Data.Colour.Matrix:determinant from colour-2.3.3, A

Average Error: 12.7 → 3.5

Time: 20.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot z - i \cdot j\\ t_2 := y \cdot z - t \cdot a\\ t_3 := \left(x \cdot t_2 + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(y, t_1, \mathsf{fma}\left(t, \mathsf{fma}\left(-a, x, b \cdot i\right), c \cdot \left(a \cdot j - z \cdot b\right)\right)\right)\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(x, t_2, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), j \cdot \mathsf{fma}\left(1, a \cdot c, i \cdot \left(-y\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, t_1, i \cdot \left(t \cdot b\right) + \left(a \cdot \left(c \cdot j - x \cdot t\right) - c \cdot \left(z \cdot b\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (+
  (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i))))
  (* j (- (* c a) (* y i)))))

↓

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* x z) (* i j)))
        (t_2 (- (* y z) (* t a)))
        (t_3
         (+
          (+ (* x t_2) (* b (- (* t i) (* z c))))
          (* j (- (* a c) (* y i))))))
   (if (<= t_3 -2e+294)
     (fma y t_1 (fma t (fma (- a) x (* b i)) (* c (- (* a j) (* z b)))))
     (if (<= t_3 5e+302)
       (fma
        x
        t_2
        (fma b (fma z (- c) (* t i)) (* j (fma 1.0 (* a c) (* i (- y))))))
       (fma
        y
        t_1
        (+ (* i (* t b)) (- (* a (- (* c j) (* x t))) (* c (* z b)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (x * z) - (i * j);
	double t_2 = (y * z) - (t * a);
	double t_3 = ((x * t_2) + (b * ((t * i) - (z * c)))) + (j * ((a * c) - (y * i)));
	double tmp;
	if (t_3 <= -2e+294) {
		tmp = fma(y, t_1, fma(t, fma(-a, x, (b * i)), (c * ((a * j) - (z * b)))));
	} else if (t_3 <= 5e+302) {
		tmp = fma(x, t_2, fma(b, fma(z, -c, (t * i)), (j * fma(1.0, (a * c), (i * -y)))));
	} else {
		tmp = fma(y, t_1, ((i * (t * b)) + ((a * ((c * j) - (x * t))) - (c * (z * b)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
end

↓

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(x * z) - Float64(i * j))
	t_2 = Float64(Float64(y * z) - Float64(t * a))
	t_3 = Float64(Float64(Float64(x * t_2) + Float64(b * Float64(Float64(t * i) - Float64(z * c)))) + Float64(j * Float64(Float64(a * c) - Float64(y * i))))
	tmp = 0.0
	if (t_3 <= -2e+294)
		tmp = fma(y, t_1, fma(t, fma(Float64(-a), x, Float64(b * i)), Float64(c * Float64(Float64(a * j) - Float64(z * b)))));
	elseif (t_3 <= 5e+302)
		tmp = fma(x, t_2, fma(b, fma(z, Float64(-c), Float64(t * i)), Float64(j * fma(1.0, Float64(a * c), Float64(i * Float64(-y))))));
	else
		tmp = fma(y, t_1, Float64(Float64(i * Float64(t * b)) + Float64(Float64(a * Float64(Float64(c * j) - Float64(x * t))) - Float64(c * Float64(z * b)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(x * z), $MachinePrecision] - N[(i * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * t$95$2), $MachinePrecision] + N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+294], N[(y * t$95$1 + N[(t * N[((-a) * x + N[(b * i), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(a * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+302], N[(x * t$95$2 + N[(b * N[(z * (-c) + N[(t * i), $MachinePrecision]), $MachinePrecision] + N[(j * N[(1.0 * N[(a * c), $MachinePrecision] + N[(i * (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * t$95$1 + N[(N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(N[(c * j), $MachinePrecision] - N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c * N[(z * b), $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

Target

Original	12.7
Target	20.8
Herbie	3.5

\[\begin{array}{l} \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < -2.00000000000000013e294

   1. Initial program 52.9

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

   2. Simplified 52.9

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

   3. Taylor expanded in x around 0 52.9

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right) + \left(\left(y \cdot z - a \cdot t\right) \cdot x + j \cdot \left(c \cdot a - y \cdot i\right)\right)} \]

   4. Simplified 22.7

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

   5. Taylor expanded in b around 0 22.7

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right) + \left(y \cdot \left(z \cdot x - i \cdot j\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)\right)} \]

   6. Simplified 15.6

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

   7. Applied egg-rr 15.9

      \[\leadsto \mathsf{fma}\left(y, z \cdot x - i \cdot j, \mathsf{fma}\left(t, \mathsf{fma}\left(-a, x, i \cdot b\right), \color{blue}{{\left(\sqrt[3]{c \cdot \left(a \cdot j - b \cdot z\right)}\right)}^{3}}\right)\right) \]

   8. Applied egg-rr 15.6

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(y, z \cdot x - j \cdot i, \mathsf{fma}\left(t, \mathsf{fma}\left(-a, x, b \cdot i\right), c \cdot \left(a \cdot j - b \cdot z\right)\right)\right)\right)}^{1}} \]

   if -2.00000000000000013e294 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < 5e302

   1. Initial program 0.8

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

   2. Simplified 0.8

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

   3. Applied egg-rr 0.8

      \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), j \cdot \color{blue}{\mathsf{fma}\left(1, a \cdot c, \left(-y\right) \cdot i\right)}\right)\right) \]

   if 5e302 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i))))

   1. Initial program 58.9

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

   2. Simplified 58.9

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

   3. Taylor expanded in x around 0 58.9

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right) + \left(\left(y \cdot z - a \cdot t\right) \cdot x + j \cdot \left(c \cdot a - y \cdot i\right)\right)} \]

   4. Simplified 25.3

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

   5. Taylor expanded in b around 0 25.3

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right) + \left(y \cdot \left(z \cdot x - i \cdot j\right) + a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)\right)} \]

   6. Simplified 12.7

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

   7. Taylor expanded in a around 0 11.2

      \[\leadsto \mathsf{fma}\left(y, z \cdot x - i \cdot j, \color{blue}{i \cdot \left(t \cdot b\right) + \left(a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right) + -1 \cdot \left(c \cdot \left(z \cdot b\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq -2 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot z - i \cdot j, \mathsf{fma}\left(t, \mathsf{fma}\left(-a, x, b \cdot i\right), c \cdot \left(a \cdot j - z \cdot b\right)\right)\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), j \cdot \mathsf{fma}\left(1, a \cdot c, i \cdot \left(-y\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot z - i \cdot j, i \cdot \left(t \cdot b\right) + \left(a \cdot \left(c \cdot j - x \cdot t\right) - c \cdot \left(z \cdot b\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 3.2113527362226803e-147) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))