Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.3 → 0.4

Time: 10.5s

Precision: binary64

Cost: 2640

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ t_2 := y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+237}:\\ \;\;\;\;\frac{y \cdot t}{\frac{1}{x - z}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-180}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-315}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+255}:\\ \;\;\;\;t_1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (* y (* t (- x z)))))
   (if (<= t_1 -2e+237)
     (/ (* y t) (/ 1.0 (- x z)))
     (if (<= t_1 -1e-180)
       (* t (* y (- x z)))
       (if (<= t_1 5e-315) t_2 (if (<= t_1 4e+255) (* t_1 t) t_2))))))

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 = y * (t * (x - z));
	double tmp;
	if (t_1 <= -2e+237) {
		tmp = (y * t) / (1.0 / (x - z));
	} else if (t_1 <= -1e-180) {
		tmp = t * (y * (x - z));
	} else if (t_1 <= 5e-315) {
		tmp = t_2;
	} else if (t_1 <= 4e+255) {
		tmp = t_1 * t;
	} else {
		tmp = t_2;
	}
	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 = y * (t * (x - z))
    if (t_1 <= (-2d+237)) then
        tmp = (y * t) / (1.0d0 / (x - z))
    else if (t_1 <= (-1d-180)) then
        tmp = t * (y * (x - z))
    else if (t_1 <= 5d-315) then
        tmp = t_2
    else if (t_1 <= 4d+255) then
        tmp = t_1 * t
    else
        tmp = t_2
    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 = y * (t * (x - z));
	double tmp;
	if (t_1 <= -2e+237) {
		tmp = (y * t) / (1.0 / (x - z));
	} else if (t_1 <= -1e-180) {
		tmp = t * (y * (x - z));
	} else if (t_1 <= 5e-315) {
		tmp = t_2;
	} else if (t_1 <= 4e+255) {
		tmp = t_1 * t;
	} else {
		tmp = t_2;
	}
	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 = y * (t * (x - z))
	tmp = 0
	if t_1 <= -2e+237:
		tmp = (y * t) / (1.0 / (x - z))
	elif t_1 <= -1e-180:
		tmp = t * (y * (x - z))
	elif t_1 <= 5e-315:
		tmp = t_2
	elif t_1 <= 4e+255:
		tmp = t_1 * t
	else:
		tmp = t_2
	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(y * Float64(t * Float64(x - z)))
	tmp = 0.0
	if (t_1 <= -2e+237)
		tmp = Float64(Float64(y * t) / Float64(1.0 / Float64(x - z)));
	elseif (t_1 <= -1e-180)
		tmp = Float64(t * Float64(y * Float64(x - z)));
	elseif (t_1 <= 5e-315)
		tmp = t_2;
	elseif (t_1 <= 4e+255)
		tmp = Float64(t_1 * t);
	else
		tmp = t_2;
	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 = y * (t * (x - z));
	tmp = 0.0;
	if (t_1 <= -2e+237)
		tmp = (y * t) / (1.0 / (x - z));
	elseif (t_1 <= -1e-180)
		tmp = t * (y * (x - z));
	elseif (t_1 <= 5e-315)
		tmp = t_2;
	elseif (t_1 <= 4e+255)
		tmp = t_1 * t;
	else
		tmp = t_2;
	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[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+237], N[(N[(y * t), $MachinePrecision] / N[(1.0 / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-180], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-315], t$95$2, If[LessEqual[t$95$1, 4e+255], N[(t$95$1 * t), $MachinePrecision], t$95$2]]]]]]

Target

Original	7.3
Target	3.2
Herbie	0.4

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -1.99999999999999988e237

   1. Initial program 33.6

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

   2. Simplified 0.5

      \[\leadsto \color{blue}{y \cdot \left(t \cdot \left(x - z\right)\right)} \]
      Proof
      (*.f64 y (*.f64 t (-.f64 x z))): 0 points increase in error, 0 points decrease in error
      (*.f64 y (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 x z) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y (-.f64 x z)) t)): 60 points increase in error, 58 points decrease in error
      (*.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 x y) (*.f64 z y))) t): 0 points increase in error, 2 points decrease in error

   3. Applied egg-rr 49.9

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

   4. Applied egg-rr 0.5

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

   5. Applied egg-rr 0.5

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

   if -1.99999999999999988e237 < (-.f64 (*.f64 x y) (*.f64 z y)) < -1e-180

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

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

   if -1e-180 < (-.f64 (*.f64 x y) (*.f64 z y)) < 5.0000000023e-315 or 3.99999999999999995e255 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 24.9

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

   2. Simplified 0.7

      \[\leadsto \color{blue}{y \cdot \left(t \cdot \left(x - z\right)\right)} \]
      Proof
      (*.f64 y (*.f64 t (-.f64 x z))): 0 points increase in error, 0 points decrease in error
      (*.f64 y (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 x z) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y (-.f64 x z)) t)): 60 points increase in error, 58 points decrease in error
      (*.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 x y) (*.f64 z y))) t): 0 points increase in error, 2 points decrease in error

   if 5.0000000023e-315 < (-.f64 (*.f64 x y) (*.f64 z y)) < 3.99999999999999995e255

   1. Initial program 0.3

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -2 \cdot 10^{+237}:\\ \;\;\;\;\frac{y \cdot t}{\frac{1}{x - z}}\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq -1 \cdot 10^{-180}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 5 \cdot 10^{-315}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 4 \cdot 10^{+255}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	19.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2837830722674 \cdot 10^{-76}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 12150.314889011725:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\frac{-1}{z}}{y}}\\ \end{array} \]

Alternative 2
Error	19.4
Cost	648

\[\begin{array}{l} t_1 := y \cdot \left(z \cdot \left(-t\right)\right)\\ \mathbf{if}\;z \leq -5.2837830722674 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 12150.314889011725:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	19.2
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2837830722674 \cdot 10^{-76}:\\ \;\;\;\;z \cdot \left(-y \cdot t\right)\\ \mathbf{elif}\;z \leq 12150.314889011725:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-t\right)\right)\\ \end{array} \]

Alternative 4
Error	19.6
Cost	648

\[\begin{array}{l} t_1 := t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{if}\;z \leq -5.2837830722674 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 12150.314889011725:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	2.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -8.348697912254973 \cdot 10^{-85}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 6
Error	29.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 10^{-15}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 7
Error	29.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq -8.348697912254973 \cdot 10^{-85}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]

Alternative 8
Error	7.3
Cost	448

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

Alternative 9
Error	31.6
Cost	320

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

Reproduce

herbie shell --seed 2022300 
(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))