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

Average Error: 11.9 → 6.8

Time: 8.6s

Precision: binary64

Math

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

↓

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

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

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

Target

Original	11.9
Target	19.4
Herbie	6.8

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

Derivation

1. Split input into 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 z around 0 27.1

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

   4. Taylor expanded in c around 0 12.4

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

   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)))) < +inf.0

   1. Initial program 6.4

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

   2. Simplified 6.4

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

   3. Applied egg-rr 6.4

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

   if +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))))

   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 a around 0 41.5

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

   4. Taylor expanded in c around 0 4.1

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

4. Final simplification 6.8

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

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)))))