Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Accuracy: 88.9% → 98.9%

Time: 12.7s

Precision: binary64

Cost: 2512

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ t_2 := t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+194}:\\ \;\;\;\;\frac{y}{\frac{\frac{1}{t}}{x - z}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-137}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+256}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

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_2 (* t (* y (- x z)))))
   (if (<= t_1 -2e+194)
     (/ y (/ (/ 1.0 t) (- x z)))
     (if (<= t_1 -2e-148)
       t_2
       (if (<= t_1 5e-137)
         (* (- x z) (* y t))
         (if (<= t_1 5e+256) t_2 (* 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);
	double t_2 = t * (y * (x - z));
	double tmp;
	if (t_1 <= -2e+194) {
		tmp = y / ((1.0 / t) / (x - z));
	} else if (t_1 <= -2e-148) {
		tmp = t_2;
	} else if (t_1 <= 5e-137) {
		tmp = (x - z) * (y * t);
	} else if (t_1 <= 5e+256) {
		tmp = t_2;
	} else {
		tmp = y * (t * (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * y) - (y * z)
    t_2 = t * (y * (x - z))
    if (t_1 <= (-2d+194)) then
        tmp = y / ((1.0d0 / t) / (x - z))
    else if (t_1 <= (-2d-148)) then
        tmp = t_2
    else if (t_1 <= 5d-137) then
        tmp = (x - z) * (y * t)
    else if (t_1 <= 5d+256) then
        tmp = t_2
    else
        tmp = y * (t * (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 t_1 = (x * y) - (y * z);
	double t_2 = t * (y * (x - z));
	double tmp;
	if (t_1 <= -2e+194) {
		tmp = y / ((1.0 / t) / (x - z));
	} else if (t_1 <= -2e-148) {
		tmp = t_2;
	} else if (t_1 <= 5e-137) {
		tmp = (x - z) * (y * t);
	} else if (t_1 <= 5e+256) {
		tmp = t_2;
	} else {
		tmp = y * (t * (x - z));
	}
	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)
	t_2 = t * (y * (x - z))
	tmp = 0
	if t_1 <= -2e+194:
		tmp = y / ((1.0 / t) / (x - z))
	elif t_1 <= -2e-148:
		tmp = t_2
	elif t_1 <= 5e-137:
		tmp = (x - z) * (y * t)
	elif t_1 <= 5e+256:
		tmp = t_2
	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(x * y) - Float64(y * z))
	t_2 = Float64(t * Float64(y * Float64(x - z)))
	tmp = 0.0
	if (t_1 <= -2e+194)
		tmp = Float64(y / Float64(Float64(1.0 / t) / Float64(x - z)));
	elseif (t_1 <= -2e-148)
		tmp = t_2;
	elseif (t_1 <= 5e-137)
		tmp = Float64(Float64(x - z) * Float64(y * t));
	elseif (t_1 <= 5e+256)
		tmp = t_2;
	else
		tmp = Float64(y * Float64(t * 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)
	t_1 = (x * y) - (y * z);
	t_2 = t * (y * (x - z));
	tmp = 0.0;
	if (t_1 <= -2e+194)
		tmp = y / ((1.0 / t) / (x - z));
	elseif (t_1 <= -2e-148)
		tmp = t_2;
	elseif (t_1 <= 5e-137)
		tmp = (x - z) * (y * t);
	elseif (t_1 <= 5e+256)
		tmp = t_2;
	else
		tmp = y * (t * (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_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+194], N[(y / N[(N[(1.0 / t), $MachinePrecision] / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-148], t$95$2, If[LessEqual[t$95$1, 5e-137], N[(N[(x - z), $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+256], t$95$2, N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	88.9%
Target	95.1%
Herbie	98.9%

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -1.99999999999999989e194

   1. Initial program 59.3%

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

   2. Simplified 97.5%

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

      [Start] 59.3	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 59.3	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 97.5	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 97.5	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   3. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{y \cdot t}{\frac{1}{x - z}}} \]
      Proof

