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

Average Error: 3.8 → 3.8

Time: 14.2s

Precision: binary64

Cost: 968

Math

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

↓

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-270}:\\ \;\;\;\;x + \frac{\frac{t_1}{z}}{-3}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-97}:\\ \;\;\;\;x + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x + t_1 \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- y (/ t y))))
   (if (<= y -1e-270)
     (+ x (/ (/ t_1 z) -3.0))
     (if (<= y 1.35e-97)
       (+ x (/ t (* 3.0 (* y z))))
       (+ x (* t_1 (/ -0.3333333333333333 z)))))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = y - (t / y);
	double tmp;
	if (y <= -1e-270) {
		tmp = x + ((t_1 / z) / -3.0);
	} else if (y <= 1.35e-97) {
		tmp = x + (t / (3.0 * (y * z)));
	} else {
		tmp = x + (t_1 * (-0.3333333333333333 / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y - (t / y)
    if (y <= (-1d-270)) then
        tmp = x + ((t_1 / z) / (-3.0d0))
    else if (y <= 1.35d-97) then
        tmp = x + (t / (3.0d0 * (y * z)))
    else
        tmp = x + (t_1 * ((-0.3333333333333333d0) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = y - (t / y);
	double tmp;
	if (y <= -1e-270) {
		tmp = x + ((t_1 / z) / -3.0);
	} else if (y <= 1.35e-97) {
		tmp = x + (t / (3.0 * (y * z)));
	} else {
		tmp = x + (t_1 * (-0.3333333333333333 / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

↓

def code(x, y, z, t):
	t_1 = y - (t / y)
	tmp = 0
	if y <= -1e-270:
		tmp = x + ((t_1 / z) / -3.0)
	elif y <= 1.35e-97:
		tmp = x + (t / (3.0 * (y * z)))
	else:
		tmp = x + (t_1 * (-0.3333333333333333 / z))
	return tmp

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

↓

function code(x, y, z, t)
	t_1 = Float64(y - Float64(t / y))
	tmp = 0.0
	if (y <= -1e-270)
		tmp = Float64(x + Float64(Float64(t_1 / z) / -3.0));
	elseif (y <= 1.35e-97)
		tmp = Float64(x + Float64(t / Float64(3.0 * Float64(y * z))));
	else
		tmp = Float64(x + Float64(t_1 * Float64(-0.3333333333333333 / z)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = y - (t / y);
	tmp = 0.0;
	if (y <= -1e-270)
		tmp = x + ((t_1 / z) / -3.0);
	elseif (y <= 1.35e-97)
		tmp = x + (t / (3.0 * (y * z)));
	else
		tmp = x + (t_1 * (-0.3333333333333333 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-270], N[(x + N[(N[(t$95$1 / z), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e-97], N[(x + N[(t / N[(3.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$1 * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.8
Target	1.5
Herbie	3.8

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

Derivation

1. Split input into 3 regimes

2. if y < -1e-270

   1. Initial program 3.4

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Simplified 3.4

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
      Proof
      (+.f64 x (*.f64 (/.f64 -1/3 z) (-.f64 y (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 -1 3)) z) (-.f64 y (/.f64 t y)))): 7 points increase in error, 0 points decrease in error
      (+.f64 x (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 3 z))) (-.f64 y (/.f64 t y)))): 0 points increase in error, 7 points decrease in error
      (+.f64 x (*.f64 (/.f64 -1 (Rewrite<= *-commutative_binary64 (*.f64 z 3))) (-.f64 y (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 -1 (*.f64 z 3)) y) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 -1 y) (*.f64 z 3))) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 2 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 y (*.f64 z 3)))) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 8 points increase in error, 2 points decrease in error
      (+.f64 x (-.f64 (*.f64 -1 (/.f64 y (*.f64 z 3))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 t) (*.f64 (*.f64 z 3) y))))): 0 points increase in error, 8 points decrease in error
      (+.f64 x (-.f64 (*.f64 -1 (/.f64 y (*.f64 z 3))) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 t (*.f64 (*.f64 z 3) y)))))): 7 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite=> distribute-lft-out--_binary64 (*.f64 -1 (-.f64 (/.f64 y (*.f64 z 3)) (/.f64 t (*.f64 (*.f64 z 3) y)))))): 0 points increase in error, 7 points decrease in error
      (+.f64 x (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (/.f64 y (*.f64 z 3)) (/.f64 t (*.f64 (*.f64 z 3) y)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 x (-.f64 (/.f64 y (*.f64 z 3)) (/.f64 t (*.f64 (*.f64 z 3) y))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y)))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 3.4

      \[\leadsto x + \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{-3}} \]

