Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Accuracy: 89.0% → 97.9%

Time: 10.3s

Precision: binary64

Cost: 1480

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

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 -2e+300)
     (* y (* t (- x z)))
     (if (<= t_1 2e+247) (* t (* y (- x z))) (* (- x z) (* y 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 <= -2e+300) {
		tmp = y * (t * (x - z));
	} else if (t_1 <= 2e+247) {
		tmp = t * (y * (x - z));
	} else {
		tmp = (x - z) * (y * t);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - (y * z)
    if (t_1 <= (-2d+300)) then
        tmp = y * (t * (x - z))
    else if (t_1 <= 2d+247) then
        tmp = t * (y * (x - z))
    else
        tmp = (x - z) * (y * t)
    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 t_1 = (x * y) - (y * z);
	double tmp;
	if (t_1 <= -2e+300) {
		tmp = y * (t * (x - z));
	} else if (t_1 <= 2e+247) {
		tmp = t * (y * (x - z));
	} else {
		tmp = (x - z) * (y * t);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = (x * y) - (y * z)
	tmp = 0
	if t_1 <= -2e+300:
		tmp = y * (t * (x - z))
	elif t_1 <= 2e+247:
		tmp = t * (y * (x - z))
	else:
		tmp = (x - z) * (y * 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 <= -2e+300)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	elseif (t_1 <= 2e+247)
		tmp = Float64(t * Float64(y * Float64(x - z)));
	else
		tmp = Float64(Float64(x - z) * Float64(y * t));
	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)
	t_1 = (x * y) - (y * z);
	tmp = 0.0;
	if (t_1 <= -2e+300)
		tmp = y * (t * (x - z));
	elseif (t_1 <= 2e+247)
		tmp = t * (y * (x - z));
	else
		tmp = (x - z) * (y * t);
	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_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+300], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+247], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	89.0%
Target	94.6%
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)) < -2.0000000000000001e300

   1. Initial program 3.3%

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

   2. Simplified 99.6%

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

      [Start] 3.3	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 3.3	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 99.6	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 99.6	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   if -2.0000000000000001e300 < (-.f64 (*.f64 x y) (*.f64 z y)) < 1.9999999999999999e247

   1. Initial program 97.7%

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

   2. Simplified 97.7%

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

      [Start] 97.7	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 97.7	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]

   if 1.9999999999999999e247 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 39.1%

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

   2. Simplified 98.9%

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

      [Start] 39.1	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 39.1	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 98.9	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 98.9	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   3. Taylor expanded in x around 0 98.9%

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

   4. Simplified 99.1%

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

      [Start] 98.9	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 98.9	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 98.6	\[ \color{blue}{\left(y \cdot t\right) \cdot x} + \left(-y \cdot \left(t \cdot z\right)\right) \]
      associate-*r* [=>] 99.1	\[ \left(y \cdot t\right) \cdot x + \left(-\color{blue}{\left(y \cdot t\right) \cdot z}\right) \]
      distribute-rgt-neg-out [<=] 99.1	\[ \left(y \cdot t\right) \cdot x + \color{blue}{\left(y \cdot t\right) \cdot \left(-z\right)} \]
      distribute-lft-in [<=] 99.1	\[ \color{blue}{\left(y \cdot t\right) \cdot \left(x + \left(-z\right)\right)} \]
      sub-neg [<=] 99.1	\[ \left(y \cdot t\right) \cdot \color{blue}{\left(x - z\right)} \]
      *-commutative [=>] 99.1	\[ \color{blue}{\left(x - z\right) \cdot \left(y \cdot t\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -2 \cdot 10^{+300}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 2 \cdot 10^{+247}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.5%
Cost	1176

\[\begin{array}{l} t_1 := t \cdot \left(x \cdot y\right)\\ t_2 := \left(y \cdot t\right) \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -6 \cdot 10^{-20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-99}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-194}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	69.4%
Cost	1176

\[\begin{array}{l} t_1 := t \cdot \left(x \cdot y\right)\\ t_2 := \left(y \cdot z\right) \cdot \left(-t\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-99}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-193}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	68.8%
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-98}:\\ \;\;\;\;t \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 4
Accuracy	87.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{+101}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	95.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	53.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 10^{+81}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 7
Accuracy	50.8%
Cost	320

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

Reproduce

herbie shell --seed 2023140 
(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))