Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Percentage Accurate: 65.2% → 99.9%

Time: 9.5s

Alternatives: 16

Speedup: 2.1×

Specification

Math

\[\begin{array}{l} \\ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))

C

double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

Python

def code(x, y):
	return 1.0 - (((1.0 - x) * y) / (y + 1.0))

Julia

function code(x, y)
	return Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
end

Wolfram

code[x_, y_] := N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))

C

double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

Python

def code(x, y):
	return 1.0 - (((1.0 - x) * y) / (y + 1.0))

Julia

function code(x, y)
	return Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
end

Wolfram

code[x_, y_] := N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -550000 \lor \neg \left(y \leq 430000\right):\\ \;\;\;\;x + \left(\frac{x + -1}{y \cdot y} - \frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -550000.0) (not (<= y 430000.0)))
   (+ x (- (/ (+ x -1.0) (* y y)) (/ (+ x -1.0) y)))
   (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -550000.0) || !(y <= 430000.0)) {
		tmp = x + (((x + -1.0) / (y * y)) - ((x + -1.0) / y));
	} else {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -550000.0) || !(y <= 430000.0))
		tmp = Float64(x + Float64(Float64(Float64(x + -1.0) / Float64(y * y)) - Float64(Float64(x + -1.0) / y)));
	else
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -550000.0], N[Not[LessEqual[y, 430000.0]], $MachinePrecision]], N[(x + N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5e5 or 4.3e5 < y

   1. Initial program 30.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{x - 1}{y} + \frac{x}{{y}^{2}}\right)\right) - \frac{1}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{x + \left(\left(-1 \cdot \frac{x - 1}{y} + \frac{x}{{y}^{2}}\right) - \frac{1}{{y}^{2}}\right)} \]

      2. associate--l+ 100.0%

         \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{x - 1}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right)} \]

      3. mul-1-neg 100.0%

         \[\leadsto x + \left(\color{blue}{\left(-\frac{x - 1}{y}\right)} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      4. sub-neg 100.0%

         \[\leadsto x + \left(\left(-\frac{\color{blue}{x + \left(-1\right)}}{y}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      5. metadata-eval 100.0%

         \[\leadsto x + \left(\left(-\frac{x + \color{blue}{-1}}{y}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      6. distribute-neg-frac 100.0%

         \[\leadsto x + \left(\color{blue}{\frac{-\left(x + -1\right)}{y}} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      7. +-commutative 100.0%

         \[\leadsto x + \left(\frac{-\color{blue}{\left(-1 + x\right)}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      8. distribute-neg-in 100.0%

         \[\leadsto x + \left(\frac{\color{blue}{\left(--1\right) + \left(-x\right)}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      9. metadata-eval 100.0%

         \[\leadsto x + \left(\frac{\color{blue}{1} + \left(-x\right)}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      10. sub-neg 100.0%

          \[\leadsto x + \left(\frac{\color{blue}{1 - x}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      11. div-sub 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \color{blue}{\frac{x - 1}{{y}^{2}}}\right) \]

      12. sub-neg 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{\color{blue}{x + \left(-1\right)}}{{y}^{2}}\right) \]

      13. metadata-eval 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{x + \color{blue}{-1}}{{y}^{2}}\right) \]

      14. +-commutative 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{\color{blue}{-1 + x}}{{y}^{2}}\right) \]

      15. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if -5.5e5 < y < 4.3e5

   1. Initial program 99.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*l/ 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. fma-def 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{1 - x}{y + 1}, y, 1\right)} \]

      6. *-rgt-identity 99.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-\frac{1 - x}{y + 1}\right) \cdot 1}, y, 1\right) \]

      7. *-rgt-identity 99.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\frac{1 - x}{y + 1}}, y, 1\right) \]

      8. distribute-frac-neg 99.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-\left(1 - x\right)}{y + 1}}, y, 1\right) \]

      9. neg-sub0 99.9%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1}, y, 1\right) \]

      10. associate--r- 99.9%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1}, y, 1\right) \]

      11. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1} + x}{y + 1}, y, 1\right) \]

      12. +-commutative 99.9%

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x + -1}}{y + 1}, y, 1\right) \]

      13. +-commutative 99.9%

          \[\leadsto \mathsf{fma}\left(\frac{x + -1}{\color{blue}{1 + y}}, y, 1\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x + -1}{1 + y}, y, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -550000 \lor \neg \left(y \leq 430000\right):\\ \;\;\;\;x + \left(\frac{x + -1}{y \cdot y} - \frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{if}\;t_0 \leq 0.1 \lor \neg \left(t_0 \leq 1.001\right):\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{y} + \frac{-1}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (- 1.0 x)) (+ y 1.0))))
   (if (or (<= t_0 0.1) (not (<= t_0 1.001)))
     (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
     (+ x (+ (/ 1.0 y) (/ -1.0 (* y y)))))))

C

double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if ((t_0 <= 0.1) || !(t_0 <= 1.001)) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * (1.0d0 - x)) / (y + 1.0d0)
    if ((t_0 <= 0.1d0) .or. (.not. (t_0 <= 1.001d0))) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = x + ((1.0d0 / y) + ((-1.0d0) / (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if ((t_0 <= 0.1) || !(t_0 <= 1.001)) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * (1.0 - x)) / (y + 1.0)
	tmp = 0
	if (t_0 <= 0.1) or not (t_0 <= 1.001):
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0))
	tmp = 0.0
	if ((t_0 <= 0.1) || !(t_0 <= 1.001))
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(Float64(1.0 / y) + Float64(-1.0 / Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * (1.0 - x)) / (y + 1.0);
	tmp = 0.0;
	if ((t_0 <= 0.1) || ~((t_0 <= 1.001)))
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.1], N[Not[LessEqual[t$95$0, 1.001]], $MachinePrecision]], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 / y), $MachinePrecision] + N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 0.10000000000000001 or 1.0009999999999999 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 84.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 84.4%

