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

Average Error: 12.0 → 2.8

Time: 9.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(c, a \cdot j - z \cdot b, i \cdot \left(t \cdot b\right) + \left(y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\right)\\ t_2 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_3 := y \cdot z - t \cdot a\\ t_4 := \mathsf{fma}\left(x, t_3, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), t_2\right)\right)\\ t_5 := \left(x \cdot t_3 + b \cdot \left(t \cdot i - z \cdot c\right)\right) + t_2\\ \mathbf{if}\;t_5 \leq -2 \cdot 10^{+301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_5 \leq -1 \cdot 10^{-129}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_5 \leq 5 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_5 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (fma
          c
          (- (* a j) (* z b))
          (+ (* i (* t b)) (- (* y (- (* x z) (* i j))) (* a (* x t))))))
        (t_2 (* j (- (* a c) (* y i))))
        (t_3 (- (* y z) (* t a)))
        (t_4 (fma x t_3 (fma b (fma z (- c) (* t i)) t_2)))
        (t_5 (+ (+ (* x t_3) (* b (- (* t i) (* z c)))) t_2)))
   (if (<= t_5 -2e+301)
     t_1
     (if (<= t_5 -1e-129)
       t_4
       (if (<= t_5 5e-205) t_1 (if (<= t_5 5e+307) t_4 t_1))))))

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 = fma(c, ((a * j) - (z * b)), ((i * (t * b)) + ((y * ((x * z) - (i * j))) - (a * (x * t)))));
	double t_2 = j * ((a * c) - (y * i));
	double t_3 = (y * z) - (t * a);
	double t_4 = fma(x, t_3, fma(b, fma(z, -c, (t * i)), t_2));
	double t_5 = ((x * t_3) + (b * ((t * i) - (z * c)))) + t_2;
	double tmp;
	if (t_5 <= -2e+301) {
		tmp = t_1;
	} else if (t_5 <= -1e-129) {
		tmp = t_4;
	} else if (t_5 <= 5e-205) {
		tmp = t_1;
	} else if (t_5 <= 5e+307) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	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 = fma(c, Float64(Float64(a * j) - Float64(z * b)), Float64(Float64(i * Float64(t * b)) + Float64(Float64(y * Float64(Float64(x * z) - Float64(i * j))) - Float64(a * Float64(x * t)))))
	t_2 = Float64(j * Float64(Float64(a * c) - Float64(y * i)))
	t_3 = Float64(Float64(y * z) - Float64(t * a))
	t_4 = fma(x, t_3, fma(b, fma(z, Float64(-c), Float64(t * i)), t_2))
	t_5 = Float64(Float64(Float64(x * t_3) + Float64(b * Float64(Float64(t * i) - Float64(z * c)))) + t_2)
	tmp = 0.0
	if (t_5 <= -2e+301)
		tmp = t_1;
	elseif (t_5 <= -1e-129)
		tmp = t_4;
	elseif (t_5 <= 5e-205)
		tmp = t_1;
	elseif (t_5 <= 5e+307)
		tmp = t_4;
	else
		tmp = t_1;
	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[(c * N[(N[(a * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision] + N[(N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision] + N[(N[(y * N[(N[(x * z), $MachinePrecision] - N[(i * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(x * t$95$3 + N[(b * N[(z * (-c) + N[(t * i), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(x * t$95$3), $MachinePrecision] + N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+301], t$95$1, If[LessEqual[t$95$5, -1e-129], t$95$4, If[LessEqual[t$95$5, 5e-205], t$95$1, If[LessEqual[t$95$5, 5e+307], t$95$4, t$95$1]]]]]]]]]

Target

Original	12.0
Target	19.7
Herbie	2.8

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

Derivation

1. Split input into 2 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)))) < -2.00000000000000011e301 or -9.9999999999999993e-130 < (+.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)))) < 5.00000000000000001e-205 or 5e307 < (+.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 54.3

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

   2. Simplified 54.3

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

   3. Taylor expanded in x around 0 54.3

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

   4. Simplified 23.7

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

   5. Taylor expanded in y around inf 11.5

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

   if -2.00000000000000011e301 < (+.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)))) < -9.9999999999999993e-130 or 5.00000000000000001e-205 < (+.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)))) < 5e307

   1. Initial program 0.4

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

   2. Simplified 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), j \cdot \left(a \cdot c - y \cdot i\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(c, a \cdot j - z \cdot b, i \cdot \left(t \cdot b\right) + \left(y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq -1 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), j \cdot \left(a \cdot c - y \cdot i\right)\right)\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq 5 \cdot 10^{-205}:\\ \;\;\;\;\mathsf{fma}\left(c, a \cdot j - z \cdot b, i \cdot \left(t \cdot b\right) + \left(y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), j \cdot \left(a \cdot c - y \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, a \cdot j - z \cdot b, i \cdot \left(t \cdot b\right) + \left(y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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)))))