   if -1e-270 < y < 1.34999999999999993e-97

   1. Initial program 10.1

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Simplified 10.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}} \]
      Proof
      (+.f64 x (*.f64 (/.f64 -1/3 z) (-.f64 y (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 -1 3)) z) (-.f64 y (/.f64 t y)))): 7 points increase in error, 0 points decrease in error
      (+.f64 x (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 3 z))) (-.f64 y (/.f64 t y)))): 0 points increase in error, 7 points decrease in error
      (+.f64 x (*.f64 (/.f64 -1 (Rewrite<= *-commutative_binary64 (*.f64 z 3))) (-.f64 y (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 -1 (*.f64 z 3)) y) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 -1 y) (*.f64 z 3))) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 2 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 y (*.f64 z 3)))) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 8 points increase in error, 2 points decrease in error
      (+.f64 x (-.f64 (*.f64 -1 (/.f64 y (*.f64 z 3))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 t) (*.f64 (*.f64 z 3) y))))): 0 points increase in error, 8 points decrease in error
      (+.f64 x (-.f64 (*.f64 -1 (/.f64 y (*.f64 z 3))) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 t (*.f64 (*.f64 z 3) y)))))): 7 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite=> distribute-lft-out--_binary64 (*.f64 -1 (-.f64 (/.f64 y (*.f64 z 3)) (/.f64 t (*.f64 (*.f64 z 3) y)))))): 0 points increase in error, 7 points decrease in error
      (+.f64 x (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (/.f64 y (*.f64 z 3)) (/.f64 t (*.f64 (*.f64 z 3) y)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 x (-.f64 (/.f64 y (*.f64 z 3)) (/.f64 t (*.f64 (*.f64 z 3) y))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around 0 10.3

      \[\leadsto \color{blue}{x} + \frac{t}{y \cdot \left(z \cdot 3\right)} \]

   4. Taylor expanded in y around 0 10.3

      \[\leadsto x + \frac{t}{\color{blue}{3 \cdot \left(y \cdot z\right)}} \]

   5. Simplified 10.3

      \[\leadsto x + \frac{t}{\color{blue}{\left(y \cdot z\right) \cdot 3}} \]
      Proof
      (+.f64 x (/.f64 t (*.f64 (*.f64 y z) 3))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 t (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 y z))))): 0 points increase in error, 2 points decrease in error

   if 1.34999999999999993e-97 < y

   1. Initial program 0.9

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Simplified 1.0

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
      Proof
      (+.f64 x (*.f64 (/.f64 -1/3 z) (-.f64 y (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 -1 3)) z) (-.f64 y (/.f64 t y)))): 7 points increase in error, 0 points decrease in error
      (+.f64 x (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 3 z))) (-.f64 y (/.f64 t y)))): 0 points increase in error, 7 points decrease in error
      (+.f64 x (*.f64 (/.f64 -1 (Rewrite<= *-commutative_binary64 (*.f64 z 3))) (-.f64 y (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 -1 (*.f64 z 3)) y) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 -1 y) (*.f64 z 3))) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 2 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 y (*.f64 z 3)))) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 8 points increase in error, 2 points decrease in error
      (+.f64 x (-.f64 (*.f64 -1 (/.f64 y (*.f64 z 3))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 t) (*.f64 (*.f64 z 3) y))))): 0 points increase in error, 8 points decrease in error
      (+.f64 x (-.f64 (*.f64 -1 (/.f64 y (*.f64 z 3))) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 t (*.f64 (*.f64 z 3) y)))))): 7 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite=> distribute-lft-out--_binary64 (*.f64 -1 (-.f64 (/.f64 y (*.f64 z 3)) (/.f64 t (*.f64 (*.f64 z 3) y)))))): 0 points increase in error, 7 points decrease in error
      (+.f64 x (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (/.f64 y (*.f64 z 3)) (/.f64 t (*.f64 (*.f64 z 3) y)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 x (-.f64 (/.f64 y (*.f64 z 3)) (/.f64 t (*.f64 (*.f64 z 3) y))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y)))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-270}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-97}:\\ \;\;\;\;x + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	30.1
Cost	1637

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ t_2 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+79}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-254}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.32 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-22} \lor \neg \left(x \leq 4.1 \cdot 10^{+51}\right) \land x \leq 2.2 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	29.9
Cost	1372

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ t_2 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+79}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-256}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+92}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	29.8
Cost	1372

\[\begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+80}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-99}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+50}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+92}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	3.7
Cost	1092

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-272}:\\ \;\;\;\;x + \frac{\frac{y - \frac{t}{y}}{z}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \end{array} \]

Alternative 5
Error	13.1
Cost	1040

\[\begin{array}{l} t_1 := x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+51}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\frac{z}{t}}}{-y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	18.5
Cost	977

\[\begin{array}{l} t_1 := x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-232}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-236} \lor \neg \left(y \leq 2 \cdot 10^{-140}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \end{array} \]

Alternative 7
Error	18.5
Cost	977

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-111}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-232}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-238} \lor \neg \left(y \leq 1.02 \cdot 10^{-139}\right):\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \end{array} \]

Alternative 8
Error	18.5
Cost	977

\[\begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-112}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-232}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{z}{\frac{t}{y}}}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-234} \lor \neg \left(y \leq 6.4 \cdot 10^{-140}\right):\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \end{array} \]

Alternative 9
Error	18.5
Cost	977

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-110}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{\frac{t}{y}}{z}}{3}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-240} \lor \neg \left(y \leq 9.2 \cdot 10^{-140}\right):\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \end{array} \]

Alternative 10
Error	9.1
Cost	972

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+74}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-271}:\\ \;\;\;\;x - \frac{\frac{\frac{t}{y}}{z}}{-3}\\ \mathbf{elif}\;y \leq 0.034:\\ \;\;\;\;x + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 11
Error	3.9
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-275} \lor \neg \left(y \leq 1.95 \cdot 10^{-97}\right):\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 12
Error	9.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+74}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;y \leq 0.04:\\ \;\;\;\;x + t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 13
Error	9.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+74}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;y \leq 0.039:\\ \;\;\;\;x + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 14
Error	9.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+74}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \mathbf{elif}\;y \leq 0.052:\\ \;\;\;\;x + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 15
Error	28.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+79}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-21}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	28.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+79}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	37.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))