Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 13.1 → 9.6

Time: 13.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := a \cdot \left(t \cdot x\right)\\ t_2 := c \cdot \left(z \cdot b\right) + t_1\\ t_3 := i \cdot \left(b \cdot a\right)\\ t_4 := c \cdot \left(j \cdot t\right)\\ t_5 := y \cdot \left(z \cdot x\right)\\ t_6 := j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{if}\;j \leq -1.5075391945027925 \cdot 10^{+131}:\\ \;\;\;\;t_6 + \mathsf{fma}\left(z, y \cdot x - c \cdot b, a \cdot \left(i \cdot b - t \cdot x\right)\right)\\ \mathbf{elif}\;j \leq 1.8594302167394335 \cdot 10^{-73}:\\ \;\;\;\;\left(t_3 + \left(t_4 + t_5\right)\right) - \left(y \cdot \left(j \cdot i\right) + t_2\right)\\ \mathbf{elif}\;j \leq 1.210859548848804 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(b, i \cdot a - c \cdot z, \mathsf{fma}\left(x, y \cdot z - t \cdot a, t_6\right)\right)\\ \mathbf{elif}\;j \leq 3.619669044580342 \cdot 10^{+60}:\\ \;\;\;\;\left(t_4 + a \cdot \left(i \cdot b\right)\right) - \left(t_1 + i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_6 + \left(\left(t_3 + t_5\right) - t_2\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (+
  (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
  (* j (- (* c t) (* i y)))))

↓

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (* t x)))
        (t_2 (+ (* c (* z b)) t_1))
        (t_3 (* i (* b a)))
        (t_4 (* c (* j t)))
        (t_5 (* y (* z x)))
        (t_6 (* j (- (* c t) (* y i)))))
   (if (<= j -1.5075391945027925e+131)
     (+ t_6 (fma z (- (* y x) (* c b)) (* a (- (* i b) (* t x)))))
     (if (<= j 1.8594302167394335e-73)
       (- (+ t_3 (+ t_4 t_5)) (+ (* y (* j i)) t_2))
       (if (<= j 1.210859548848804e+46)
         (fma b (- (* i a) (* c z)) (fma x (- (* y z) (* t a)) t_6))
         (if (<= j 3.619669044580342e+60)
           (- (+ t_4 (* a (* i b))) (+ t_1 (* i (* j y))))
           (+ t_6 (- (+ t_3 t_5) 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) - (i * a)))) + (j * ((c * t) - (i * y)));
}

↓

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 = (c * (z * b)) + t_1;
	double t_3 = i * (b * a);
	double t_4 = c * (j * t);
	double t_5 = y * (z * x);
	double t_6 = j * ((c * t) - (y * i));
	double tmp;
	if (j <= -1.5075391945027925e+131) {
		tmp = t_6 + fma(z, ((y * x) - (c * b)), (a * ((i * b) - (t * x))));
	} else if (j <= 1.8594302167394335e-73) {
		tmp = (t_3 + (t_4 + t_5)) - ((y * (j * i)) + t_2);
	} else if (j <= 1.210859548848804e+46) {
		tmp = fma(b, ((i * a) - (c * z)), fma(x, ((y * z) - (t * a)), t_6));
	} else if (j <= 3.619669044580342e+60) {
		tmp = (t_4 + (a * (i * b))) - (t_1 + (i * (j * y)));
	} else {
		tmp = t_6 + ((t_3 + t_5) - 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(i * a)))) + Float64(j * Float64(Float64(c * t) - Float64(i * y))))
end

