Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Accuracy: 89.0% → 97.9%

Time: 12.7s

Precision: binary64

Cost: 14728

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+283}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(y, -z, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

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))))
   (if (<= t_1 -1e+283)
     (* (- x z) (* y t))
     (if (<= t_1 2e+231)
       (fma (* y (- x z)) t (* t (fma y (- z) (* y z))))
       (* y (* (- x z) t))))))

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);
	double tmp;
	if (t_1 <= -1e+283) {
		tmp = (x - z) * (y * t);
	} else if (t_1 <= 2e+231) {
		tmp = fma((y * (x - z)), t, (t * fma(y, -z, (y * z))));
	} else {
		tmp = y * ((x - z) * t);
	}
	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(x * y) - Float64(y * z))
	tmp = 0.0
	if (t_1 <= -1e+283)
		tmp = Float64(Float64(x - z) * Float64(y * t));
	elseif (t_1 <= 2e+231)
		tmp = fma(Float64(y * Float64(x - z)), t, Float64(t * fma(y, Float64(-z), Float64(y * z))));
	else
		tmp = Float64(y * Float64(Float64(x - z) * t));
	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[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+283], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+231], N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t + N[(t * N[(y * (-z) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	89.0%
Target	94.7%
Herbie	97.9%

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -9.99999999999999955e282

   1. Initial program 19.3%

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

   2. Taylor expanded in x around 0 99.5%

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

   3. Simplified 99.6%

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

      [Start] 99.5	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      associate-*r* [=>] 99.6	\[ \color{blue}{\left(y \cdot t\right) \cdot x} + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      *-commutative [<=] 99.6	\[ \color{blue}{\left(t \cdot y\right)} \cdot x + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 99.6	\[ \left(t \cdot y\right) \cdot x + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 99.6	\[ \left(t \cdot y\right) \cdot x + \left(-\color{blue}{\left(y \cdot t\right) \cdot z}\right) \]
      *-commutative [<=] 99.6	\[ \left(t \cdot y\right) \cdot x + \left(-\color{blue}{\left(t \cdot y\right)} \cdot z\right) \]
      distribute-rgt-neg-in [=>] 99.6	\[ \left(t \cdot y\right) \cdot x + \color{blue}{\left(t \cdot y\right) \cdot \left(-z\right)} \]
      distribute-lft-in [<=] 99.6	\[ \color{blue}{\left(t \cdot y\right) \cdot \left(x + \left(-z\right)\right)} \]
      sub-neg [<=] 99.6	\[ \left(t \cdot y\right) \cdot \color{blue}{\left(x - z\right)} \]
      *-commutative [=>] 99.6	\[ \color{blue}{\left(x - z\right) \cdot \left(t \cdot y\right)} \]
      *-commutative [=>] 99.6	\[ \left(x - z\right) \cdot \color{blue}{\left(y \cdot t\right)} \]

   if -9.99999999999999955e282 < (-.f64 (*.f64 x y) (*.f64 z y)) < 2.0000000000000001e231

   1. Initial program 97.5%

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

   2. Applied egg-rr 97.7%

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

      [Start] 97.5	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      *-commutative [=>] 97.5	\[ \color{blue}{t \cdot \left(x \cdot y - z \cdot y\right)} \]
      prod-diff [=>] 97.5	\[ t \cdot \color{blue}{\left(\mathsf{fma}\left(x, y, -y \cdot z\right) + \mathsf{fma}\left(-y, z, y \cdot z\right)\right)} \]
      *-commutative [<=] 97.5	\[ t \cdot \left(\mathsf{fma}\left(x, y, -\color{blue}{z \cdot y}\right) + \mathsf{fma}\left(-y, z, y \cdot z\right)\right) \]
      fma-neg [<=] 97.5	\[ t \cdot \left(\color{blue}{\left(x \cdot y - z \cdot y\right)} + \mathsf{fma}\left(-y, z, y \cdot z\right)\right) \]
      distribute-rgt-in [=>] 97.6	\[ \color{blue}{\left(x \cdot y - z \cdot y\right) \cdot t + \mathsf{fma}\left(-y, z, y \cdot z\right) \cdot t} \]
      fma-def [=>] 97.7	\[ \color{blue}{\mathsf{fma}\left(x \cdot y - z \cdot y, t, \mathsf{fma}\left(-y, z, y \cdot z\right) \cdot t\right)} \]
      distribute-rgt-out-- [=>] 97.7	\[ \mathsf{fma}\left(\color{blue}{y \cdot \left(x - z\right)}, t, \mathsf{fma}\left(-y, z, y \cdot z\right) \cdot t\right) \]
      *-commutative [<=] 97.7	\[ \mathsf{fma}\left(y \cdot \left(x - z\right), t, \mathsf{fma}\left(-y, z, \color{blue}{z \cdot y}\right) \cdot t\right) \]
      fma-udef [=>] 97.5	\[ \mathsf{fma}\left(y \cdot \left(x - z\right), t, \color{blue}{\left(\left(-y\right) \cdot z + z \cdot y\right)} \cdot t\right) \]
      distribute-lft-neg-in [<=] 97.5	\[ \mathsf{fma}\left(y \cdot \left(x - z\right), t, \left(\color{blue}{\left(-y \cdot z\right)} + z \cdot y\right) \cdot t\right) \]
      distribute-rgt-neg-in [=>] 97.5	\[ \mathsf{fma}\left(y \cdot \left(x - z\right), t, \left(\color{blue}{y \cdot \left(-z\right)} + z \cdot y\right) \cdot t\right) \]
      fma-def [=>] 97.7	\[ \mathsf{fma}\left(y \cdot \left(x - z\right), t, \color{blue}{\mathsf{fma}\left(y, -z, z \cdot y\right)} \cdot t\right) \]
      *-commutative [=>] 97.7	\[ \mathsf{fma}\left(y \cdot \left(x - z\right), t, \mathsf{fma}\left(y, -z, \color{blue}{y \cdot z}\right) \cdot t\right) \]

   if 2.0000000000000001e231 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 44.2%

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

   2. Simplified 98.2%

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

      [Start] 44.2	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 44.2	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 98.2	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -1 \cdot 10^{+283}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 2 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(y, -z, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.7%
Cost	1864

\[\begin{array}{l} t_1 := \left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;t_1 \leq 10^{+291}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 2
Accuracy	68.6%
Cost	1176

\[\begin{array}{l} t_1 := \left(x \cdot y\right) \cdot t\\ t_2 := y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-139}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-107}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	68.9%
Cost	912

\[\begin{array}{l} t_1 := \left(x \cdot y\right) \cdot t\\ t_2 := z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-107}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	68.9%
Cost	912

\[\begin{array}{l} t_1 := \left(x \cdot y\right) \cdot t\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-107}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-15}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	87.5%
Cost	648

\[\begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{+196}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+223}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 6
Accuracy	95.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+47}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	55.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{+38}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 8
Accuracy	53.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-300}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 9
Accuracy	51.2%
Cost	320

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

Reproduce

herbie shell --seed 2023151 
(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))