Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.0 → 1.1

Time: 10.0s

Precision: binary64

Cost: 8588

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

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \left(y \cdot \left(x - z\right)\right)\\ t_2 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot t, z \cdot \left(y \cdot \left(-t\right)\right)\right)\\ \mathbf{elif}\;t_2 \leq 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (* y (- x z)))) (t_2 (- (* x y) (* y z))))
   (if (<= t_2 (- INFINITY))
     (* (- x z) (* y t))
     (if (<= t_2 -1e-102)
       t_1
       (if (<= t_2 1e-146)
         (fma y (* x t) (* z (* y (- t))))
         (if (<= t_2 1e+164) t_1 (* 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 = t * (y * (x - z));
	double t_2 = (x * y) - (y * z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (x - z) * (y * t);
	} else if (t_2 <= -1e-102) {
		tmp = t_1;
	} else if (t_2 <= 1e-146) {
		tmp = fma(y, (x * t), (z * (y * -t)));
	} else if (t_2 <= 1e+164) {
		tmp = t_1;
	} 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(t * Float64(y * Float64(x - z)))
	t_2 = Float64(Float64(x * y) - Float64(y * z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(x - z) * Float64(y * t));
	elseif (t_2 <= -1e-102)
		tmp = t_1;
	elseif (t_2 <= 1e-146)
		tmp = fma(y, Float64(x * t), Float64(z * Float64(y * Float64(-t))));
	elseif (t_2 <= 1e+164)
		tmp = t_1;
	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[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-102], t$95$1, If[LessEqual[t$95$2, 1e-146], N[(y * N[(x * t), $MachinePrecision] + N[(z * N[(y * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+164], t$95$1, N[(y * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	7.0
Target	3.4
Herbie	1.1

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.3

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

      [Start] 64.0	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 64.0	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 0.3	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 0.3	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   3. Taylor expanded in x around 0 0.3

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

   4. Simplified 0.3

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

      [Start] 0.3	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 0.3	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 0.3	\[ \color{blue}{\left(y \cdot t\right) \cdot x} + \left(-y \cdot \left(t \cdot z\right)\right) \]
      associate-*r* [=>] 0.3	\[ \left(y \cdot t\right) \cdot x + \left(-\color{blue}{\left(y \cdot t\right) \cdot z}\right) \]
      distribute-rgt-neg-out [<=] 0.3	\[ \left(y \cdot t\right) \cdot x + \color{blue}{\left(y \cdot t\right) \cdot \left(-z\right)} \]
      distribute-lft-in [<=] 0.3	\[ \color{blue}{\left(y \cdot t\right) \cdot \left(x + \left(-z\right)\right)} \]
      sub-neg [<=] 0.3	\[ \left(y \cdot t\right) \cdot \color{blue}{\left(x - z\right)} \]
      *-commutative [=>] 0.3	\[ \color{blue}{\left(x - z\right) \cdot \left(y \cdot t\right)} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z y)) < -9.99999999999999933e-103 or 1.00000000000000003e-146 < (-.f64 (*.f64 x y) (*.f64 z y)) < 1e164

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 0.2	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]

   if -9.99999999999999933e-103 < (-.f64 (*.f64 x y) (*.f64 z y)) < 1.00000000000000003e-146

   1. Initial program 5.4

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

   2. Simplified 2.7

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

      [Start] 5.4	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 5.4	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 2.7	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 2.7	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   3. Taylor expanded in x around 0 2.7

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

   4. Simplified 2.8

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

      [Start] 2.7	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 2.7	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 3.2	\[ \color{blue}{\left(y \cdot t\right) \cdot x} + \left(-y \cdot \left(t \cdot z\right)\right) \]
      associate-*r* [=>] 2.8	\[ \left(y \cdot t\right) \cdot x + \left(-\color{blue}{\left(y \cdot t\right) \cdot z}\right) \]
      distribute-rgt-neg-out [<=] 2.8	\[ \left(y \cdot t\right) \cdot x + \color{blue}{\left(y \cdot t\right) \cdot \left(-z\right)} \]
      distribute-lft-in [<=] 2.8	\[ \color{blue}{\left(y \cdot t\right) \cdot \left(x + \left(-z\right)\right)} \]
      sub-neg [<=] 2.8	\[ \left(y \cdot t\right) \cdot \color{blue}{\left(x - z\right)} \]
      *-commutative [=>] 2.8	\[ \color{blue}{\left(x - z\right) \cdot \left(y \cdot t\right)} \]

   5. Taylor expanded in x around 0 2.7

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

   6. Simplified 2.9

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

      [Start] 2.7	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 2.7	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 2.9	\[ y \cdot \left(t \cdot x\right) + \left(-\color{blue}{\left(y \cdot t\right) \cdot z}\right) \]
      distribute-rgt-neg-out [<=] 2.9	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(y \cdot t\right) \cdot \left(-z\right)} \]
      *-lft-identity [<=] 2.9	\[ y \cdot \left(t \cdot x\right) + \color{blue}{1 \cdot \left(\left(y \cdot t\right) \cdot \left(-z\right)\right)} \]
      *-commutative [<=] 2.9	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(\left(y \cdot t\right) \cdot \left(-z\right)\right) \cdot 1} \]
      fma-def [=>] 2.9	\[ \color{blue}{\mathsf{fma}\left(y, t \cdot x, \left(\left(y \cdot t\right) \cdot \left(-z\right)\right) \cdot 1\right)} \]
      *-commutative [=>] 2.9	\[ \mathsf{fma}\left(y, t \cdot x, \color{blue}{1 \cdot \left(\left(y \cdot t\right) \cdot \left(-z\right)\right)}\right) \]
      *-lft-identity [=>] 2.9	\[ \mathsf{fma}\left(y, t \cdot x, \color{blue}{\left(y \cdot t\right) \cdot \left(-z\right)}\right) \]

   if 1e164 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 24.0

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

   2. Simplified 2.3

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

      [Start] 24.0	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 24.0	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 2.3	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 2.3	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -\infty:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq -1 \cdot 10^{-102}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot t, z \cdot \left(y \cdot \left(-t\right)\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 10^{+164}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	2512

\[\begin{array}{l} t_1 := t \cdot \left(y \cdot \left(x - z\right)\right)\\ t_2 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{-181}:\\ \;\;\;\;y \cdot \frac{t}{\frac{1}{x - z}}\\ \mathbf{elif}\;t_2 \leq 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

Alternative 2
Error	0.6
Cost	2512

\[\begin{array}{l} t_1 := t \cdot \left(y \cdot \left(x - z\right)\right)\\ t_2 := x \cdot y - y \cdot z\\ t_3 := \left(x - z\right) \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{-181}:\\ \;\;\;\;\frac{y}{\frac{1}{t_3}}\\ \mathbf{elif}\;t_2 \leq 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot t_3\\ \end{array} \]

Alternative 3
Error	20.2
Cost	912

\[\begin{array}{l} t_1 := z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+64}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+286}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 4
Error	19.8
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-12} \lor \neg \left(z \leq 3.9 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 5
Error	2.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 6
Error	2.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{+50}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 7
Error	29.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 8
Error	7.3
Cost	448

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

Alternative 9
Error	31.9
Cost	320

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

Reproduce

herbie shell --seed 2023060 
(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))