↓

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * Float64(t * x))
	t_2 = Float64(Float64(c * Float64(z * b)) + t_1)
	t_3 = Float64(i * Float64(b * a))
	t_4 = Float64(c * Float64(j * t))
	t_5 = Float64(y * Float64(z * x))
	t_6 = Float64(j * Float64(Float64(c * t) - Float64(y * i)))
	tmp = 0.0
	if (j <= -1.5075391945027925e+131)
		tmp = Float64(t_6 + fma(z, Float64(Float64(y * x) - Float64(c * b)), Float64(a * Float64(Float64(i * b) - Float64(t * x)))));
	elseif (j <= 1.8594302167394335e-73)
		tmp = Float64(Float64(t_3 + Float64(t_4 + t_5)) - Float64(Float64(y * Float64(j * i)) + t_2));
	elseif (j <= 1.210859548848804e+46)
		tmp = fma(b, Float64(Float64(i * a) - Float64(c * z)), fma(x, Float64(Float64(y * z) - Float64(t * a)), t_6));
	elseif (j <= 3.619669044580342e+60)
		tmp = Float64(Float64(t_4 + Float64(a * Float64(i * b))) - Float64(t_1 + Float64(i * Float64(j * y))));
	else
		tmp = Float64(t_6 + Float64(Float64(t_3 + t_5) - 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[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $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[(N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(c * N[(j * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(j * N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.5075391945027925e+131], N[(t$95$6 + N[(z * N[(N[(y * x), $MachinePrecision] - N[(c * b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(i * b), $MachinePrecision] - N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.8594302167394335e-73], N[(N[(t$95$3 + N[(t$95$4 + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[(y * N[(j * i), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.210859548848804e+46], N[(b * N[(N[(i * a), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 3.619669044580342e+60], N[(N[(t$95$4 + N[(a * N[(i * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 + N[(i * N[(j * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$6 + N[(N[(t$95$3 + t$95$5), $MachinePrecision] - 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	13.1
Target	16.7
Herbie	9.6

\[\begin{array}{l} \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array} \]

Derivation

1. Split input into 5 regimes

2. if j < -1.5075391945027925e131

   1. Initial program 6.2

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

   2. Simplified 6.2

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

   3. Taylor expanded in b around 0 22.6

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

   4. Simplified 5.6

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

   if -1.5075391945027925e131 < j < 1.85943021673943348e-73

   1. Initial program 15.5

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

   2. Simplified 15.5

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

   3. Taylor expanded in b around 0 9.5

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

   if 1.85943021673943348e-73 < j < 1.21085954884880389e46

   1. Initial program 10.8

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

   2. Simplified 10.8

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

   3. Taylor expanded in t around -inf 9.4

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

   4. Taylor expanded in x around 0 8.7

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

   5. Simplified 10.8

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

   if 1.21085954884880389e46 < j < 3.61966904458034195e60

   1. Initial program 10.5

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

   2. Simplified 10.5

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

   3. Taylor expanded in z around 0 25.1

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

   if 3.61966904458034195e60 < j

   1. Initial program 8.6

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

   2. Simplified 8.6

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

   3. Taylor expanded in b around 0 21.7

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

   4. Simplified 6.7

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

   5. Taylor expanded in z around -inf 10.2

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

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.5075391945027925 \cdot 10^{+131}:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \mathsf{fma}\left(z, y \cdot x - c \cdot b, a \cdot \left(i \cdot b - t \cdot x\right)\right)\\ \mathbf{elif}\;j \leq 1.8594302167394335 \cdot 10^{-73}:\\ \;\;\;\;\left(i \cdot \left(b \cdot a\right) + \left(c \cdot \left(j \cdot t\right) + y \cdot \left(z \cdot x\right)\right)\right) - \left(y \cdot \left(j \cdot i\right) + \left(c \cdot \left(z \cdot b\right) + a \cdot \left(t \cdot x\right)\right)\right)\\ \mathbf{elif}\;j \leq 1.210859548848804 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(b, i \cdot a - c \cdot z, \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \left(c \cdot t - y \cdot i\right)\right)\right)\\ \mathbf{elif}\;j \leq 3.619669044580342 \cdot 10^{+60}:\\ \;\;\;\;\left(c \cdot \left(j \cdot t\right) + a \cdot \left(i \cdot b\right)\right) - \left(a \cdot \left(t \cdot x\right) + i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\left(i \cdot \left(b \cdot a\right) + y \cdot \left(z \cdot x\right)\right) - \left(c \cdot \left(z \cdot b\right) + a \cdot \left(t \cdot x\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0))) (+ (* c t) (* i y)))) (if (< t -7.633533346031584e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))