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

Average Error: 12.5 → 9.2

Time: 18.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := a \cdot \left(t \cdot x\right)\\ t_2 := t \cdot \left(i \cdot b - a \cdot x\right)\\ t_3 := c \cdot \left(j \cdot a\right) + y \cdot \left(x \cdot z\right)\\ t_4 := y \cdot x - c \cdot b\\ t_5 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_6 := c \cdot \left(b \cdot z\right)\\ \mathbf{if}\;t \leq -1.0553404975127396 \cdot 10^{+57}:\\ \;\;\;\;t_5 + \left(t_2 - t_6\right)\\ \mathbf{elif}\;t \leq -1.2414725876838894 \cdot 10^{-249}:\\ \;\;\;\;\mathsf{fma}\left(b, t \cdot i - c \cdot z, t_3 - \left(i \cdot \left(j \cdot y\right) + t_1\right)\right)\\ \mathbf{elif}\;t \leq -6.438775900591724 \cdot 10^{-308}:\\ \;\;\;\;t_5 + \mathsf{fma}\left(i, t \cdot b, z \cdot t_4\right)\\ \mathbf{elif}\;t \leq 31.517030764292915:\\ \;\;\;\;\left(t_3 + i \cdot \left(t \cdot b\right)\right) - \left(y \cdot \left(j \cdot i\right) + \left(t_6 + t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5 + \mathsf{fma}\left(z, t_4, t_2\right)\\ \end{array} \]

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 (* a (* t x)))
        (t_2 (* t (- (* i b) (* a x))))
        (t_3 (+ (* c (* j a)) (* y (* x z))))
        (t_4 (- (* y x) (* c b)))
        (t_5 (* j (- (* c a) (* y i))))
        (t_6 (* c (* b z))))
   (if (<= t -1.0553404975127396e+57)
     (+ t_5 (- t_2 t_6))
     (if (<= t -1.2414725876838894e-249)
       (fma b (- (* t i) (* c z)) (- t_3 (+ (* i (* j y)) t_1)))
       (if (<= t -6.438775900591724e-308)
         (+ t_5 (fma i (* t b) (* z t_4)))
         (if (<= t 31.517030764292915)
           (- (+ t_3 (* i (* t b))) (+ (* y (* j i)) (+ t_6 t_1)))
           (+ t_5 (fma z t_4 t_2))))))))

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 = a * (t * x);
	double t_2 = t * ((i * b) - (a * x));
	double t_3 = (c * (j * a)) + (y * (x * z));
	double t_4 = (y * x) - (c * b);
	double t_5 = j * ((c * a) - (y * i));
	double t_6 = c * (b * z);
	double tmp;
	if (t <= -1.0553404975127396e+57) {
		tmp = t_5 + (t_2 - t_6);
	} else if (t <= -1.2414725876838894e-249) {
		tmp = fma(b, ((t * i) - (c * z)), (t_3 - ((i * (j * y)) + t_1)));
	} else if (t <= -6.438775900591724e-308) {
		tmp = t_5 + fma(i, (t * b), (z * t_4));
	} else if (t <= 31.517030764292915) {
		tmp = (t_3 + (i * (t * b))) - ((y * (j * i)) + (t_6 + t_1));
	} else {
		tmp = t_5 + fma(z, t_4, t_2);
	}
	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(a * Float64(t * x))
	t_2 = Float64(t * Float64(Float64(i * b) - Float64(a * x)))
	t_3 = Float64(Float64(c * Float64(j * a)) + Float64(y * Float64(x * z)))
	t_4 = Float64(Float64(y * x) - Float64(c * b))
	t_5 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_6 = Float64(c * Float64(b * z))
	tmp = 0.0
	if (t <= -1.0553404975127396e+57)
		tmp = Float64(t_5 + Float64(t_2 - t_6));
	elseif (t <= -1.2414725876838894e-249)
		tmp = fma(b, Float64(Float64(t * i) - Float64(c * z)), Float64(t_3 - Float64(Float64(i * Float64(j * y)) + t_1)));
	elseif (t <= -6.438775900591724e-308)
		tmp = Float64(t_5 + fma(i, Float64(t * b), Float64(z * t_4)));
	elseif (t <= 31.517030764292915)
		tmp = Float64(Float64(t_3 + Float64(i * Float64(t * b))) - Float64(Float64(y * Float64(j * i)) + Float64(t_6 + t_1)));
	else
		tmp = Float64(t_5 + fma(z, t_4, t_2));
	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[(a * N[(t * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(N[(i * b), $MachinePrecision] - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * N[(j * a), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y * x), $MachinePrecision] - N[(c * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(c * N[(b * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.0553404975127396e+57], N[(t$95$5 + N[(t$95$2 - t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.2414725876838894e-249], N[(b * N[(N[(t * i), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[(N[(i * N[(j * y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -6.438775900591724e-308], N[(t$95$5 + N[(i * N[(t * b), $MachinePrecision] + N[(z * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 31.517030764292915], N[(N[(t$95$3 + N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y * N[(j * i), $MachinePrecision]), $MachinePrecision] + N[(t$95$6 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$5 + N[(z * t$95$4 + t$95$2), $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	12.5
Target	20.1
Herbie	9.2