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

      2. associate-*l/ 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. neg-sub0 99.8%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 99.8%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   if 0.10000000000000001 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.0009999999999999

   1. Initial program 7.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{x - 1}{y} + \frac{x}{{y}^{2}}\right)\right) - \frac{1}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{x + \left(\left(-1 \cdot \frac{x - 1}{y} + \frac{x}{{y}^{2}}\right) - \frac{1}{{y}^{2}}\right)} \]

      2. associate--l+ 100.0%

         \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{x - 1}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right)} \]

      3. mul-1-neg 100.0%

         \[\leadsto x + \left(\color{blue}{\left(-\frac{x - 1}{y}\right)} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      4. sub-neg 100.0%

         \[\leadsto x + \left(\left(-\frac{\color{blue}{x + \left(-1\right)}}{y}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      5. metadata-eval 100.0%

         \[\leadsto x + \left(\left(-\frac{x + \color{blue}{-1}}{y}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      6. distribute-neg-frac 100.0%

         \[\leadsto x + \left(\color{blue}{\frac{-\left(x + -1\right)}{y}} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      7. +-commutative 100.0%

         \[\leadsto x + \left(\frac{-\color{blue}{\left(-1 + x\right)}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      8. distribute-neg-in 100.0%

         \[\leadsto x + \left(\frac{\color{blue}{\left(--1\right) + \left(-x\right)}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      9. metadata-eval 100.0%

         \[\leadsto x + \left(\frac{\color{blue}{1} + \left(-x\right)}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      10. sub-neg 100.0%

          \[\leadsto x + \left(\frac{\color{blue}{1 - x}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      11. div-sub 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \color{blue}{\frac{x - 1}{{y}^{2}}}\right) \]

      12. sub-neg 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{\color{blue}{x + \left(-1\right)}}{{y}^{2}}\right) \]

      13. metadata-eval 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{x + \color{blue}{-1}}{{y}^{2}}\right) \]

      14. +-commutative 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{\color{blue}{-1 + x}}{{y}^{2}}\right) \]

      15. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto x + \color{blue}{\left(\frac{1}{y} - \frac{1}{{y}^{2}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

   7. Simplified 100.0%

      \[\leadsto x + \color{blue}{\left(\frac{1}{y} - \frac{1}{y \cdot y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 0.1 \lor \neg \left(\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 1.001\right):\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{y} + \frac{-1}{y \cdot y}\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{if}\;t_0 \leq 0.1:\\ \;\;\;\;1 + \frac{x + -1}{1 + \frac{1}{y}}\\ \mathbf{elif}\;t_0 \leq 1.001:\\ \;\;\;\;x + \left(\frac{1}{y} + \frac{-1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (- 1.0 x)) (+ y 1.0))))
   (if (<= t_0 0.1)
     (+ 1.0 (/ (+ x -1.0) (+ 1.0 (/ 1.0 y))))
     (if (<= t_0 1.001)
       (+ x (+ (/ 1.0 y) (/ -1.0 (* y y))))
       (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))))))

C

double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = 1.0 + ((x + -1.0) / (1.0 + (1.0 / y)));
	} else if (t_0 <= 1.001) {
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)));
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * (1.0d0 - x)) / (y + 1.0d0)
    if (t_0 <= 0.1d0) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / (1.0d0 + (1.0d0 / y)))
    else if (t_0 <= 1.001d0) then
        tmp = x + ((1.0d0 / y) + ((-1.0d0) / (y * y)))
    else
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = 1.0 + ((x + -1.0) / (1.0 + (1.0 / y)));
	} else if (t_0 <= 1.001) {
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)));
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * (1.0 - x)) / (y + 1.0)
	tmp = 0
	if t_0 <= 0.1:
		tmp = 1.0 + ((x + -1.0) / (1.0 + (1.0 / y)))
	elif t_0 <= 1.001:
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)))
	else:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / Float64(1.0 + Float64(1.0 / y))));
	elseif (t_0 <= 1.001)
		tmp = Float64(x + Float64(Float64(1.0 / y) + Float64(-1.0 / Float64(y * y))));
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * (1.0 - x)) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= 0.1)
		tmp = 1.0 + ((x + -1.0) / (1.0 + (1.0 / y)));
	elseif (t_0 <= 1.001)
		tmp = x + ((1.0 / y) + (-1.0 / (y * y)));
	else
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.001], N[(x + N[(N[(1.0 / y), $MachinePrecision] + N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 0.10000000000000001

   1. Initial program 86.5%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 86.5%

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

      2. metadata-eval 86.5%

         \[\leadsto \color{blue}{\left(1 - 0\right)} + \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right) \]

      3. associate--r- 86.5%

         \[\leadsto \color{blue}{1 - \left(0 - \left(-\frac{\left(1 - x\right) \cdot y}{y + 1}\right)\right)} \]

      4. neg-sub0 86.5%

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

      5. remove-double-neg 86.5%

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

      6. associate-/l* 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   if 0.10000000000000001 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.0009999999999999

   1. Initial program 7.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{x - 1}{y} + \frac{x}{{y}^{2}}\right)\right) - \frac{1}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{x + \left(\left(-1 \cdot \frac{x - 1}{y} + \frac{x}{{y}^{2}}\right) - \frac{1}{{y}^{2}}\right)} \]

      2. associate--l+ 100.0%

         \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{x - 1}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right)} \]

