Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.9 → 3.5

Time: 34.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot z - t \cdot a\\ t_2 := \left(x \cdot t_1 + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, -b \cdot c\right) + \mathsf{fma}\left(t, c \cdot j, i \cdot \left(a \cdot b - y \cdot j\right)\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(x, t_1, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, a \cdot i\right), j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(b \cdot i\right) + \left(c \cdot \left(t \cdot j\right) + y \cdot \left(x \cdot z\right)\right)\right) - \left(y \cdot \left(i \cdot j\right) + \left(c \cdot \left(z \cdot b\right) + a \cdot \left(x \cdot t\right)\right)\right)\\ \end{array} \]

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 (- (* y z) (* t a)))
        (t_2
         (+
          (+ (* x t_1) (* b (- (* a i) (* z c))))
          (* j (- (* t c) (* y i))))))
   (if (<= t_2 (- INFINITY))
     (+ (* z (fma y x (- (* b c)))) (fma t (* c j) (* i (- (* a b) (* y j)))))
     (if (<= t_2 2e+300)
       (fma x t_1 (fma b (fma z (- c) (* a i)) (* j (fma i (- y) (* t c)))))
       (-
        (+ (* a (* b i)) (+ (* c (* t j)) (* y (* x z))))
        (+ (* y (* i j)) (+ (* c (* z b)) (* 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) - (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 = (y * z) - (t * a);
	double t_2 = ((x * t_1) + (b * ((a * i) - (z * c)))) + (j * ((t * c) - (y * i)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (z * fma(y, x, -(b * c))) + fma(t, (c * j), (i * ((a * b) - (y * j))));
	} else if (t_2 <= 2e+300) {
		tmp = fma(x, t_1, fma(b, fma(z, -c, (a * i)), (j * fma(i, -y, (t * c)))));
	} else {
		tmp = ((a * (b * i)) + ((c * (t * j)) + (y * (x * z)))) - ((y * (i * j)) + ((c * (z * b)) + (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(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(y * z) - Float64(t * a))
	t_2 = Float64(Float64(Float64(x * t_1) + 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 = Float64(Float64(z * fma(y, x, Float64(-Float64(b * c)))) + fma(t, Float64(c * j), Float64(i * Float64(Float64(a * b) - Float64(y * j)))));
	elseif (t_2 <= 2e+300)
		tmp = fma(x, t_1, fma(b, fma(z, Float64(-c), Float64(a * i)), Float64(j * fma(i, Float64(-y), Float64(t * c)))));
	else
		tmp = Float64(Float64(Float64(a * Float64(b * i)) + Float64(Float64(c * Float64(t * j)) + Float64(y * Float64(x * z)))) - Float64(Float64(y * Float64(i * j)) + Float64(Float64(c * Float64(z * b)) + 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[(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[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * t$95$1), $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[(N[(z * N[(y * x + (-N[(b * c), $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + N[(t * N[(c * j), $MachinePrecision] + N[(i * N[(N[(a * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+300], N[(x * t$95$1 + 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[(N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision] + N[(N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y * N[(i * j), $MachinePrecision]), $MachinePrecision] + N[(N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision] + 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	11.9
Target	15.6
Herbie	3.5

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

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 x around 0 13.1

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

   4. Simplified 23.9

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

   5. Taylor expanded in x around 0 18.6

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

   6. Simplified 19.1

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

   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)))) < 2.0000000000000001e300

   1. Initial program 0.7

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

   2. Simplified 0.7

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

   if 2.0000000000000001e300 < (+.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 58.0

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

   2. Simplified 58.0

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

   3. Taylor expanded in x around 0 12.8

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

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right) \leq -\infty:\\ \;\;\;\;z \cdot \mathsf{fma}\left(y, x, -b \cdot c\right) + \mathsf{fma}\left(t, c \cdot j, i \cdot \left(a \cdot b - y \cdot j\right)\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right) \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, a \cdot i\right), j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(b \cdot i\right) + \left(c \cdot \left(t \cdot j\right) + y \cdot \left(x \cdot z\right)\right)\right) - \left(y \cdot \left(i \cdot j\right) + \left(c \cdot \left(z \cdot b\right) + a \cdot \left(x \cdot t\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (x y z t a b c i j)
  :name "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)))))