Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.6% → 97.2%

Time: 7.4s

Precision: binary64

Cost: 580

• Unsound rule application detected in e-graph. Results may not be sound.

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+36}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1e+36) (* y (* (- x z) t)) (* t (* y (- 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 tmp;
	if (y <= -1e+36) {
		tmp = y * ((x - z) * t);
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * y) - (z * y)) * t
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1d+36)) then
        tmp = y * ((x - z) * t)
    else
        tmp = t * (y * (x - z))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1e+36) {
		tmp = y * ((x - z) * t);
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return ((x * y) - (z * y)) * t

↓

def code(x, y, z, t):
	tmp = 0
	if y <= -1e+36:
		tmp = y * ((x - z) * t)
	else:
		tmp = t * (y * (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)
	tmp = 0.0
	if (y <= -1e+36)
		tmp = Float64(y * Float64(Float64(x - z) * t));
	else
		tmp = Float64(t * Float64(y * Float64(x - z)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * y) - (z * y)) * t;
end

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1e+36)
		tmp = y * ((x - z) * t);
	else
		tmp = t * (y * (x - z));
	end
	tmp_2 = 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_] := If[LessEqual[y, -1e+36], N[(y * N[(N[(x - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	90.6%
Target	96.4%
Herbie	97.2%

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

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000004e36

   1. Initial program 81.3%

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

   2. Simplified 99.8%

      \[\leadsto \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      Step-by-step derivation

      [Start] 81.3%	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 89.8%	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 99.8%	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]

   if -1.00000000000000004e36 < y

   1. Initial program 93.3%

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

   2. Simplified 94.3%

      \[\leadsto \color{blue}{\left(y \cdot \left(x - z\right)\right) \cdot t} \]
      Step-by-step derivation

      [Start] 93.3%	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 94.3%	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]

3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+36}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+36}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	73.1%
Cost	912

\[\begin{array}{l} t_1 := t \cdot \left(y \cdot x\right)\\ t_2 := z \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{if}\;x \leq -6.4 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+128}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 3
Accuracy	72.5%
Cost	912

\[\begin{array}{l} t_1 := t \cdot \left(y \cdot x\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+134}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 4
Accuracy	72.6%
Cost	912

\[\begin{array}{l} t_1 := t \cdot \left(y \cdot x\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+37}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+128}:\\ \;\;\;\;z \cdot \left(t \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 5
Accuracy	97.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 6
Accuracy	56.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+39}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 7
Accuracy	92.5%
Cost	448

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

Alternative 8
Accuracy	54.1%
Cost	320

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

Reproduce

herbie shell --seed 2023171 
(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))