      3. mul-1-neg 100.0%

         \[\leadsto x + \left(\color{blue}{\left(-\frac{x - 1}{y}\right)} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      4. sub-neg 100.0%

         \[\leadsto x + \left(\left(-\frac{\color{blue}{x + \left(-1\right)}}{y}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      5. metadata-eval 100.0%

         \[\leadsto x + \left(\left(-\frac{x + \color{blue}{-1}}{y}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      6. distribute-neg-frac 100.0%

         \[\leadsto x + \left(\color{blue}{\frac{-\left(x + -1\right)}{y}} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      7. +-commutative 100.0%

         \[\leadsto x + \left(\frac{-\color{blue}{\left(-1 + x\right)}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      8. distribute-neg-in 100.0%

         \[\leadsto x + \left(\frac{\color{blue}{\left(--1\right) + \left(-x\right)}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      9. metadata-eval 100.0%

         \[\leadsto x + \left(\frac{\color{blue}{1} + \left(-x\right)}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      10. sub-neg 100.0%

          \[\leadsto x + \left(\frac{\color{blue}{1 - x}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      11. div-sub 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \color{blue}{\frac{x - 1}{{y}^{2}}}\right) \]

      12. sub-neg 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{\color{blue}{x + \left(-1\right)}}{{y}^{2}}\right) \]

      13. metadata-eval 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{x + \color{blue}{-1}}{{y}^{2}}\right) \]

      14. +-commutative 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{\color{blue}{-1 + x}}{{y}^{2}}\right) \]

      15. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto x + \color{blue}{\left(\frac{1}{y} - \frac{1}{{y}^{2}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

   7. Simplified 100.0%

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

   if 1.0009999999999999 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 77.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 77.8%

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

      2. associate-*l/ 99.5%

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

      3. distribute-lft-neg-in 99.5%

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

      4. distribute-frac-neg 99.5%

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

      5. neg-sub0 99.5%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 99.5%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{1 + \frac{x + -1}{1 + y} \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 0.1:\\ \;\;\;\;1 + \frac{x + -1}{1 + \frac{1}{y}}\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 1.001:\\ \;\;\;\;x + \left(\frac{1}{y} + \frac{-1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -370000 \lor \neg \left(y \leq 510000\right):\\ \;\;\;\;x + \left(\frac{x + -1}{y \cdot y} - \frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -370000.0) (not (<= y 510000.0)))
   (+ x (- (/ (+ x -1.0) (* y y)) (/ (+ x -1.0) y)))
   (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -370000.0) || !(y <= 510000.0)) {
		tmp = x + (((x + -1.0) / (y * y)) - ((x + -1.0) / y));
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-370000.0d0)) .or. (.not. (y <= 510000.0d0))) then
        tmp = x + (((x + (-1.0d0)) / (y * y)) - ((x + (-1.0d0)) / y))
    else
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -370000.0) || !(y <= 510000.0)) {
		tmp = x + (((x + -1.0) / (y * y)) - ((x + -1.0) / y));
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -370000.0) or not (y <= 510000.0):
		tmp = x + (((x + -1.0) / (y * y)) - ((x + -1.0) / y))
	else:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -370000.0) || !(y <= 510000.0))
		tmp = Float64(x + Float64(Float64(Float64(x + -1.0) / Float64(y * y)) - Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -370000.0) || ~((y <= 510000.0)))
		tmp = x + (((x + -1.0) / (y * y)) - ((x + -1.0) / y));
	else
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -370000.0], N[Not[LessEqual[y, 510000.0]], $MachinePrecision]], N[(x + N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7e5 or 5.1e5 < y

   1. Initial program 30.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{x - 1}{y} + \frac{x}{{y}^{2}}\right)\right) - \frac{1}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{x + \left(\left(-1 \cdot \frac{x - 1}{y} + \frac{x}{{y}^{2}}\right) - \frac{1}{{y}^{2}}\right)} \]

      2. associate--l+ 100.0%

         \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{x - 1}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right)} \]

      3. mul-1-neg 100.0%

         \[\leadsto x + \left(\color{blue}{\left(-\frac{x - 1}{y}\right)} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      4. sub-neg 100.0%

         \[\leadsto x + \left(\left(-\frac{\color{blue}{x + \left(-1\right)}}{y}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      5. metadata-eval 100.0%

         \[\leadsto x + \left(\left(-\frac{x + \color{blue}{-1}}{y}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      6. distribute-neg-frac 100.0%

         \[\leadsto x + \left(\color{blue}{\frac{-\left(x + -1\right)}{y}} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      7. +-commutative 100.0%

         \[\leadsto x + \left(\frac{-\color{blue}{\left(-1 + x\right)}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      8. distribute-neg-in 100.0%

         \[\leadsto x + \left(\frac{\color{blue}{\left(--1\right) + \left(-x\right)}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      9. metadata-eval 100.0%

         \[\leadsto x + \left(\frac{\color{blue}{1} + \left(-x\right)}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      10. sub-neg 100.0%

          \[\leadsto x + \left(\frac{\color{blue}{1 - x}}{y} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right) \]

      11. div-sub 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \color{blue}{\frac{x - 1}{{y}^{2}}}\right) \]

      12. sub-neg 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{\color{blue}{x + \left(-1\right)}}{{y}^{2}}\right) \]

      13. metadata-eval 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{x + \color{blue}{-1}}{{y}^{2}}\right) \]

      14. +-commutative 100.0%

          \[\leadsto x + \left(\frac{1 - x}{y} + \frac{\color{blue}{-1 + x}}{{y}^{2}}\right) \]

