Average Error: 5.7 → 1.7
Time: 9.6s
Precision: binary64
\[ \begin{array}{c}[j, k] = \mathsf{sort}([j, k])\\ \end{array} \]
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
↓
\[\begin{array}{l} t_1 := i \cdot \left(x \cdot -4\right)\\ t_2 := t \cdot \left(a \cdot -4\right)\\ t_3 := \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) + t_2\right) + b \cdot c\right) + t_1\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;t \leq -8.151055569265776 \cdot 10^{-26}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 10^{-20}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(t \cdot z\right)\right) + t_2\right)\right) + t_1\right) + \left(j \cdot k\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
(FPCore (x y z t a b c i j k)
 :precision binary64
 (-
  (-
   (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
   (* (* x 4.0) i))
  (* (* j 27.0) k)))
↓
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1 (* i (* x -4.0)))
        (t_2 (* t (* a -4.0)))
        (t_3
         (+
          (+ (+ (+ (* t (* (* (* x 18.0) y) z)) t_2) (* b c)) t_1)
          (* k (* j -27.0)))))
   (if (<= t -8.151055569265776e-26)
     t_3
     (if (<= t 1e-20)
       (+
        (+ (+ (* b c) (+ (* (* x 18.0) (* y (* t z))) t_2)) t_1)
        (* (* j k) -27.0))
       t_3))))
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}
↓
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = i * (x * -4.0);
	double t_2 = t * (a * -4.0);
	double t_3 = ((((t * (((x * 18.0) * y) * z)) + t_2) + (b * c)) + t_1) + (k * (j * -27.0));
	double tmp;
	if (t <= -8.151055569265776e-26) {
		tmp = t_3;
	} else if (t <= 1e-20) {
		tmp = (((b * c) + (((x * 18.0) * (y * (t * z))) + t_2)) + t_1) + ((j * k) * -27.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

real(8) function code(x, y, z, t, a, b, c, i, j, k)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    code = (((((((x * 18.0d0) * y) * z) * t) - ((a * 4.0d0) * t)) + (b * c)) - ((x * 4.0d0) * i)) - ((j * 27.0d0) * k)
end function
↓
real(8) function code(x, y, z, t, a, b, c, i, j, k)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = i * (x * (-4.0d0))
    t_2 = t * (a * (-4.0d0))
    t_3 = ((((t * (((x * 18.0d0) * y) * z)) + t_2) + (b * c)) + t_1) + (k * (j * (-27.0d0)))
    if (t <= (-8.151055569265776d-26)) then
        tmp = t_3
    else if (t <= 1d-20) then
        tmp = (((b * c) + (((x * 18.0d0) * (y * (t * z))) + t_2)) + t_1) + ((j * k) * (-27.0d0))
    else
        tmp = t_3
    end if
    code = tmp
end function

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}
↓
public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = i * (x * -4.0);
	double t_2 = t * (a * -4.0);
	double t_3 = ((((t * (((x * 18.0) * y) * z)) + t_2) + (b * c)) + t_1) + (k * (j * -27.0));
	double tmp;
	if (t <= -8.151055569265776e-26) {
		tmp = t_3;
	} else if (t <= 1e-20) {
		tmp = (((b * c) + (((x * 18.0) * (y * (t * z))) + t_2)) + t_1) + ((j * k) * -27.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

def code(x, y, z, t, a, b, c, i, j, k):
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k)
↓
def code(x, y, z, t, a, b, c, i, j, k):
	t_1 = i * (x * -4.0)
	t_2 = t * (a * -4.0)
	t_3 = ((((t * (((x * 18.0) * y) * z)) + t_2) + (b * c)) + t_1) + (k * (j * -27.0))
	tmp = 0
	if t <= -8.151055569265776e-26:
		tmp = t_3
	elif t <= 1e-20:
		tmp = (((b * c) + (((x * 18.0) * (y * (t * z))) + t_2)) + t_1) + ((j * k) * -27.0)
	else:
		tmp = t_3
	return tmp

function code(x, y, z, t, a, b, c, i, j, k)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
end
↓
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(i * Float64(x * -4.0))
	t_2 = Float64(t * Float64(a * -4.0))
	t_3 = Float64(Float64(Float64(Float64(Float64(t * Float64(Float64(Float64(x * 18.0) * y) * z)) + t_2) + Float64(b * c)) + t_1) + Float64(k * Float64(j * -27.0)))
	tmp = 0.0
	if (t <= -8.151055569265776e-26)
		tmp = t_3;
	elseif (t <= 1e-20)
		tmp = Float64(Float64(Float64(Float64(b * c) + Float64(Float64(Float64(x * 18.0) * Float64(y * Float64(t * z))) + t_2)) + t_1) + Float64(Float64(j * k) * -27.0));
	else
		tmp = t_3;
	end
	return tmp
end

function tmp = code(x, y, z, t, a, b, c, i, j, k)
	tmp = (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
end
↓
function tmp_2 = code(x, y, z, t, a, b, c, i, j, k)
	t_1 = i * (x * -4.0);
	t_2 = t * (a * -4.0);
	t_3 = ((((t * (((x * 18.0) * y) * z)) + t_2) + (b * c)) + t_1) + (k * (j * -27.0));
	tmp = 0.0;
	if (t <= -8.151055569265776e-26)
		tmp = t_3;
	elseif (t <= 1e-20)
		tmp = (((b * c) + (((x * 18.0) * (y * (t * z))) + t_2)) + t_1) + ((j * k) * -27.0);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(i * N[(x * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(t * N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.151055569265776e-26], t$95$3, If[LessEqual[t, 1e-20], N[(N[(N[(N[(b * c), $MachinePrecision] + N[(N[(N[(x * 18.0), $MachinePrecision] * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(N[(j * k), $MachinePrecision] * -27.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k
↓
\begin{array}{l}
t_1 := i \cdot \left(x \cdot -4\right)\\
t_2 := t \cdot \left(a \cdot -4\right)\\
t_3 := \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) + t_2\right) + b \cdot c\right) + t_1\right) + k \cdot \left(j \cdot -27\right)\\
\mathbf{if}\;t \leq -8.151055569265776 \cdot 10^{-26}:\\
\;\;\;\;t_3\\

\mathbf{elif}\;t \leq 10^{-20}:\\
\;\;\;\;\left(\left(b \cdot c + \left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(t \cdot z\right)\right) + t_2\right)\right) + t_1\right) + \left(j \cdot k\right) \cdot -27\\

\mathbf{else}:\\
\;\;\;\;t_3\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.7
Target1.7
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;t < -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t < 165.68027943805222:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if t < -8.15105556926577638e-26 or 9.99999999999999945e-21 < t

    1. Initial program 2.0

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

    if -8.15105556926577638e-26 < t < 9.99999999999999945e-21

    1. Initial program 8.1

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

    2. Taylor expanded in j around 0 8.1

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{27 \cdot \left(k \cdot j\right)} \]

    3. Applied egg-rr1.6

      \[\leadsto \left(\left(\left(\color{blue}{{\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right)\right)}^{1}} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.151055569265776 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{elif}\;t \leq 10^{-20}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(t \cdot z\right)\right) + t \cdot \left(a \cdot -4\right)\right)\right) + i \cdot \left(x \cdot -4\right)\right) + \left(j \cdot k\right) \cdot -27\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))