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

Derivation

1. Split input into 5 regimes

2. if t < -1.05534049751273956e57

   1. Initial program 17.8

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

   2. Simplified 17.8

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

   3. Taylor expanded in b around 0 17.8

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

   4. Simplified 8.0

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

   5. Taylor expanded in y around 0 22.0

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

   6. Simplified 11.3

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

   if -1.05534049751273956e57 < t < -1.24147258768388942e-249

   1. Initial program 10.0

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

   2. Simplified 10.0

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

   3. Taylor expanded in x around 0 9.1

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

   if -1.24147258768388942e-249 < t < -6.4387759005917241e-308

   1. Initial program 8.4

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

   2. Simplified 8.4

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

   3. Taylor expanded in b around 0 11.0

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

   4. Simplified 16.7

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

   5. Taylor expanded in a around 0 13.3

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

   6. Simplified 13.1

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

   if -6.4387759005917241e-308 < t < 31.5170307642929153

   1. Initial program 9.8

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

   2. Simplified 9.8

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

   3. Taylor expanded in b around 0 9.4

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

   if 31.5170307642929153 < t

   1. Initial program 17.9

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

   2. Simplified 17.9

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

   3. Taylor expanded in b around 0 16.5

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

   4. Simplified 6.4

      \[\leadsto \color{blue}{j \cdot \left(c \cdot a - y \cdot i\right) + \mathsf{fma}\left(z, y \cdot x - c \cdot b, t \cdot \left(i \cdot b - a \cdot x\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.0553404975127396 \cdot 10^{+57}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(t \cdot \left(i \cdot b - a \cdot x\right) - c \cdot \left(b \cdot z\right)\right)\\ \mathbf{elif}\;t \leq -1.2414725876838894 \cdot 10^{-249}:\\ \;\;\;\;\mathsf{fma}\left(b, t \cdot i - c \cdot z, \left(c \cdot \left(j \cdot a\right) + y \cdot \left(x \cdot z\right)\right) - \left(i \cdot \left(j \cdot y\right) + a \cdot \left(t \cdot x\right)\right)\right)\\ \mathbf{elif}\;t \leq -6.438775900591724 \cdot 10^{-308}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \mathsf{fma}\left(i, t \cdot b, z \cdot \left(y \cdot x - c \cdot b\right)\right)\\ \mathbf{elif}\;t \leq 31.517030764292915:\\ \;\;\;\;\left(\left(c \cdot \left(j \cdot a\right) + y \cdot \left(x \cdot z\right)\right) + i \cdot \left(t \cdot b\right)\right) - \left(y \cdot \left(j \cdot i\right) + \left(c \cdot \left(b \cdot z\right) + a \cdot \left(t \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \mathsf{fma}\left(z, y \cdot x - c \cdot b, t \cdot \left(i \cdot b - a \cdot x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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)))))