Linear.Projection:perspective from linear-1.19.1.3, B

Average Accuracy: 76.2% → 99.7%

Time: 4.1s

Precision: binary64

Cost: 841

Math

\[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+34} \lor \neg \left(x \leq 2.65 \cdot 10^{-20}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{x}{y} + -1}}{0.5}\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7.2e+34) (not (<= x 2.65e-20)))
   (* y (/ (* x 2.0) (- x y)))
   (/ (/ x (+ (/ x y) -1.0)) 0.5)))

C

double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

↓

double code(double x, double y) {
	double tmp;
	if ((x <= -7.2e+34) || !(x <= 2.65e-20)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = (x / ((x / y) + -1.0)) / 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) * y) / (x - y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-7.2d+34)) .or. (.not. (x <= 2.65d-20))) then
        tmp = y * ((x * 2.0d0) / (x - y))
    else
        tmp = (x / ((x / y) + (-1.0d0))) / 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}

↓

public static double code(double x, double y) {
	double tmp;
	if ((x <= -7.2e+34) || !(x <= 2.65e-20)) {
		tmp = y * ((x * 2.0) / (x - y));
	} else {
		tmp = (x / ((x / y) + -1.0)) / 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	return ((x * 2.0) * y) / (x - y)

↓

def code(x, y):
	tmp = 0
	if (x <= -7.2e+34) or not (x <= 2.65e-20):
		tmp = y * ((x * 2.0) / (x - y))
	else:
		tmp = (x / ((x / y) + -1.0)) / 0.5
	return tmp

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) * y) / Float64(x - y))
end

↓

function code(x, y)
	tmp = 0.0
	if ((x <= -7.2e+34) || !(x <= 2.65e-20))
		tmp = Float64(y * Float64(Float64(x * 2.0) / Float64(x - y)));
	else
		tmp = Float64(Float64(x / Float64(Float64(x / y) + -1.0)) / 0.5);
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) * y) / (x - y);
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -7.2e+34) || ~((x <= 2.65e-20)))
		tmp = y * ((x * 2.0) / (x - y));
	else
		tmp = (x / ((x / y) + -1.0)) / 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := If[Or[LessEqual[x, -7.2e+34], N[Not[LessEqual[x, 2.65e-20]], $MachinePrecision]], N[(y * N[(N[(x * 2.0), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]]

Target

Original	76.2%
Target	99.4%
Herbie	99.7%

\[\begin{array}{l} \mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x < 83645045635564430:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -7.2000000000000001e34 or 2.6500000000000001e-20 < x

   1. Initial program 73.9%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]

   2. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
      Proof

      [Start] 73.9	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-*l/ [<=] 99.7	\[ \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]

   if -7.2000000000000001e34 < x < 2.6500000000000001e-20

   1. Initial program 78.6%

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]

   2. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
      Proof

      [Start] 78.6	\[ \frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
      associate-*l/ [<=] 78.0	\[ \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]

   3. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{0 + y \cdot \left(2 \cdot \frac{x}{x - y}\right)} \]
      Proof

      [Start] 78.0	\[ \frac{x \cdot 2}{x - y} \cdot y \]
      add-log-exp [=>] 6.9	\[ \color{blue}{\log \left(e^{\frac{x \cdot 2}{x - y} \cdot y}\right)} \]
      *-un-lft-identity [=>] 6.9	\[ \log \color{blue}{\left(1 \cdot e^{\frac{x \cdot 2}{x - y} \cdot y}\right)} \]
      log-prod [=>] 6.9	\[ \color{blue}{\log 1 + \log \left(e^{\frac{x \cdot 2}{x - y} \cdot y}\right)} \]
      metadata-eval [=>] 6.9	\[ \color{blue}{0} + \log \left(e^{\frac{x \cdot 2}{x - y} \cdot y}\right) \]
      add-log-exp [<=] 78.0	\[ 0 + \color{blue}{\frac{x \cdot 2}{x - y} \cdot y} \]
      *-commutative [=>] 78.0	\[ 0 + \color{blue}{y \cdot \frac{x \cdot 2}{x - y}} \]
      *-commutative [=>] 78.0	\[ 0 + y \cdot \frac{\color{blue}{2 \cdot x}}{x - y} \]
      *-un-lft-identity [=>] 78.0	\[ 0 + y \cdot \frac{2 \cdot x}{\color{blue}{1 \cdot \left(x - y\right)}} \]
      times-frac [=>] 77.9	\[ 0 + y \cdot \color{blue}{\left(\frac{2}{1} \cdot \frac{x}{x - y}\right)} \]
      metadata-eval [=>] 77.9	\[ 0 + y \cdot \left(\color{blue}{2} \cdot \frac{x}{x - y}\right) \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{x}{y} + -1}}{0.5}} \]
      Proof

