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

Average Error: 12.6 → 4.2

Time: 15.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := c \cdot \left(z \cdot b\right)\\ t_2 := i \cdot \left(t \cdot b\right)\\ t_3 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_4 := y \cdot z - t \cdot a\\ t_5 := \left(x \cdot t_4 + b \cdot \left(t \cdot i - z \cdot c\right)\right) + t_3\\ t_6 := c \cdot \left(a \cdot j\right)\\ \mathbf{if}\;t_5 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, t \cdot \left(-a\right), \left(t_2 + t_6\right) - \left(i \cdot \left(y \cdot j\right) + t_1\right)\right)\\ \mathbf{elif}\;t_5 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(x, t_4, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), t_3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 + \left(t_6 + y \cdot \left(x \cdot z\right)\right)\right) - \left(y \cdot \left(i \cdot j\right) + \left(t_1 + a \cdot \left(x \cdot t\right)\right)\right)\\ \end{array} \]

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 (* c (* z b)))
        (t_2 (* i (* t b)))
        (t_3 (* j (- (* a c) (* y i))))
        (t_4 (- (* y z) (* t a)))
        (t_5 (+ (+ (* x t_4) (* b (- (* t i) (* z c)))) t_3))
        (t_6 (* c (* a j))))
   (if (<= t_5 (- INFINITY))
     (fma x (* t (- a)) (- (+ t_2 t_6) (+ (* i (* y j)) t_1)))
     (if (<= t_5 5e+295)
       (fma x t_4 (fma b (fma z (- c) (* t i)) t_3))
       (-
        (+ t_2 (+ t_6 (* y (* x z))))
        (+ (* y (* i j)) (+ t_1 (* a (* x t)))))))))

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 = c * (z * b);
	double t_2 = i * (t * b);
	double t_3 = j * ((a * c) - (y * i));
	double t_4 = (y * z) - (t * a);
	double t_5 = ((x * t_4) + (b * ((t * i) - (z * c)))) + t_3;
	double t_6 = c * (a * j);
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = fma(x, (t * -a), ((t_2 + t_6) - ((i * (y * j)) + t_1)));
	} else if (t_5 <= 5e+295) {
		tmp = fma(x, t_4, fma(b, fma(z, -c, (t * i)), t_3));
	} else {
		tmp = (t_2 + (t_6 + (y * (x * z)))) - ((y * (i * j)) + (t_1 + (a * (x * t))));
	}
	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(c * Float64(z * b))
	t_2 = Float64(i * Float64(t * b))
	t_3 = Float64(j * Float64(Float64(a * c) - Float64(y * i)))
	t_4 = Float64(Float64(y * z) - Float64(t * a))
	t_5 = Float64(Float64(Float64(x * t_4) + Float64(b * Float64(Float64(t * i) - Float64(z * c)))) + t_3)
	t_6 = Float64(c * Float64(a * j))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = fma(x, Float64(t * Float64(-a)), Float64(Float64(t_2 + t_6) - Float64(Float64(i * Float64(y * j)) + t_1)));
	elseif (t_5 <= 5e+295)
		tmp = fma(x, t_4, fma(b, fma(z, Float64(-c), Float64(t * i)), t_3));
	else
		tmp = Float64(Float64(t_2 + Float64(t_6 + Float64(y * Float64(x * z)))) - Float64(Float64(y * Float64(i * j)) + Float64(t_1 + Float64(a * Float64(x * t)))));
	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[(z * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(x * t$95$4), $MachinePrecision] + N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[(c * N[(a * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(x * N[(t * (-a)), $MachinePrecision] + N[(N[(t$95$2 + t$95$6), $MachinePrecision] - N[(N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 5e+295], N[(x * t$95$4 + N[(b * N[(z * (-c) + N[(t * i), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[(t$95$6 + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y * N[(i * j), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(a * N[(x * t), $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	12.6
Target	19.4
Herbie	4.2

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

Derivation

1. Split input into 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)))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded in b around 0 23.2

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

   4. Simplified 23.2

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

   5. Taylor expanded in y around 0 24.3

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

   6. Simplified 24.3

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

   7. Taylor expanded in c around inf 24.4

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

   if -inf.0 < (+.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)))) < 4.99999999999999991e295

   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

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

   if 4.99999999999999991e295 < (+.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 55.7

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

   2. Simplified 55.7

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

   3. Taylor expanded in x around 0 12.7

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

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, t \cdot \left(-a\right), \left(i \cdot \left(t \cdot b\right) + c \cdot \left(a \cdot j\right)\right) - \left(i \cdot \left(y \cdot j\right) + c \cdot \left(z \cdot b\right)\right)\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), j \cdot \left(a \cdot c - y \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \left(t \cdot b\right) + \left(c \cdot \left(a \cdot j\right) + y \cdot \left(x \cdot z\right)\right)\right) - \left(y \cdot \left(i \cdot j\right) + \left(c \cdot \left(z \cdot b\right) + a \cdot \left(x \cdot t\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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)))))