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

Average Error: 12.8 → 2.6

Time: 12.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := c \cdot \left(z \cdot b\right)\\ t_2 := i \cdot \left(t \cdot b\right)\\ t_3 := c \cdot \left(a \cdot j\right)\\ t_4 := j \cdot \left(a \cdot c - y \cdot i\right)\\ t_5 := i \cdot \left(y \cdot j\right)\\ t_6 := \left(t_2 + \left(t_3 + y \cdot \left(x \cdot z\right)\right)\right) - \left(t_5 + \left(a \cdot \left(x \cdot t\right) + t_1\right)\right)\\ t_7 := y \cdot z - t \cdot a\\ t_8 := \mathsf{fma}\left(x, t_7, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), t_4\right)\right)\\ t_9 := \left(x \cdot t_7 + b \cdot \left(t \cdot i - z \cdot c\right)\right) + t_4\\ \mathbf{if}\;t_9 \leq -\infty:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t_9 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;t_8\\ \mathbf{elif}\;t_9 \leq 10^{-210}:\\ \;\;\;\;\mathsf{fma}\left(x, t_7, \left(t_2 + t_3\right) - \left(t_5 + t_1\right)\right)\\ \mathbf{elif}\;t_9 \leq 10^{+308}:\\ \;\;\;\;t_8\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array} \]

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 (* c (* a j)))
        (t_4 (* j (- (* a c) (* y i))))
        (t_5 (* i (* y j)))
        (t_6 (- (+ t_2 (+ t_3 (* y (* x z)))) (+ t_5 (+ (* a (* x t)) t_1))))
        (t_7 (- (* y z) (* t a)))
        (t_8 (fma x t_7 (fma b (fma z (- c) (* t i)) t_4)))
        (t_9 (+ (+ (* x t_7) (* b (- (* t i) (* z c)))) t_4)))
   (if (<= t_9 (- INFINITY))
     t_6
     (if (<= t_9 -5e+36)
       t_8
       (if (<= t_9 1e-210)
         (fma x t_7 (- (+ t_2 t_3) (+ t_5 t_1)))
         (if (<= t_9 1e+308) t_8 t_6))))))

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 = c * (a * j);
	double t_4 = j * ((a * c) - (y * i));
	double t_5 = i * (y * j);
	double t_6 = (t_2 + (t_3 + (y * (x * z)))) - (t_5 + ((a * (x * t)) + t_1));
	double t_7 = (y * z) - (t * a);
	double t_8 = fma(x, t_7, fma(b, fma(z, -c, (t * i)), t_4));
	double t_9 = ((x * t_7) + (b * ((t * i) - (z * c)))) + t_4;
	double tmp;
	if (t_9 <= -((double) INFINITY)) {
		tmp = t_6;
	} else if (t_9 <= -5e+36) {
		tmp = t_8;
	} else if (t_9 <= 1e-210) {
		tmp = fma(x, t_7, ((t_2 + t_3) - (t_5 + t_1)));
	} else if (t_9 <= 1e+308) {
		tmp = t_8;
	} else {
		tmp = t_6;
	}
	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(c * Float64(a * j))
	t_4 = Float64(j * Float64(Float64(a * c) - Float64(y * i)))
	t_5 = Float64(i * Float64(y * j))
	t_6 = Float64(Float64(t_2 + Float64(t_3 + Float64(y * Float64(x * z)))) - Float64(t_5 + Float64(Float64(a * Float64(x * t)) + t_1)))
	t_7 = Float64(Float64(y * z) - Float64(t * a))
	t_8 = fma(x, t_7, fma(b, fma(z, Float64(-c), Float64(t * i)), t_4))
	t_9 = Float64(Float64(Float64(x * t_7) + Float64(b * Float64(Float64(t * i) - Float64(z * c)))) + t_4)
	tmp = 0.0
	if (t_9 <= Float64(-Inf))
		tmp = t_6;
	elseif (t_9 <= -5e+36)
		tmp = t_8;
	elseif (t_9 <= 1e-210)
		tmp = fma(x, t_7, Float64(Float64(t_2 + t_3) - Float64(t_5 + t_1)));
	elseif (t_9 <= 1e+308)
		tmp = t_8;
	else
		tmp = t_6;
	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[(c * N[(a * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$2 + N[(t$95$3 + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$5 + N[(N[(a * N[(x * t), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(x * t$95$7 + N[(b * N[(z * (-c) + N[(t * i), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[(x * t$95$7), $MachinePrecision] + N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$9, (-Infinity)], t$95$6, If[LessEqual[t$95$9, -5e+36], t$95$8, If[LessEqual[t$95$9, 1e-210], N[(x * t$95$7 + N[(N[(t$95$2 + t$95$3), $MachinePrecision] - N[(t$95$5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$9, 1e+308], t$95$8, t$95$6]]]]]]]]]]]]]

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	12.8
Target	20.5
Herbie	2.6

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

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 or 1e308 < (+.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 63.9

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

   2. Simplified 63.9

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

   3. Taylor expanded in y around 0 10.7

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

   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)))) < -4.99999999999999977e36 or 1e-210 < (+.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)))) < 1e308

   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

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

   if -4.99999999999999977e36 < (+.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)))) < 1e-210

   1. Initial program 3.9

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

   2. Simplified 3.9

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

   3. Taylor expanded in b around 0 2.6

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq -\infty:\\ \;\;\;\;\left(i \cdot \left(t \cdot b\right) + \left(c \cdot \left(a \cdot j\right) + y \cdot \left(x \cdot z\right)\right)\right) - \left(i \cdot \left(y \cdot j\right) + \left(a \cdot \left(x \cdot t\right) + c \cdot \left(z \cdot b\right)\right)\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), j \cdot \left(a \cdot c - y \cdot i\right)\right)\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq 10^{-210}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \left(i \cdot \left(t \cdot b\right) + c \cdot \left(a \cdot j\right)\right) - \left(i \cdot \left(y \cdot j\right) + c \cdot \left(z \cdot b\right)\right)\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, t \cdot i\right), j \cdot \left(a \cdot c - y \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \left(t \cdot b\right) + \left(c \cdot \left(a \cdot j\right) + y \cdot \left(x \cdot z\right)\right)\right) - \left(i \cdot \left(y \cdot j\right) + \left(a \cdot \left(x \cdot t\right) + c \cdot \left(z \cdot b\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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)))))