      15. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if -3.7e5 < y < 5.1e5

   1. Initial program 99.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l/ 99.9%

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

      3. distribute-lft-neg-in 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. neg-sub0 99.9%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 99.9%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{x + -1}{1 + y} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -370000 \lor \neg \left(y \leq 510000\right):\\ \;\;\;\;x + \left(\frac{x + -1}{y \cdot y} - \frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 5: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -94000000 \lor \neg \left(y \leq 240000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -94000000.0) (not (<= y 240000000.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -94000000.0) || !(y <= 240000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-94000000.0d0)) .or. (.not. (y <= 240000000.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -94000000.0) || !(y <= 240000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -94000000.0) or not (y <= 240000000.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -94000000.0) || !(y <= 240000000.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -94000000.0) || ~((y <= 240000000.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -94000000.0], N[Not[LessEqual[y, 240000000.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.4e7 or 2.4e8 < y

   1. Initial program 29.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Taylor expanded in y around -inf 99.8%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   3. Step-by-step derivation

      1. mul-1-neg 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-neg-in 99.8%

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

      7. metadata-eval 99.8%

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

      8. sub-neg 99.8%

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

   4. Simplified 99.8%

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

   if -9.4e7 < y < 2.4e8

   1. Initial program 99.6%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. distribute-lft-neg-in 99.6%

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

      4. distribute-frac-neg 99.6%

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

      5. neg-sub0 99.6%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 99.6%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \frac{x + -1}{1 + y} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -94000000 \lor \neg \left(y \leq 240000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 6: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1 \lor \neg \left(y \leq 1.62\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.06e+99)
   x
   (if (<= y -1.08e+46)
     (/ 1.0 y)
     (if (or (<= y -1.0) (not (<= y 1.62))) (- x (/ x y)) (+ 1.0 (* y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+99) {
		tmp = x;
	} else if (y <= -1.08e+46) {
		tmp = 1.0 / y;
	} else if ((y <= -1.0) || !(y <= 1.62)) {
		tmp = x - (x / y);
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.06d+99)) then
        tmp = x
    else if (y <= (-1.08d+46)) then
        tmp = 1.0d0 / y
    else if ((y <= (-1.0d0)) .or. (.not. (y <= 1.62d0))) then
        tmp = x - (x / y)
    else
        tmp = 1.0d0 + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+99) {
		tmp = x;
	} else if (y <= -1.08e+46) {
		tmp = 1.0 / y;
	} else if ((y <= -1.0) || !(y <= 1.62)) {
		tmp = x - (x / y);
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.06e+99:
		tmp = x
	elif y <= -1.08e+46:
		tmp = 1.0 / y
	elif (y <= -1.0) or not (y <= 1.62):
		tmp = x - (x / y)
	else:
		tmp = 1.0 + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.06e+99)
		tmp = x;
	elseif (y <= -1.08e+46)
		tmp = Float64(1.0 / y);
	elseif ((y <= -1.0) || !(y <= 1.62))
		tmp = Float64(x - Float64(x / y));
	else
		tmp = Float64(1.0 + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.06e+99)
		tmp = x;
	elseif (y <= -1.08e+46)
		tmp = 1.0 / y;
	elseif ((y <= -1.0) || ~((y <= 1.62)))
		tmp = x - (x / y);
	else
		tmp = 1.0 + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.06e+99], x, If[LessEqual[y, -1.08e+46], N[(1.0 / y), $MachinePrecision], If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.62]], $MachinePrecision]], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.05999999999999999e99

   1. Initial program 24.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 24.0%

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

      2. associate-*l/ 53.5%

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

      3. distribute-lft-neg-in 53.5%

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

      4. distribute-frac-neg 53.5%

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

      5. neg-sub0 53.5%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 53.5%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 53.5%

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

      8. +-commutative 53.5%

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

      9. +-commutative 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in y around inf 91.9%

      \[\leadsto \color{blue}{x} \]

   if -1.05999999999999999e99 < y < -1.07999999999999994e46

   1. Initial program 33.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 33.7%

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

      2. associate-*l/ 33.3%

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

      3. distribute-lft-neg-in 33.3%

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

      4. distribute-frac-neg 33.3%

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

      5. neg-sub0 33.3%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 33.3%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 33.3%

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

      8. +-commutative 33.3%

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

      9. +-commutative 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in x around 0 3.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. neg-mul-1 3.4%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y}{1 + y}\right)} \]

      2. sub-neg 3.4%

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

   6. Simplified 3.4%

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

   7. Taylor expanded in y around inf 70.1%

      \[\leadsto \color{blue}{\frac{1}{y}} \]

   if -1.07999999999999994e46 < y < -1 or 1.6200000000000001 < y

   1. Initial program 35.6%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 35.6%

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

      2. associate-*l/ 60.7%

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

      3. distribute-lft-neg-in 60.7%

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

      4. distribute-frac-neg 60.7%

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

      5. neg-sub0 60.7%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 60.7%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 60.7%

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

      8. +-commutative 60.7%

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

      9. +-commutative 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in x around inf 52.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + y}} \]

   5. Taylor expanded in y around inf 75.3%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x}{y}} \]
   6. Step-by-step derivation

      1. neg-mul-1 75.3%

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

      2. unsub-neg 75.3%

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

   7. Simplified 75.3%

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

   if -1 < y < 1.6200000000000001

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 100.0%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto 1 + \color{blue}{\frac{x}{1 + y}} \cdot y \]

   5. Taylor expanded in y around 0 98.5%

      \[\leadsto 1 + \color{blue}{x \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1 \lor \neg \left(y \leq 1.62\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]

