Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.9 → 3.7

Time: 18.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(c, j, x \cdot \left(-a\right)\right)\\ t_2 := i \cdot \left(a \cdot b\right)\\ t_3 := y \cdot z - t \cdot a\\ t_4 := \left(x \cdot t_3 + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(t, t_1, z \cdot \left(x \cdot y - b \cdot c\right) + \left(t_2 - i \cdot \left(y \cdot j\right)\right)\right)\\ \mathbf{elif}\;t_4 \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(x, t_3, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, a \cdot i\right), j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, t_1, t_2 + \left(y \cdot \left(x \cdot z - i \cdot j\right) - c \cdot \left(z \cdot b\right)\right)\right)\\ \end{array} \]

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 (fma c j (* x (- a))))
        (t_2 (* i (* a b)))
        (t_3 (- (* y z) (* t a)))
        (t_4
         (+
          (+ (* x t_3) (* b (- (* a i) (* z c))))
          (* j (- (* t c) (* y i))))))
   (if (<= t_4 -2e+294)
     (fma t t_1 (+ (* z (- (* x y) (* b c))) (- t_2 (* i (* y j)))))
     (if (<= t_4 1e+302)
       (fma x t_3 (fma b (fma z (- c) (* a i)) (* j (fma i (- y) (* t c)))))
       (fma t t_1 (+ t_2 (- (* y (- (* x z) (* i j))) (* c (* z b)))))))))

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 = fma(c, j, (x * -a));
	double t_2 = i * (a * b);
	double t_3 = (y * z) - (t * a);
	double t_4 = ((x * t_3) + (b * ((a * i) - (z * c)))) + (j * ((t * c) - (y * i)));
	double tmp;
	if (t_4 <= -2e+294) {
		tmp = fma(t, t_1, ((z * ((x * y) - (b * c))) + (t_2 - (i * (y * j)))));
	} else if (t_4 <= 1e+302) {
		tmp = fma(x, t_3, fma(b, fma(z, -c, (a * i)), (j * fma(i, -y, (t * c)))));
	} else {
		tmp = fma(t, t_1, (t_2 + ((y * ((x * z) - (i * j))) - (c * (z * b)))));
	}
	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 = fma(c, j, Float64(x * Float64(-a)))
	t_2 = Float64(i * Float64(a * b))
	t_3 = Float64(Float64(y * z) - Float64(t * a))
	t_4 = Float64(Float64(Float64(x * t_3) + Float64(b * Float64(Float64(a * i) - Float64(z * c)))) + Float64(j * Float64(Float64(t * c) - Float64(y * i))))
	tmp = 0.0
	if (t_4 <= -2e+294)
		tmp = fma(t, t_1, Float64(Float64(z * Float64(Float64(x * y) - Float64(b * c))) + Float64(t_2 - Float64(i * Float64(y * j)))));
	elseif (t_4 <= 1e+302)
		tmp = fma(x, t_3, fma(b, fma(z, Float64(-c), Float64(a * i)), Float64(j * fma(i, Float64(-y), Float64(t * c)))));
	else
		tmp = fma(t, t_1, Float64(t_2 + Float64(Float64(y * Float64(Float64(x * z) - Float64(i * j))) - Float64(c * Float64(z * b)))));
	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[(c * j + N[(x * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * t$95$3), $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$4, -2e+294], N[(t * t$95$1 + N[(N[(z * N[(N[(x * y), $MachinePrecision] - N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+302], N[(x * t$95$3 + N[(b * N[(z * (-c) + N[(a * i), $MachinePrecision]), $MachinePrecision] + N[(j * N[(i * (-y) + N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * t$95$1 + N[(t$95$2 + N[(N[(y * N[(N[(x * z), $MachinePrecision] - N[(i * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c * N[(z * b), $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	12.9
Target	16.8
Herbie	3.7

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

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)))) < -2.00000000000000013e294

   1. Initial program 52.5

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

   2. Simplified 52.5

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

   3. Taylor expanded in x around 0 52.5

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

   4. Simplified 21.1

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

   5. Applied egg-rr 21.2

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

   6. Taylor expanded in y around 0 21.1

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

   7. Simplified 21.1

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

   8. Taylor expanded in z around 0 17.3

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

   if -2.00000000000000013e294 < (+.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)))) < 1.0000000000000001e302

   1. Initial program 0.9

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

   2. Simplified 0.9

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

   if 1.0000000000000001e302 < (+.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 59.3

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

   2. Simplified 59.3

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

   3. Taylor expanded in x around 0 59.3

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

   4. Simplified 25.8

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

   5. Applied egg-rr 25.9

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

   6. Taylor expanded in y around 0 25.8

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

   7. Simplified 25.8

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

   8. Taylor expanded in c around -inf 10.4

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right) \leq -2 \cdot 10^{+294}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(c, j, x \cdot \left(-a\right)\right), z \cdot \left(x \cdot y - b \cdot c\right) + \left(i \cdot \left(a \cdot b\right) - i \cdot \left(y \cdot j\right)\right)\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right) \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, \mathsf{fma}\left(z, -c, a \cdot i\right), j \cdot \mathsf{fma}\left(i, -y, t \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(c, j, x \cdot \left(-a\right)\right), i \cdot \left(a \cdot b\right) + \left(y \cdot \left(x \cdot z - i \cdot j\right) - c \cdot \left(z \cdot b\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(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)))))