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

Average Error: 5.7 → 2.0

Time: 20.1s

Precision: binary64

Cost: 5320

Math

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

↓

\[\begin{array}{l} t_1 := \left(a \cdot 4\right) \cdot t\\ t_2 := \left(x \cdot 4\right) \cdot i\\ t_3 := \left(\left(\left(y \cdot \left(\left(z \cdot \left(x \cdot 18\right)\right) \cdot t\right) - t_1\right) + b \cdot c\right) - t_2\right) - \left(j \cdot 27\right) \cdot k\\ t_4 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - t_1\right) + b \cdot c\right) - t_2\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_4 \leq 5 \cdot 10^{+251}:\\ \;\;\;\;t_4 - \left(k \cdot j\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

FPCore

(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 (* (* a 4.0) t))
        (t_2 (* (* x 4.0) i))
        (t_3
         (-
          (- (+ (- (* y (* (* z (* x 18.0)) t)) t_1) (* b c)) t_2)
          (* (* j 27.0) k)))
        (t_4 (- (+ (- (* (* (* (* x 18.0) y) z) t) t_1) (* b c)) t_2)))
   (if (<= t_4 (- INFINITY))
     t_3
     (if (<= t_4 5e+251) (- t_4 (* (* k j) 27.0)) t_3))))

C

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 = (a * 4.0) * t;
	double t_2 = (x * 4.0) * i;
	double t_3 = ((((y * ((z * (x * 18.0)) * t)) - t_1) + (b * c)) - t_2) - ((j * 27.0) * k);
	double t_4 = ((((((x * 18.0) * y) * z) * t) - t_1) + (b * c)) - t_2;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_4 <= 5e+251) {
		tmp = t_4 - ((k * j) * 27.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

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 = (a * 4.0) * t;
	double t_2 = (x * 4.0) * i;
	double t_3 = ((((y * ((z * (x * 18.0)) * t)) - t_1) + (b * c)) - t_2) - ((j * 27.0) * k);
	double t_4 = ((((((x * 18.0) * y) * z) * t) - t_1) + (b * c)) - t_2;
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else if (t_4 <= 5e+251) {
		tmp = t_4 - ((k * j) * 27.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

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 = (a * 4.0) * t
	t_2 = (x * 4.0) * i
	t_3 = ((((y * ((z * (x * 18.0)) * t)) - t_1) + (b * c)) - t_2) - ((j * 27.0) * k)
	t_4 = ((((((x * 18.0) * y) * z) * t) - t_1) + (b * c)) - t_2
	tmp = 0
	if t_4 <= -math.inf:
		tmp = t_3
	elif t_4 <= 5e+251:
		tmp = t_4 - ((k * j) * 27.0)
	else:
		tmp = t_3
	return tmp

Julia

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(Float64(a * 4.0) * t)
	t_2 = Float64(Float64(x * 4.0) * i)
	t_3 = Float64(Float64(Float64(Float64(Float64(y * Float64(Float64(z * Float64(x * 18.0)) * t)) - t_1) + Float64(b * c)) - t_2) - Float64(Float64(j * 27.0) * k))
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - t_1) + Float64(b * c)) - t_2)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_4 <= 5e+251)
		tmp = Float64(t_4 - Float64(Float64(k * j) * 27.0));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

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 = (a * 4.0) * t;
	t_2 = (x * 4.0) * i;
	t_3 = ((((y * ((z * (x * 18.0)) * t)) - t_1) + (b * c)) - t_2) - ((j * 27.0) * k);
	t_4 = ((((((x * 18.0) * y) * z) * t) - t_1) + (b * c)) - t_2;
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = t_3;
	elseif (t_4 <= 5e+251)
		tmp = t_4 - ((k * j) * 27.0);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

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[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(y * N[(N[(z * N[(x * 18.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - t$95$1), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$3, If[LessEqual[t$95$4, 5e+251], N[(t$95$4 - N[(N[(k * j), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Target

Original	5.7
Target	1.5
Herbie	2.0

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) < -inf.0 or 5.0000000000000005e251 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i))

   1. Initial program 33.1

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

   2. Applied egg-rr 10.8

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

   if -inf.0 < (-.f64 (+.f64 (-.f64 (*.f64 (*.f64 (*.f64 (*.f64 x 18) y) z) t) (*.f64 (*.f64 a 4) t)) (*.f64 b c)) (*.f64 (*.f64 x 4) i)) < 5.0000000000000005e251

   1. Initial program 0.4

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

   2. Taylor expanded in j around 0 0.3

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

   3. Simplified 0.3

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

Alternatives

Alternative 1
Error	2.1
Cost	5320

\[\begin{array}{l} t_1 := \left(a \cdot 4\right) \cdot t\\ t_2 := \left(x \cdot 4\right) \cdot i\\ t_3 := \left(\left(\left(y \cdot \left(\left(z \cdot \left(x \cdot 18\right)\right) \cdot t\right) - t_1\right) + b \cdot c\right) - t_2\right) - \left(j \cdot 27\right) \cdot k\\ t_4 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - t_1\right) + b \cdot c\right) - t_2\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_4 \leq 5 \cdot 10^{+251}:\\ \;\;\;\;t_4 - \left(27 \cdot k\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	2.2
Cost	5320

\[\begin{array}{l} t_1 := \left(a \cdot 4\right) \cdot t\\ t_2 := \left(j \cdot 27\right) \cdot k\\ t_3 := \left(x \cdot 4\right) \cdot i\\ t_4 := \left(\left(\left(y \cdot \left(\left(z \cdot \left(x \cdot 18\right)\right) \cdot t\right) - t_1\right) + b \cdot c\right) - t_3\right) - t_2\\ t_5 := \left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - t_1\right) + b \cdot c\right) - t_3\\ \mathbf{if}\;t_5 \leq -\infty:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_5 \leq 10^{+241}:\\ \;\;\;\;t_5 - t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 3
Error	4.9
Cost	2248

\[\begin{array}{l} t_1 := \left(x \cdot 4\right) \cdot i\\ t_2 := \left(j \cdot 27\right) \cdot k\\ t_3 := \left(a \cdot 4\right) \cdot t\\ t_4 := \left(\left(\left(y \cdot \left(\left(z \cdot \left(x \cdot 18\right)\right) \cdot t\right) - t_3\right) + b \cdot c\right) - t_1\right) - t_2\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+42}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-239}:\\ \;\;\;\;\left(\left(\left(\left(\left(y \cdot x\right) \cdot 18\right) \cdot \left(z \cdot t\right) - t_3\right) + b \cdot c\right) - t_1\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 4
Error	4.9
Cost	1984

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

Reproduce

herbie shell --seed 2023010 
(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)))