Alternative 7: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.9:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.06e+99)
   x
   (if (<= y -5.5e+45)
     (/ 1.0 y)
     (if (<= y -1.0) (* y (/ x y)) (if (<= y 1.9) (+ 1.0 (* y x)) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+99) {
		tmp = x;
	} else if (y <= -5.5e+45) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = y * (x / y);
	} else if (y <= 1.9) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.06d+99)) then
        tmp = x
    else if (y <= (-5.5d+45)) then
        tmp = 1.0d0 / y
    else if (y <= (-1.0d0)) then
        tmp = y * (x / y)
    else if (y <= 1.9d0) then
        tmp = 1.0d0 + (y * x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+99) {
		tmp = x;
	} else if (y <= -5.5e+45) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = y * (x / y);
	} else if (y <= 1.9) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.06e+99:
		tmp = x
	elif y <= -5.5e+45:
		tmp = 1.0 / y
	elif y <= -1.0:
		tmp = y * (x / y)
	elif y <= 1.9:
		tmp = 1.0 + (y * x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.06e+99)
		tmp = x;
	elseif (y <= -5.5e+45)
		tmp = Float64(1.0 / y);
	elseif (y <= -1.0)
		tmp = Float64(y * Float64(x / y));
	elseif (y <= 1.9)
		tmp = Float64(1.0 + Float64(y * x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.06e+99)
		tmp = x;
	elseif (y <= -5.5e+45)
		tmp = 1.0 / y;
	elseif (y <= -1.0)
		tmp = y * (x / y);
	elseif (y <= 1.9)
		tmp = 1.0 + (y * x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.06e+99], x, If[LessEqual[y, -5.5e+45], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, -1.0], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.05999999999999999e99 or 1.8999999999999999 < y

   1. Initial program 27.5%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 27.5%

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

      2. associate-*l/ 55.3%

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

      3. distribute-lft-neg-in 55.3%

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

      4. distribute-frac-neg 55.3%

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

      5. neg-sub0 55.3%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 55.3%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 55.3%

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

      8. +-commutative 55.3%

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

      9. +-commutative 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in y around inf 80.5%

      \[\leadsto \color{blue}{x} \]

   if -1.05999999999999999e99 < y < -5.5000000000000001e45

   1. Initial program 33.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 33.7%

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

      2. associate-*l/ 33.3%

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

      3. distribute-lft-neg-in 33.3%

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

      4. distribute-frac-neg 33.3%

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

      5. neg-sub0 33.3%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 33.3%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 33.3%

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

      8. +-commutative 33.3%

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

      9. +-commutative 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in x around 0 3.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. neg-mul-1 3.4%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y}{1 + y}\right)} \]

      2. sub-neg 3.4%

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

   6. Simplified 3.4%

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

   7. Taylor expanded in y around inf 70.1%

      \[\leadsto \color{blue}{\frac{1}{y}} \]

   if -5.5000000000000001e45 < y < -1

   1. Initial program 70.1%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 70.1%

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

      2. associate-*l/ 84.9%

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

      3. distribute-lft-neg-in 84.9%

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

      4. distribute-frac-neg 84.9%

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

      5. neg-sub0 84.9%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 84.9%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 84.9%

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

      8. +-commutative 84.9%

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

      9. +-commutative 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in x around inf 60.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + y}} \]
   5. Step-by-step derivation

      1. associate-/l* 76.0%

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

      2. associate-/r/ 75.9%

         \[\leadsto \color{blue}{\frac{x}{1 + y} \cdot y} \]

   6. Applied egg-rr 75.9%

      \[\leadsto \color{blue}{\frac{x}{1 + y} \cdot y} \]

   7. Taylor expanded in y around inf 71.6%

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

   if -1 < y < 1.8999999999999999

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 100.0%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto 1 + \color{blue}{\frac{x}{1 + y}} \cdot y \]

   5. Taylor expanded in y around 0 98.5%

      \[\leadsto 1 + \color{blue}{x \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.9:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.9:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.06e+99)
   x
   (if (<= y -1.1e+46)
     (/ 1.0 y)
     (if (<= y -5e-8)
       (* y (/ x (+ y 1.0)))
       (if (<= y 1.9) (+ 1.0 (* y x)) (- x (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+99) {
		tmp = x;
	} else if (y <= -1.1e+46) {
		tmp = 1.0 / y;
	} else if (y <= -5e-8) {
		tmp = y * (x / (y + 1.0));
	} else if (y <= 1.9) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.06d+99)) then
        tmp = x
    else if (y <= (-1.1d+46)) then
        tmp = 1.0d0 / y
    else if (y <= (-5d-8)) then
        tmp = y * (x / (y + 1.0d0))
    else if (y <= 1.9d0) then
        tmp = 1.0d0 + (y * x)
    else
        tmp = x - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+99) {
		tmp = x;
	} else if (y <= -1.1e+46) {
		tmp = 1.0 / y;
	} else if (y <= -5e-8) {
		tmp = y * (x / (y + 1.0));
	} else if (y <= 1.9) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.06e+99:
		tmp = x
	elif y <= -1.1e+46:
		tmp = 1.0 / y
	elif y <= -5e-8:
		tmp = y * (x / (y + 1.0))
	elif y <= 1.9:
		tmp = 1.0 + (y * x)
	else:
		tmp = x - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.06e+99)
		tmp = x;
	elseif (y <= -1.1e+46)
		tmp = Float64(1.0 / y);
	elseif (y <= -5e-8)
		tmp = Float64(y * Float64(x / Float64(y + 1.0)));
	elseif (y <= 1.9)
		tmp = Float64(1.0 + Float64(y * x));
	else
		tmp = Float64(x - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.06e+99)
		tmp = x;
	elseif (y <= -1.1e+46)
		tmp = 1.0 / y;
	elseif (y <= -5e-8)
		tmp = y * (x / (y + 1.0));
	elseif (y <= 1.9)
		tmp = 1.0 + (y * x);
	else
		tmp = x - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.06e+99], x, If[LessEqual[y, -1.1e+46], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, -5e-8], N[(y * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.05999999999999999e99

