\frac{t_1}{z \cdot c}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-203}:\\ \;\;\;\;\frac{t_1}{z} \cdot \frac{1}{c}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 12
Error	34.9
Cost	977

\[\begin{array}{l} \mathbf{if}\;b \leq -1400000:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-62} \lor \neg \left(b \leq 8 \cdot 10^{+131}\right) \land b \leq 6.6 \cdot 10^{+202}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 13
Error	35.0
Cost	976

\[\begin{array}{l} t_1 := -4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{if}\;b \leq -47000:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{+131}:\\ \;\;\;\;b \cdot \frac{1}{z \cdot c}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{+203}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 14
Error	35.0
Cost	976

\[\begin{array}{l} t_1 := -4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{if}\;b \leq -4500000:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b}{c}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+131}:\\ \;\;\;\;b \cdot \frac{1}{z \cdot c}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+204}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 15
Error	34.9
Cost	976

\[\begin{array}{l} \mathbf{if}\;b \leq -0.175:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b}{c}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-64}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+131}:\\ \;\;\;\;b \cdot \frac{1}{z \cdot c}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+202}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 16
Error	19.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-25} \lor \neg \left(z \leq 4.6 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \end{array} \]

Alternative 17
Error	43.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;c \leq 2.3 \cdot 10^{-258} \lor \neg \left(c \leq 3.1 \cdot 10^{+234}\right):\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 18
Error	43.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;c \leq 1.8 \cdot 10^{-255}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{+264}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \end{array} \]

Alternative 19
Error	43.6
Cost	320

\[\frac{b}{z \cdot c} \]

Reproduce

herbie shell --seed 2022340 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -1.100156740804105e-171) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 0.0) (/ (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9.0 (/ y c)) (/ x z)) (/ b (* c z))) (* 4.0 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (- (+ (* 9.0 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4.0 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))