Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.0 → 4.2

Time: 14.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot z - i \cdot j\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, t_1, t \cdot \left(c \cdot j - x \cdot a\right)\right)\\ \mathbf{elif}\;t_2 \leq 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(t \cdot a - t \cdot a\right) + \mathsf{fma}\left(y, z, a \cdot \left(-t\right)\right), \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, a \cdot i\right), j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j - z \cdot b\right) + \mathsf{fma}\left(y, t_1, a \cdot \left(b \cdot i - x \cdot t\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* x z) (* i j)))
        (t_2
         (+
          (+ (* x (- (* y z) (* t a))) (* b (- (* a i) (* z c))))
          (* j (- (* t c) (* y i))))))
   (if (<= t_2 (- INFINITY))
     (fma y t_1 (* t (- (* c j) (* x a))))
     (if (<= t_2 1e+299)
       (fma
        x
        (+ (- (* t a) (* t a)) (fma y z (* a (- t))))
        (fma b (fma z (- c) (* a i)) (* j (fma i (- y) (* t c)))))
       (+ (* c (- (* t j) (* z b))) (fma y t_1 (* a (- (* b i) (* 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) - (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 = (x * z) - (i * j);
	double t_2 = ((x * ((y * z) - (t * a))) + (b * ((a * i) - (z * c)))) + (j * ((t * c) - (y * i)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(y, t_1, (t * ((c * j) - (x * a))));
	} else if (t_2 <= 1e+299) {
		tmp = fma(x, (((t * a) - (t * a)) + fma(y, z, (a * -t))), fma(b, fma(z, -c, (a * i)), (j * fma(i, -y, (t * c)))));
	} else {
		tmp = (c * ((t * j) - (z * b))) + fma(y, t_1, (a * ((b * i) - (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(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(Float64(x * z) - Float64(i * j))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) + Float64(b * Float64(Float64(a * i) - Float64(z * c)))) + Float64(j * Float64(Float64(t * c) - Float64(y * i))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(y, t_1, Float64(t * Float64(Float64(c * j) - Float64(x * a))));
	elseif (t_2 <= 1e+299)
		tmp = fma(x, Float64(Float64(Float64(t * a) - Float64(t * a)) + fma(y, z, Float64(a * Float64(-t)))), fma(b, fma(z, Float64(-c), Float64(a * i)), Float64(j * fma(i, Float64(-y), Float64(t * c)))));
	else
		tmp = Float64(Float64(c * Float64(Float64(t * j) - Float64(z * b))) + fma(y, t_1, Float64(a * Float64(Float64(b * i) - 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[(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[(N[(x * z), $MachinePrecision] - N[(i * j), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y * t$95$1 + N[(t * N[(N[(c * j), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+299], N[(x * N[(N[(N[(t * a), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(y * z + N[(a * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(z * (-c) + N[(a * i), $MachinePrecision]), $MachinePrecision] + N[(j * N[(i * (-y) + N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(t * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * t$95$1 + N[(a * N[(N[(b * i), $MachinePrecision] - 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	12.0
Target	16.0
Herbie	4.2

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded in a around 0 27.3

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

   4. Taylor expanded in b around 0 27.9

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

   5. Simplified 25.9

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

   if -inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < 1.0000000000000001e299

   1. Initial program 0.8

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

   2. Simplified 0.8

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

   3. Applied egg-rr 1.0

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, z, -\sqrt[3]{t \cdot a} \cdot {\left(\sqrt[3]{t \cdot a}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{t \cdot a}, {\left(\sqrt[3]{t \cdot a}\right)}^{2}, \sqrt[3]{t \cdot a} \cdot {\left(\sqrt[3]{t \cdot a}\right)}^{2}\right)}, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, a \cdot i\right), j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\right)\right) \]

   4. Applied egg-rr 0.8

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

   if 1.0000000000000001e299 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y))))

   1. Initial program 57.3

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

   2. Simplified 57.3

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

   3. Taylor expanded in x around 0 13.8

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

   4. Simplified 13.8

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

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot z - i \cdot j, t \cdot \left(c \cdot j - x \cdot a\right)\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right) \leq 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(t \cdot a - t \cdot a\right) + \mathsf{fma}\left(y, z, a \cdot \left(-t\right)\right), \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, a \cdot i\right), j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j - z \cdot b\right) + \mathsf{fma}\left(y, x \cdot z - i \cdot j, a \cdot \left(b \cdot i - x \cdot t\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(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)))))