   1. Initial program 24.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 24.0%

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

      2. associate-*l/ 53.5%

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

      3. distribute-lft-neg-in 53.5%

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

      4. distribute-frac-neg 53.5%

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

      5. neg-sub0 53.5%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 53.5%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 53.5%

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

      8. +-commutative 53.5%

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

      9. +-commutative 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in y around inf 91.9%

      \[\leadsto \color{blue}{x} \]

   if -1.05999999999999999e99 < y < -1.1e46

   1. Initial program 33.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 33.7%

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

      2. associate-*l/ 33.3%

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

      3. distribute-lft-neg-in 33.3%

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

      4. distribute-frac-neg 33.3%

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

      5. neg-sub0 33.3%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 33.3%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 33.3%

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

      8. +-commutative 33.3%

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

      9. +-commutative 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in x around 0 3.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. neg-mul-1 3.4%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y}{1 + y}\right)} \]

      2. sub-neg 3.4%

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

   6. Simplified 3.4%

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

   7. Taylor expanded in y around inf 70.1%

      \[\leadsto \color{blue}{\frac{1}{y}} \]

   if -1.1e46 < y < -4.9999999999999998e-8

   1. Initial program 74.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 74.3%

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

      2. associate-*l/ 86.9%

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

      3. distribute-lft-neg-in 86.9%

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

      4. distribute-frac-neg 86.9%

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

      5. neg-sub0 86.9%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 86.9%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 86.9%

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

      8. +-commutative 86.9%

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

      9. +-commutative 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in x around inf 66.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + y}} \]
   5. Step-by-step derivation

      1. associate-/l* 79.1%

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

      2. associate-/r/ 78.9%

         \[\leadsto \color{blue}{\frac{x}{1 + y} \cdot y} \]

   6. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\frac{x}{1 + y} \cdot y} \]

   if -4.9999999999999998e-8 < y < 1.8999999999999999

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 100.0%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto 1 + \color{blue}{\frac{x}{1 + y}} \cdot y \]

   5. Taylor expanded in y around 0 99.2%

      \[\leadsto 1 + \color{blue}{x \cdot y} \]

   if 1.8999999999999999 < y

   1. Initial program 29.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 29.4%

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

      2. associate-*l/ 56.3%

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

      3. distribute-lft-neg-in 56.3%

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

      4. distribute-frac-neg 56.3%

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

      5. neg-sub0 56.3%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 56.3%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 56.3%

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

      8. +-commutative 56.3%

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

      9. +-commutative 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in x around inf 50.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + y}} \]

   5. Taylor expanded in y around inf 75.5%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x}{y}} \]
   6. Step-by-step derivation

      1. neg-mul-1 75.5%

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

      2. unsub-neg 75.5%

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

   7. Simplified 75.5%

      \[\leadsto \color{blue}{x - \frac{x}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.9:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \]

Alternative 9: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -420 \lor \neg \left(y \leq 150000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -420.0) (not (<= y 150000.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* y (/ x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -420.0) || !(y <= 150000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * (x / (y + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-420.0d0)) .or. (.not. (y <= 150000.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + (y * (x / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -420.0) || !(y <= 150000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * (x / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -420.0) or not (y <= 150000.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + (y * (x / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -420.0) || !(y <= 150000.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -420.0) || ~((y <= 150000.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + (y * (x / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -420.0], N[Not[LessEqual[y, 150000.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -420 or 1.5e5 < y

   1. Initial program 30.6%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Taylor expanded in y around -inf 98.8%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   3. Step-by-step derivation

      1. mul-1-neg 98.8%

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

      2. sub-neg 98.8%

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

      3. metadata-eval 98.8%

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

      4. distribute-neg-frac 98.8%

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

      5. +-commutative 98.8%

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

      6. distribute-neg-in 98.8%

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

      7. metadata-eval 98.8%

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

      8. sub-neg 98.8%

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

   4. Simplified 98.8%

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

   if -420 < y < 1.5e5

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 100.0%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.6%

      \[\leadsto 1 + \color{blue}{\frac{x}{1 + y}} \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -420 \lor \neg \left(y \leq 150000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 10: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e+99)
   x
   (if (<= y -1.1e+46)
     (/ 1.0 y)
     (if (<= y -1.0) x (if (<= y 9.5e-12) (- 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+99) {
		tmp = x;
	} else if (y <= -1.1e+46) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = x;
	} else if (y <= 9.5e-12) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.4d+99)) then
        tmp = x
    else if (y <= (-1.1d+46)) then
        tmp = 1.0d0 / y
    else if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 9.5d-12) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4e+99) {
		tmp = x;
	} else if (y <= -1.1e+46) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = x;
	} else if (y <= 9.5e-12) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e+99:
		tmp = x
	elif y <= -1.1e+46:
		tmp = 1.0 / y
	elif y <= -1.0:
		tmp = x
	elif y <= 9.5e-12:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e+99)
		tmp = x;
	elseif (y <= -1.1e+46)
		tmp = Float64(1.0 / y);
	elseif (y <= -1.0)
		tmp = x;
	elseif (y <= 9.5e-12)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4e+99)
		tmp = x;
	elseif (y <= -1.1e+46)
		tmp = 1.0 / y;
	elseif (y <= -1.0)
		tmp = x;
	elseif (y <= 9.5e-12)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e+99], x, If[LessEqual[y, -1.1e+46], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, -1.0], x, If[LessEqual[y, 9.5e-12], N[(1.0 - y), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4e99 or -1.1e46 < y < -1 or 9.4999999999999995e-12 < y

