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

Average Error: 5.5 → 1.2

Time: 9.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(x, y \cdot \left(t \cdot \left(18 \cdot z\right)\right), \mathsf{fma}\left(k, j \cdot -27, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;t_1 \leq 1.0105893170236064 \cdot 10^{+298}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right) + \left(-4 \cdot \left(x \cdot i\right) + \left(-4 \cdot \left(t \cdot a\right) + -27 \cdot \left(j \cdot k\right)\right)\right)\\ \end{array} \]

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
         (+
          (+
           (+ (+ (* (* (* (* x 18.0) y) z) t) (* t (* a -4.0))) (* b c))
           (* i (* x -4.0)))
          (* k (* j -27.0)))))
   (if (<= t_1 (- INFINITY))
     (fma
      c
      b
      (fma
       x
       (* y (* t (* 18.0 z)))
       (fma k (* j -27.0) (* -4.0 (fma i x (* t a))))))
     (if (<= t_1 1.0105893170236064e+298)
       t_1
       (+
        (+ (* b c) (* 18.0 (* y (* t (* x z)))))
        (+ (* -4.0 (* x i)) (+ (* -4.0 (* t a)) (* -27.0 (* j k)))))))))

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 = (((((((x * 18.0) * y) * z) * t) + (t * (a * -4.0))) + (b * c)) + (i * (x * -4.0))) + (k * (j * -27.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(c, b, fma(x, (y * (t * (18.0 * z))), fma(k, (j * -27.0), (-4.0 * fma(i, x, (t * a))))));
	} else if (t_1 <= 1.0105893170236064e+298) {
		tmp = t_1;
	} else {
		tmp = ((b * c) + (18.0 * (y * (t * (x * z))))) + ((-4.0 * (x * i)) + ((-4.0 * (t * a)) + (-27.0 * (j * k))));
	}
	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(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) + Float64(t * Float64(a * -4.0))) + Float64(b * c)) + Float64(i * Float64(x * -4.0))) + Float64(k * Float64(j * -27.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(c, b, fma(x, Float64(y * Float64(t * Float64(18.0 * z))), fma(k, Float64(j * -27.0), Float64(-4.0 * fma(i, x, Float64(t * a))))));
	elseif (t_1 <= 1.0105893170236064e+298)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(b * c) + Float64(18.0 * Float64(y * Float64(t * Float64(x * z))))) + Float64(Float64(-4.0 * Float64(x * i)) + Float64(Float64(-4.0 * Float64(t * a)) + Float64(-27.0 * Float64(j * k)))));
	end
	return 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[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] + N[(t * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] + N[(i * N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(k * N[(j * -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(c * b + N[(x * N[(y * N[(t * N[(18.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(k * N[(j * -27.0), $MachinePrecision] + N[(-4.0 * N[(i * x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0105893170236064e+298], t$95$1, N[(N[(N[(b * c), $MachinePrecision] + N[(18.0 * N[(y * N[(t * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(-27.0 * N[(j * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Bits error versus j

Bits error versus k

Target

Original	5.5
Target	1.6
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.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)) (*.f64 (*.f64 j 27) k)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 41.5

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

   3. Taylor expanded in t around 0 11.4

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

   4. Simplified 6.1

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

   if -inf.0 < (-.f64 (-.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)) (*.f64 (*.f64 j 27) k)) < 1.01058931702360637e298

   1. Initial program 0.3

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

   if 1.01058931702360637e298 < (-.f64 (-.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)) (*.f64 (*.f64 j 27) k))

   1. Initial program 44.3

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

   2. Simplified 29.9

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

   3. Taylor expanded in t around 0 11.5

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(x, y \cdot \left(t \cdot \left(18 \cdot z\right)\right), \mathsf{fma}\left(k, j \cdot -27, -4 \cdot \mathsf{fma}\left(i, x, t \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right) \leq 1.0105893170236064 \cdot 10^{+298}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t + t \cdot \left(a \cdot -4\right)\right) + b \cdot c\right) + i \cdot \left(x \cdot -4\right)\right) + k \cdot \left(j \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c + 18 \cdot \left(y \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)\right) + \left(-4 \cdot \left(x \cdot i\right) + \left(-4 \cdot \left(t \cdot a\right) + -27 \cdot \left(j \cdot k\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))