      [Start] 97.5	\[ y \cdot \left(t \cdot \left(x - z\right)\right) \]
      associate-*r* [=>] 99.1	\[ \color{blue}{\left(y \cdot t\right) \cdot \left(x - z\right)} \]
      flip3-- [=>] 24.0	\[ \left(y \cdot t\right) \cdot \color{blue}{\frac{{x}^{3} - {z}^{3}}{x \cdot x + \left(z \cdot z + x \cdot z\right)}} \]
      associate-*r/ [=>] 17.0	\[ \color{blue}{\frac{\left(y \cdot t\right) \cdot \left({x}^{3} - {z}^{3}\right)}{x \cdot x + \left(z \cdot z + x \cdot z\right)}} \]
      associate-/l* [=>] 24.0	\[ \color{blue}{\frac{y \cdot t}{\frac{x \cdot x + \left(z \cdot z + x \cdot z\right)}{{x}^{3} - {z}^{3}}}} \]
      *-un-lft-identity [=>] 24.0	\[ \frac{y \cdot t}{\frac{\color{blue}{1 \cdot \left(x \cdot x + \left(z \cdot z + x \cdot z\right)\right)}}{{x}^{3} - {z}^{3}}} \]
      associate-/l* [=>] 24.0	\[ \frac{y \cdot t}{\color{blue}{\frac{1}{\frac{{x}^{3} - {z}^{3}}{x \cdot x + \left(z \cdot z + x \cdot z\right)}}}} \]
      flip3-- [<=] 99.0	\[ \frac{y \cdot t}{\frac{1}{\color{blue}{x - z}}} \]

   4. Applied egg-rr 98.9%

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

      [Start] 99.0	\[ \frac{y \cdot t}{\frac{1}{x - z}} \]
      frac-2neg [=>] 99.0	\[ \frac{y \cdot t}{\color{blue}{\frac{-1}{-\left(x - z\right)}}} \]
      associate-/r/ [=>] 99.1	\[ \color{blue}{\frac{y \cdot t}{-1} \cdot \left(-\left(x - z\right)\right)} \]
      associate-/l* [=>] 98.9	\[ \color{blue}{\frac{y}{\frac{-1}{t}}} \cdot \left(-\left(x - z\right)\right) \]
      metadata-eval [=>] 98.9	\[ \frac{y}{\frac{\color{blue}{-1}}{t}} \cdot \left(-\left(x - z\right)\right) \]

   5. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\frac{y}{\frac{\frac{1}{t}}{x - z}}} \]
      Proof

      [Start] 98.9	\[ \frac{y}{\frac{-1}{t}} \cdot \left(-\left(x - z\right)\right) \]
      associate-*l/ [=>] 59.2	\[ \color{blue}{\frac{y \cdot \left(-\left(x - z\right)\right)}{\frac{-1}{t}}} \]
      associate-/l* [=>] 97.6	\[ \color{blue}{\frac{y}{\frac{\frac{-1}{t}}{-\left(x - z\right)}}} \]
      neg-mul-1 [=>] 97.6	\[ \frac{y}{\frac{\frac{-1}{t}}{\color{blue}{-1 \cdot \left(x - z\right)}}} \]
      associate-/r* [=>] 97.6	\[ \frac{y}{\color{blue}{\frac{\frac{\frac{-1}{t}}{-1}}{x - z}}} \]
      associate-/r* [<=] 97.6	\[ \frac{y}{\frac{\color{blue}{\frac{-1}{t \cdot -1}}}{x - z}} \]
      metadata-eval [<=] 97.6	\[ \frac{y}{\frac{\frac{-1}{t \cdot \color{blue}{\frac{1}{-1}}}}{x - z}} \]
      div-inv [<=] 97.6	\[ \frac{y}{\frac{\frac{-1}{\color{blue}{\frac{t}{-1}}}}{x - z}} \]
      clear-num [=>] 97.6	\[ \frac{y}{\frac{\frac{-1}{\color{blue}{\frac{1}{\frac{-1}{t}}}}}{x - z}} \]
      associate-/r/ [=>] 97.6	\[ \frac{y}{\frac{\frac{-1}{\color{blue}{\frac{1}{-1} \cdot t}}}{x - z}} \]
      metadata-eval [=>] 97.6	\[ \frac{y}{\frac{\frac{-1}{\color{blue}{-1} \cdot t}}{x - z}} \]
      associate-/r* [=>] 97.6	\[ \frac{y}{\frac{\color{blue}{\frac{\frac{-1}{-1}}{t}}}{x - z}} \]
      metadata-eval [=>] 97.6	\[ \frac{y}{\frac{\frac{\color{blue}{1}}{t}}{x - z}} \]

