Data.Colour.Matrix:inverse from colour-2.3.3, B

Average Error: 7.7 → 0.4

Time: 14.9s

Precision: binary64

Cost: 9744

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

\[\frac{x \cdot y - z \cdot t}{a} \]

↓

\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ t_2 := x \cdot y - z \cdot t\\ t_3 := \frac{t_2}{a} + \mathsf{fma}\left(z, -t, z \cdot t\right) \cdot \frac{1}{a}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-175}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;y \cdot \frac{x}{a} - \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+224}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (/ (- (* x y) (* z t)) a))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (/ x (/ a y)) (/ z (/ a t))))
        (t_2 (- (* x y) (* z t)))
        (t_3 (+ (/ t_2 a) (* (fma z (- t) (* z t)) (/ 1.0 a)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -2e-175)
       t_3
       (if (<= t_2 0.0)
         (- (* y (/ x a)) (/ t (/ a z)))
         (if (<= t_2 5e+224) t_3 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (z * t)) / a;
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x / (a / y)) - (z / (a / t));
	double t_2 = (x * y) - (z * t);
	double t_3 = (t_2 / a) + (fma(z, -t, (z * t)) * (1.0 / a));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -2e-175) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = (y * (x / a)) - (t / (a / z));
	} else if (t_2 <= 5e+224) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	t_3 = Float64(Float64(t_2 / a) + Float64(fma(z, Float64(-t), Float64(z * t)) * Float64(1.0 / a)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -2e-175)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(y * Float64(x / a)) - Float64(t / Float64(a / z)));
	elseif (t_2 <= 5e+224)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 / a), $MachinePrecision] + N[(N[(z * (-t) + N[(z * t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -2e-175], t$95$3, If[LessEqual[t$95$2, 0.0], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] - N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+224], t$95$3, t$95$1]]]]]]]

Target

Original	7.7
Target	5.6
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 4.99999999999999964e224 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 43.7

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Applied egg-rr 0.7

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < -2e-175 or -0.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.99999999999999964e224

   1. Initial program 0.3

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(x \cdot y - z \cdot t\right)} \]

   3. Applied egg-rr 0.3

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

   if -2e-175 < (-.f64 (*.f64 x y) (*.f64 z t)) < -0.0

   1. Initial program 8.3

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in a around 0 8.3

      \[\leadsto \color{blue}{\frac{y \cdot x - t \cdot z}{a}} \]

   3. Simplified 1.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)} \]
      Proof
      (fma.f64 -1 (/.f64 t (/.f64 a z)) (/.f64 y (/.f64 a x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t z) a)) (/.f64 y (/.f64 a x))): 31 points increase in error, 29 points decrease in error
      (fma.f64 -1 (/.f64 (*.f64 t z) a) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) a))): 32 points increase in error, 34 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 t z) a)) (/.f64 (*.f64 y x) a))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 (*.f64 y x) a) (*.f64 -1 (/.f64 (*.f64 t z) a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y x) a) (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 (*.f64 t z) a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> unsub-neg_binary64 (-.f64 (/.f64 (*.f64 y x) a) (/.f64 (*.f64 t z) a))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 y x) (*.f64 t z)) a)): 1 points increase in error, 2 points decrease in error

   4. Applied egg-rr 4.7

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{1}{a} \cdot \frac{t}{\frac{1}{z}}}, \frac{y}{\frac{a}{x}}\right) \]

   5. Applied egg-rr 1.5

      \[\leadsto \color{blue}{\frac{-t}{\frac{a}{z}} + y \cdot \frac{x}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq -2 \cdot 10^{-175}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a} + \mathsf{fma}\left(z, -t, z \cdot t\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 0:\\ \;\;\;\;y \cdot \frac{x}{a} - \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+224}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a} + \mathsf{fma}\left(z, -t, z \cdot t\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	2768

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{t_1}{a}\\ t_3 := \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+229}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+224}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	0.4
Cost	2768

\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{t_1}{a}\\ t_3 := \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+229}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-175}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;y \cdot \frac{x}{a} - \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+224}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	4.5
Cost	1864

\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-\frac{z}{a}\right)\\ \end{array} \]

Alternative 4
Error	23.7
Cost	912

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{x}}\\ t_2 := z \cdot \frac{-t}{a}\\ \mathbf{if}\;t \leq -1.1741227309619492 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.738702837920677 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.2953648376784832 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 2.3435658089956605 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	24.2
Cost	912

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;t \leq -1.1741227309619492 \cdot 10^{-172}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;t \leq 3.738702837920677 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.2953648376784832 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 2.3435658089956605 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 6
Error	24.7
Cost	912

\[\begin{array}{l} t_1 := t \cdot \left(-\frac{z}{a}\right)\\ \mathbf{if}\;z \leq -7.6 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+145}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.224447568023676 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 7
Error	32.6
Cost	584

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;t \leq 3.738702837920677 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	32.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq 4.0394047797803134 \cdot 10^{-281}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 9
Error	32.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq 4.0394047797803134 \cdot 10^{-281}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 10
Error	32.5
Cost	320

\[\frac{y}{\frac{a}{x}} \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

  (/ (- (* x y) (* z t)) a))