   1. Initial program 33.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 33.2%

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

      2. associate-*l/ 59.2%

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

      3. distribute-lft-neg-in 59.2%

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

      4. distribute-frac-neg 59.2%

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

      5. neg-sub0 59.2%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 59.2%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 59.2%

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

      8. +-commutative 59.2%

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

      9. +-commutative 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in y around inf 78.3%

      \[\leadsto \color{blue}{x} \]

   if -1.4e99 < y < -1.1e46

   1. Initial program 33.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 33.7%

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

      2. associate-*l/ 33.3%

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

      3. distribute-lft-neg-in 33.3%

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

      4. distribute-frac-neg 33.3%

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

      5. neg-sub0 33.3%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 33.3%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 33.3%

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

      8. +-commutative 33.3%

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

      9. +-commutative 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in x around 0 3.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. neg-mul-1 3.4%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y}{1 + y}\right)} \]

      2. sub-neg 3.4%

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

   6. Simplified 3.4%

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

   7. Taylor expanded in y around inf 70.1%

      \[\leadsto \color{blue}{\frac{1}{y}} \]

   if -1 < y < 9.4999999999999995e-12

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 100.0%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 74.1%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. neg-mul-1 74.1%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y}{1 + y}\right)} \]

      2. sub-neg 74.1%

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

   6. Simplified 74.1%

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

   7. Taylor expanded in y around 0 74.1%

      \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   8. Step-by-step derivation

      1. neg-mul-1 74.1%

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

      2. sub-neg 74.1%

         \[\leadsto \color{blue}{1 - y} \]

   9. Simplified 74.1%

      \[\leadsto \color{blue}{1 - y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.75e+99)
   x
   (if (<= y -1.08e+46)
     (/ 1.0 y)
     (if (<= y -1.0) (* y (/ x y)) (if (<= y 9.5e-12) (- 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.75e+99) {
		tmp = x;
	} else if (y <= -1.08e+46) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = y * (x / y);
	} else if (y <= 9.5e-12) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.75d+99)) then
        tmp = x
    else if (y <= (-1.08d+46)) then
        tmp = 1.0d0 / y
    else if (y <= (-1.0d0)) then
        tmp = y * (x / y)
    else if (y <= 9.5d-12) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.75e+99) {
		tmp = x;
	} else if (y <= -1.08e+46) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = y * (x / y);
	} else if (y <= 9.5e-12) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.75e+99:
		tmp = x
	elif y <= -1.08e+46:
		tmp = 1.0 / y
	elif y <= -1.0:
		tmp = y * (x / y)
	elif y <= 9.5e-12:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.75e+99)
		tmp = x;
	elseif (y <= -1.08e+46)
		tmp = Float64(1.0 / y);
	elseif (y <= -1.0)
		tmp = Float64(y * Float64(x / y));
	elseif (y <= 9.5e-12)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.75e+99)
		tmp = x;
	elseif (y <= -1.08e+46)
		tmp = 1.0 / y;
	elseif (y <= -1.0)
		tmp = y * (x / y);
	elseif (y <= 9.5e-12)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.75e+99], x, If[LessEqual[y, -1.08e+46], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, -1.0], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e-12], N[(1.0 - y), $MachinePrecision], x]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.7499999999999999e99 or 9.4999999999999995e-12 < y

   1. Initial program 28.9%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 28.9%

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

      2. associate-*l/ 56.2%

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

      3. distribute-lft-neg-in 56.2%

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

      4. distribute-frac-neg 56.2%

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

      5. neg-sub0 56.2%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 56.2%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 56.2%

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

      8. +-commutative 56.2%

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

      9. +-commutative 56.2%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in y around inf 79.1%

      \[\leadsto \color{blue}{x} \]

   if -1.7499999999999999e99 < y < -1.07999999999999994e46

   1. Initial program 33.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 33.7%

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

      2. associate-*l/ 33.3%

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

      3. distribute-lft-neg-in 33.3%

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

      4. distribute-frac-neg 33.3%

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

      5. neg-sub0 33.3%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 33.3%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 33.3%

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

      8. +-commutative 33.3%

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

      9. +-commutative 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in x around 0 3.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. neg-mul-1 3.4%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y}{1 + y}\right)} \]

      2. sub-neg 3.4%

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

   6. Simplified 3.4%

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

   7. Taylor expanded in y around inf 70.1%

      \[\leadsto \color{blue}{\frac{1}{y}} \]

   if -1.07999999999999994e46 < y < -1

   1. Initial program 70.1%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 70.1%

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

      2. associate-*l/ 84.9%

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

      3. distribute-lft-neg-in 84.9%

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

      4. distribute-frac-neg 84.9%

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

      5. neg-sub0 84.9%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 84.9%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 84.9%

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

      8. +-commutative 84.9%

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

      9. +-commutative 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in x around inf 60.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + y}} \]
   5. Step-by-step derivation

      1. associate-/l* 76.0%

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

      2. associate-/r/ 75.9%

         \[\leadsto \color{blue}{\frac{x}{1 + y} \cdot y} \]

   6. Applied egg-rr 75.9%

      \[\leadsto \color{blue}{\frac{x}{1 + y} \cdot y} \]

   7. Taylor expanded in y around inf 71.6%

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

   if -1 < y < 9.4999999999999995e-12

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 100.0%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 74.1%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. neg-mul-1 74.1%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y}{1 + y}\right)} \]

      2. sub-neg 74.1%

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

   6. Simplified 74.1%

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

   7. Taylor expanded in y around 0 74.1%

      \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   8. Step-by-step derivation

      1. neg-mul-1 74.1%

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

      2. sub-neg 74.1%

         \[\leadsto \color{blue}{1 - y} \]