   if -1.99999999999999989e194 < (-.f64 (*.f64 x y) (*.f64 z y)) < -1.99999999999999987e-148 or 5.00000000000000001e-137 < (-.f64 (*.f64 x y) (*.f64 z y)) < 5.00000000000000015e256

   1. Initial program 99.6%

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

   2. Simplified 99.6%

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

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

   if -1.99999999999999987e-148 < (-.f64 (*.f64 x y) (*.f64 z y)) < 5.00000000000000001e-137

   1. Initial program 91.8%

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

   2. Simplified 97.1%

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

      [Start] 91.8	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 91.8	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 97.1	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 97.1	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

   3. Taylor expanded in x around 0 97.1%

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

   4. Simplified 97.0%

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

      [Start] 97.1	\[ y \cdot \left(t \cdot x\right) + -1 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]
      mul-1-neg [=>] 97.1	\[ y \cdot \left(t \cdot x\right) + \color{blue}{\left(-y \cdot \left(t \cdot z\right)\right)} \]
      associate-*r* [=>] 96.5	\[ \color{blue}{\left(y \cdot t\right) \cdot x} + \left(-y \cdot \left(t \cdot z\right)\right) \]
      associate-*r* [=>] 97.0	\[ \left(y \cdot t\right) \cdot x + \left(-\color{blue}{\left(y \cdot t\right) \cdot z}\right) \]
      distribute-rgt-neg-out [<=] 97.0	\[ \left(y \cdot t\right) \cdot x + \color{blue}{\left(y \cdot t\right) \cdot \left(-z\right)} \]
      distribute-lft-in [<=] 97.0	\[ \color{blue}{\left(y \cdot t\right) \cdot \left(x + \left(-z\right)\right)} \]
      sub-neg [<=] 97.0	\[ \left(y \cdot t\right) \cdot \color{blue}{\left(x - z\right)} \]
      *-commutative [=>] 97.0	\[ \color{blue}{\left(x - z\right) \cdot \left(y \cdot t\right)} \]

   if 5.00000000000000015e256 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 31.7%

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

   2. Simplified 99.4%

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

      [Start] 31.7	\[ \left(x \cdot y - z \cdot y\right) \cdot t \]
      distribute-rgt-out-- [=>] 31.7	\[ \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t \]
      associate-*l* [=>] 99.4	\[ \color{blue}{y \cdot \left(\left(x - z\right) \cdot t\right)} \]
      *-commutative [=>] 99.4	\[ y \cdot \color{blue}{\left(t \cdot \left(x - z\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -2 \cdot 10^{+194}:\\ \;\;\;\;\frac{y}{\frac{\frac{1}{t}}{x - z}}\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq -2 \cdot 10^{-148}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 5 \cdot 10^{-137}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 5 \cdot 10^{+256}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	2512

\[\begin{array}{l} t_1 := t \cdot \left(y \cdot \left(x - z\right)\right)\\ t_2 := x \cdot y - y \cdot z\\ t_3 := \left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+304}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-137}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+256}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	69.7%
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -13.5:\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+221}:\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-t\right)\\ \end{array} \]

Alternative 3
Accuracy	69.4%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -31.5 \lor \neg \left(z \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 4
Accuracy	69.6%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -16 \lor \neg \left(z \leq 2.3 \cdot 10^{-6}\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 5
Accuracy	87.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+146}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 6
Accuracy	96.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{+47}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	54.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 10^{+24}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 8
Accuracy	51.4%
Cost	320

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

Reproduce

herbie shell --seed 2023137 
(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))