Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.2 → 1.6

Time: 47.3s

Precision: binary64

Cost: 14984

Math

\[\left(x \cdot y - z \cdot y\right) \cdot t \]

↓

\[\begin{array}{l} t_1 := \left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+296}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(y, -z, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- (* x y) (* y z)) t)))
   (if (<= t_1 -5e+296)
     (* (- x z) (* y t))
     (if (<= t_1 2e+283)
       (fma (* y (- x z)) t (* t (fma y (- z) (* y z))))
       (* y (* t (- x z)))))))

C

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

↓

double code(double x, double y, double z, double t) {
	double t_1 = ((x * y) - (y * z)) * t;
	double tmp;
	if (t_1 <= -5e+296) {
		tmp = (x - z) * (y * t);
	} else if (t_1 <= 2e+283) {
		tmp = fma((y * (x - z)), t, (t * fma(y, -z, (y * z))));
	} else {
		tmp = y * (t * (x - z));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * y) - Float64(y * z)) * t)
	tmp = 0.0
	if (t_1 <= -5e+296)
		tmp = Float64(Float64(x - z) * Float64(y * t));
	elseif (t_1 <= 2e+283)
		tmp = fma(Float64(y * Float64(x - z)), t, Float64(t * fma(y, Float64(-z), Float64(y * z))));
	else
		tmp = Float64(y * Float64(t * Float64(x - z)));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+296], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+283], N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t + N[(t * N[(y * (-z) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.2
Target	3.1
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < -5.0000000000000001e296

   1. Initial program 56.2

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]

   2. Taylor expanded in x around 0 3.9

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

   3. Simplified 1.5

      \[\leadsto \color{blue}{\left(x - z\right) \cdot \left(y \cdot t\right)} \]
      Proof
      (*.f64 (-.f64 x z) (*.f64 y t)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x z) (Rewrite<= *-commutative_binary64 (*.f64 t y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 t y) (-.f64 x z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 t y) (Rewrite=> sub-neg_binary64 (+.f64 x (neg.f64 z)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (*.f64 t y) x) (*.f64 (*.f64 t y) (neg.f64 z)))): 2 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 y t)) x) (*.f64 (*.f64 t y) (neg.f64 z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 t x))) (*.f64 (*.f64 t y) (neg.f64 z))): 37 points increase in error, 35 points decrease in error
      (+.f64 (*.f64 y (*.f64 t x)) (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (*.f64 t y) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 y (*.f64 t x)) (neg.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 y t)) z))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 y (*.f64 t x)) (neg.f64 (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 t z))))): 28 points increase in error, 33 points decrease in error
      (+.f64 (*.f64 y (*.f64 t x)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 y (*.f64 t z))))): 0 points increase in error, 0 points decrease in error

   if -5.0000000000000001e296 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < 1.99999999999999991e283

   1. Initial program 1.5

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]

   2. Applied egg-rr 1.4

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

   if 1.99999999999999991e283 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t)

   1. Initial program 52.6

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]

   2. Simplified 4.6

      \[\leadsto \color{blue}{y \cdot \left(t \cdot \left(x - z\right)\right)} \]
      Proof
      (*.f64 y (*.f64 t (-.f64 x z))): 0 points increase in error, 0 points decrease in error
      (*.f64 y (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 x z) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y (-.f64 x z)) t)): 51 points increase in error, 68 points decrease in error
      (*.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 x y) (*.f64 z y))) t): 1 points increase in error, 2 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq -5 \cdot 10^{+296}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(y, -z, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	7428

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+238}:\\ \;\;\;\;\frac{x - z}{\frac{{t}^{-1}}{y}}\\ \mathbf{elif}\;t_1 \leq 10^{+204}:\\ \;\;\;\;t_1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 2
Error	1.4
Cost	1608

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+268}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;t_1 \leq 10^{+204}:\\ \;\;\;\;t_1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 3
Error	7.9
Cost	844

\[\begin{array}{l} t_1 := y \cdot \left(t \cdot \left(x - z\right)\right)\\ t_2 := \left(y \cdot z\right) \cdot \left(-t\right)\\ \mathbf{if}\;z \leq -1.26 \cdot 10^{+171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+212}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	19.4
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 5
Error	19.4
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00025:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-59}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 6
Error	4.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 7
Error	30.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{+88}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 8
Error	30.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 9
Error	31.3
Cost	320

\[x \cdot \left(y \cdot t\right) \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -9.231879582886777e-80) (* (* y t) (- x z)) (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t)))

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