   9. Simplified 74.1%

      \[\leadsto \color{blue}{1 - y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+99}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.22\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.22))) (+ x (/ (- 1.0 x) y)) (+ 1.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.22)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.22d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.22)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.22):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.22))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.22)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.22]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.21999999999999997 < y

   1. Initial program 32.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Taylor expanded in y around -inf 97.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   3. Step-by-step derivation

      1. mul-1-neg 97.6%

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

      2. sub-neg 97.6%

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

      3. metadata-eval 97.6%

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

      4. distribute-neg-frac 97.6%

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

      5. +-commutative 97.6%

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

      6. distribute-neg-in 97.6%

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

      7. metadata-eval 97.6%

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

      8. sub-neg 97.6%

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

   4. Simplified 97.6%

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

   if -1 < y < 1.21999999999999997

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 100.0%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto 1 + \color{blue}{\frac{x}{1 + y}} \cdot y \]

   5. Taylor expanded in y around 0 98.5%

      \[\leadsto 1 + \color{blue}{x \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.22\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]

Alternative 13: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (- (* y x) y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y * x) - y);
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y * x) - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + ((y * x) - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * x) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + ((y * x) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 32.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Taylor expanded in y around -inf 97.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   3. Step-by-step derivation

      1. mul-1-neg 97.6%

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

      2. sub-neg 97.6%

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

      3. metadata-eval 97.6%

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

      4. distribute-neg-frac 97.6%

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

      5. +-commutative 97.6%

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

      6. distribute-neg-in 97.6%

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

      7. metadata-eval 97.6%

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

      8. sub-neg 97.6%

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

   4. Simplified 97.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]

   2. Taylor expanded in y around 0 98.9%

      \[\leadsto 1 - \color{blue}{y \cdot \left(1 - x\right)} \]
   3. Step-by-step derivation

      1. *-commutative 98.9%

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

      2. /-rgt-identity 98.9%

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

      3. associate-/r/ 98.8%

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

      4. div-sub 98.8%

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

      5. remove-double-div 98.8%

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

      6. associate-/r/ 98.9%

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

      7. /-rgt-identity 98.9%

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

      8. *-commutative 98.9%

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

   4. Simplified 98.9%

      \[\leadsto 1 - \color{blue}{\left(y - y \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot x - y\right)\\ \end{array} \]

Alternative 14: 74.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 9.5e-12) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 9.5e-12) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 9.5d-12) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 9.5e-12) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 9.5e-12:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 9.5e-12)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 9.5e-12)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 9.5e-12], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 9.4999999999999995e-12 < y

   1. Initial program 33.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 33.3%

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

      2. associate-*l/ 56.0%

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

      3. distribute-lft-neg-in 56.0%

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

      4. distribute-frac-neg 56.0%

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

      5. neg-sub0 56.0%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 56.0%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 56.0%

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

      8. +-commutative 56.0%

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

      9. +-commutative 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in y around inf 72.9%

      \[\leadsto \color{blue}{x} \]

   if -1 < y < 9.4999999999999995e-12

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 100.0%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 74.1%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. neg-mul-1 74.1%

         \[\leadsto 1 + \color{blue}{\left(-\frac{y}{1 + y}\right)} \]

      2. sub-neg 74.1%

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

   6. Simplified 74.1%

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

   7. Taylor expanded in y around 0 74.1%

      \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   8. Step-by-step derivation

      1. neg-mul-1 74.1%

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

      2. sub-neg 74.1%

         \[\leadsto \color{blue}{1 - y} \]

   9. Simplified 74.1%

      \[\leadsto \color{blue}{1 - y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15: 73.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 9.5e-12) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 9.5e-12) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 9.5d-12) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 9.5e-12) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 9.5e-12:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 9.5e-12)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 9.5e-12)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 9.5e-12], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 9.4999999999999995e-12 < y

   1. Initial program 33.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 33.3%

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

      2. associate-*l/ 56.0%

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

      3. distribute-lft-neg-in 56.0%

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

      4. distribute-frac-neg 56.0%

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

      5. neg-sub0 56.0%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 56.0%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 56.0%

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

      8. +-commutative 56.0%

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

      9. +-commutative 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in y around inf 72.9%

      \[\leadsto \color{blue}{x} \]

   if -1 < y < 9.4999999999999995e-12

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

         \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

      6. associate--r- 100.0%

         \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 73.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16: 38.0% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 65.8%

   \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
2. Step-by-step derivation

   1. sub-neg 65.8%

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

   2. associate-*l/ 77.5%

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

   3. distribute-lft-neg-in 77.5%

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

   4. distribute-frac-neg 77.5%

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

   5. neg-sub0 77.5%

      \[\leadsto 1 + \frac{\color{blue}{0 - \left(1 - x\right)}}{y + 1} \cdot y \]

   6. associate--r- 77.5%

      \[\leadsto 1 + \frac{\color{blue}{\left(0 - 1\right) + x}}{y + 1} \cdot y \]

   7. metadata-eval 77.5%

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

   8. +-commutative 77.5%

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

   9. +-commutative 77.5%

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

3. Simplified 77.5%

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

4. Taylor expanded in y around 0 37.9%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 37.9%

   \[\leadsto 1 \]

Developer target: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 y) (- (/ x y) x))))
   (if (< y -3693.8482788297247)
     t_0
     (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / y) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / y) - ((x / y) - x)
	tmp = 0
	if y < -3693.8482788297247:
		tmp = t_0
	elif y < 6799310503.41891:
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / y) - Float64(Float64(x / y) - x))
	tmp = 0.0
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / y) - ((x / y) - x);
	tmp = 0.0;
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))