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

Average Error: 3.9 → 1.7

Time: 16.3s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -1.75 \cdot 10^{-96}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{\frac{z}{t_1}}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-137}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;x + t_1 \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

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 -1.75e-96)
     (+ x (/ -0.3333333333333333 (/ z t_1)))
     (if (<= y 3e-137)
       (+ x (/ (/ t z) (* y 3.0)))
       (+ 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 <= -1.75e-96) {
		tmp = x + (-0.3333333333333333 / (z / t_1));
	} else if (y <= 3e-137) {
		tmp = x + ((t / z) / (y * 3.0));
	} 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 <= (-1.75d-96)) then
        tmp = x + ((-0.3333333333333333d0) / (z / t_1))
    else if (y <= 3d-137) then
        tmp = x + ((t / z) / (y * 3.0d0))
    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 <= -1.75e-96) {
		tmp = x + (-0.3333333333333333 / (z / t_1));
	} else if (y <= 3e-137) {
		tmp = x + ((t / z) / (y * 3.0));
	} 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 <= -1.75e-96:
		tmp = x + (-0.3333333333333333 / (z / t_1))
	elif y <= 3e-137:
		tmp = x + ((t / z) / (y * 3.0))
	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 <= -1.75e-96)
		tmp = Float64(x + Float64(-0.3333333333333333 / Float64(z / t_1)));
	elseif (y <= 3e-137)
		tmp = Float64(x + Float64(Float64(t / z) / Float64(y * 3.0)));
	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 <= -1.75e-96)
		tmp = x + (-0.3333333333333333 / (z / t_1));
	elseif (y <= 3e-137)
		tmp = x + ((t / z) / (y * 3.0));
	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, -1.75e-96], N[(x + N[(-0.3333333333333333 / N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e-137], N[(x + N[(N[(t / z), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$1 * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.9
Target	1.7
Herbie	1.7

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

Derivation

1. Split input into 3 regimes

2. if y < -1.7499999999999999e-96

   1. Initial program 1.0

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

   2. Simplified 0.8

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
      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)))): 0 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)))): 23 points increase in error, 17 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, 3 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, 19 points decrease in error
      (+.f64 x (-.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 y)) (*.f64 z 3)) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 y (*.f64 z 3)))) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 t) (*.f64 (*.f64 z 3) y))))): 14 points increase in error, 31 points decrease in error
      (+.f64 x (-.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 t)) (*.f64 (*.f64 z 3) y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 t (*.f64 (*.f64 z 3) y)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (neg.f64 (neg.f64 (/.f64 t (*.f64 (*.f64 z 3) y))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (/.f64 y (*.f64 z 3)) (neg.f64 (/.f64 t (*.f64 (*.f64 z 3) y))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (neg.f64 (Rewrite<= sub-neg_binary64 (-.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 0.8

      \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{\frac{z}{y - \frac{t}{y}}}} \]

   if -1.7499999999999999e-96 < y < 2.9999999999999998e-137

   1. Initial program 11.4

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

   2. Simplified 2.5

      \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z}}{y \cdot 3}} \]
      Proof
      (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 (/.f64 t z) (*.f64 y 3))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 (/.f64 t z) (Rewrite<= *-commutative_binary64 (*.f64 3 y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (Rewrite<= associate-/r*_binary64 (/.f64 t (*.f64 z (*.f64 3 y))))): 12 points increase in error, 11 points decrease in error
      (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z 3) y)))): 9 points increase in error, 7 points decrease in error

   3. Taylor expanded in x around inf 3.2

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

   if 2.9999999999999998e-137 < y

   1. Initial program 1.1

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

   2. Simplified 1.4

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
      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)))): 0 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)))): 23 points increase in error, 17 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, 3 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, 19 points decrease in error
      (+.f64 x (-.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 y)) (*.f64 z 3)) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 y (*.f64 z 3)))) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 t) (*.f64 (*.f64 z 3) y))))): 14 points increase in error, 31 points decrease in error
      (+.f64 x (-.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 t)) (*.f64 (*.f64 z 3) y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 t (*.f64 (*.f64 z 3) y)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (neg.f64 (neg.f64 (/.f64 t (*.f64 (*.f64 z 3) y))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (/.f64 y (*.f64 z 3)) (neg.f64 (/.f64 t (*.f64 (*.f64 z 3) y))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (neg.f64 (Rewrite<= sub-neg_binary64 (-.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 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-96}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{\frac{z}{y - \frac{t}{y}}}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-137}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	35.7
Cost	2036

\[\begin{array}{l} t_1 := \frac{y}{z} \cdot -0.3333333333333333\\ t_2 := \frac{y}{z \cdot -3}\\ t_3 := 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-114}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{-211}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-256}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-245}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+29}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+186}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+215}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	34.7
Cost	2036

\[\begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ t_2 := 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ t_3 := \frac{y}{z} \cdot -0.3333333333333333\\ t_4 := \frac{\frac{t}{z}}{y} \cdot 0.3333333333333333\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{+54}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-21}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-211}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-261}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-242}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+188}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+213}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	34.6
Cost	2036

\[\begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ t_2 := \frac{y}{z} \cdot -0.3333333333333333\\ t_3 := \frac{\frac{t}{z}}{y} \cdot 0.3333333333333333\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-212}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-257}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-242}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+184}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+213}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	18.5
Cost	1632

\[\begin{array}{l} t_1 := \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ t_2 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\frac{\frac{t}{z} \cdot 0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-258}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-243}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{y}{\frac{t}{z}}}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	17.3
Cost	1240

\[\begin{array}{l} t_1 := x + y \cdot \frac{-0.3333333333333333}{z}\\ t_2 := \frac{\frac{t}{z}}{y} \cdot 0.3333333333333333\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-211}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-259}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-242}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	17.4
Cost	1240

\[\begin{array}{l} t_1 := x + \frac{-0.3333333333333333}{\frac{z}{y}}\\ t_2 := \frac{\frac{t}{z}}{y} \cdot 0.3333333333333333\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-212}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-261}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-244}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 7
Error	17.3
Cost	1240

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ t_2 := \frac{\frac{t}{z}}{y} \cdot 0.3333333333333333\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{-212}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-256}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-245}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	17.3
Cost	1240

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-212}:\\ \;\;\;\;\frac{\frac{t}{z}}{y} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-260}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-246}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{y}{\frac{t}{z}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	17.3
Cost	1240

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{z}{\frac{t}{y}}}\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-212}:\\ \;\;\;\;\frac{\frac{t}{z}}{y} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq -2.65 \cdot 10^{-259}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.04 \cdot 10^{-244}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{y}{\frac{t}{z}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	17.8
Cost	1240

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{z}{\frac{t}{y}}}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-211}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{y \cdot z}{t}}\\ \mathbf{elif}\;y \leq -7.4 \cdot 10^{-258}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-243}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{y}{\frac{t}{z}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	17.3
Cost	1240

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{z}{\frac{t}{y}}}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-211}:\\ \;\;\;\;\frac{\frac{t}{z} \cdot 0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-259}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-246}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{y}{\frac{t}{z}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	10.7
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;\left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+25}:\\ \;\;\;\;x + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	7.5
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;\left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq -1.06 \cdot 10^{-29}:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+27}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	7.5
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{z}{\frac{t}{y} - y}}\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{-29}:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+25}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	1.7
Cost	968

\[\begin{array}{l} t_1 := x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-137}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	1.7
Cost	960

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

Alternative 17
Error	1.8
Cost	832

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

Alternative 18
Error	27.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-34}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3800:\\ \;\;\;\;\frac{y}{z} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 19
Error	27.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 32:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 20
Error	37.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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))))