      [Start] 77.9	\[ 0 + y \cdot \left(2 \cdot \frac{x}{x - y}\right) \]
      +-lft-identity [=>] 77.9	\[ \color{blue}{y \cdot \left(2 \cdot \frac{x}{x - y}\right)} \]
      *-commutative [=>] 77.9	\[ \color{blue}{\left(2 \cdot \frac{x}{x - y}\right) \cdot y} \]
      *-commutative [<=] 77.9	\[ \color{blue}{\left(\frac{x}{x - y} \cdot 2\right)} \cdot y \]
      metadata-eval [<=] 77.9	\[ \left(\frac{x}{x - y} \cdot \color{blue}{\frac{1}{0.5}}\right) \cdot y \]
      times-frac [<=] 78.0	\[ \color{blue}{\frac{x \cdot 1}{\left(x - y\right) \cdot 0.5}} \cdot y \]
      *-rgt-identity [=>] 78.0	\[ \frac{\color{blue}{x}}{\left(x - y\right) \cdot 0.5} \cdot y \]
      associate-/r/ [<=] 99.7	\[ \color{blue}{\frac{x}{\frac{\left(x - y\right) \cdot 0.5}{y}}} \]
      associate-/l* [<=] 78.6	\[ \color{blue}{\frac{x \cdot y}{\left(x - y\right) \cdot 0.5}} \]
      associate-/r* [=>] 78.6	\[ \color{blue}{\frac{\frac{x \cdot y}{x - y}}{0.5}} \]
      associate-/l* [=>] 99.7	\[ \frac{\color{blue}{\frac{x}{\frac{x - y}{y}}}}{0.5} \]
      div-sub [=>] 99.7	\[ \frac{\frac{x}{\color{blue}{\frac{x}{y} - \frac{y}{y}}}}{0.5} \]
      sub-neg [=>] 99.7	\[ \frac{\frac{x}{\color{blue}{\frac{x}{y} + \left(-\frac{y}{y}\right)}}}{0.5} \]
      *-inverses [=>] 99.7	\[ \frac{\frac{x}{\frac{x}{y} + \left(-\color{blue}{1}\right)}}{0.5} \]
      metadata-eval [=>] 99.7	\[ \frac{\frac{x}{\frac{x}{y} + \color{blue}{-1}}}{0.5} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+34} \lor \neg \left(x \leq 2.65 \cdot 10^{-20}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{x}{y} + -1}}{0.5}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	71.4%
Cost	986

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+163}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;x \leq -2.75 \cdot 10^{+132} \lor \neg \left(x \leq -9 \cdot 10^{+28}\right) \land \left(x \leq -1.85 \cdot 10^{-19} \lor \neg \left(x \leq -1.65 \cdot 10^{-118}\right) \land x \leq 1.38 \cdot 10^{+57}\right):\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]

Alternative 2
Accuracy	94.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-171} \lor \neg \left(x \leq 1.7 \cdot 10^{-153}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+38} \lor \neg \left(x \leq 10^{-20}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 2}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \end{array} \]

Alternative 4
Accuracy	50.6%
Cost	192

\[x \cdot -2 \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))

  (/ (* (* x 2.0) y) (- x y)))