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

Percentage Accurate: 66.0% → 99.9%

Time: 12.0s

Alternatives: 12

Speedup: 1.0×

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: 66.0% 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t\_0 \leq 0.9 \lor \neg \left(t\_0 \leq 1.1\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) + \left(\frac{x + -1}{{y}^{2}} + \frac{1 - x}{{y}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (or (<= t_0 0.9) (not (<= t_0 1.1)))
     (- 1.0 (* (- 1.0 x) (/ y (+ 1.0 y))))
     (+
      (+ x (/ (- 1.0 x) y))
      (+ (/ (+ x -1.0) (pow y 2.0)) (/ (- 1.0 x) (pow y 3.0)))))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if ((t_0 <= 0.9) || !(t_0 <= 1.1)) {
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	} else {
		tmp = (x + ((1.0 - x) / y)) + (((x + -1.0) / pow(y, 2.0)) + ((1.0 - x) / pow(y, 3.0)));
	}
	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 - x) * y) / (1.0d0 + y)
    if ((t_0 <= 0.9d0) .or. (.not. (t_0 <= 1.1d0))) then
        tmp = 1.0d0 - ((1.0d0 - x) * (y / (1.0d0 + y)))
    else
        tmp = (x + ((1.0d0 - x) / y)) + (((x + (-1.0d0)) / (y ** 2.0d0)) + ((1.0d0 - x) / (y ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if ((t_0 <= 0.9) || !(t_0 <= 1.1)) {
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	} else {
		tmp = (x + ((1.0 - x) / y)) + (((x + -1.0) / Math.pow(y, 2.0)) + ((1.0 - x) / Math.pow(y, 3.0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (1.0 + y)
	tmp = 0
	if (t_0 <= 0.9) or not (t_0 <= 1.1):
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)))
	else:
		tmp = (x + ((1.0 - x) / y)) + (((x + -1.0) / math.pow(y, 2.0)) + ((1.0 - x) / math.pow(y, 3.0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	tmp = 0.0
	if ((t_0 <= 0.9) || !(t_0 <= 1.1))
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) * Float64(y / Float64(1.0 + y))));
	else
		tmp = Float64(Float64(x + Float64(Float64(1.0 - x) / y)) + Float64(Float64(Float64(x + -1.0) / (y ^ 2.0)) + Float64(Float64(1.0 - x) / (y ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (1.0 + y);
	tmp = 0.0;
	if ((t_0 <= 0.9) || ~((t_0 <= 1.1)))
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	else
		tmp = (x + ((1.0 - x) / y)) + (((x + -1.0) / (y ^ 2.0)) + ((1.0 - x) / (y ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.9], N[Not[LessEqual[t$95$0, 1.1]], $MachinePrecision]], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 0.900000000000000022 or 1.1000000000000001 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 84.5%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 0.900000000000000022 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.1000000000000001

   1. Initial program 9.4%

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

      1. associate-/l* 9.5%

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

      2. remove-double-neg 9.5%

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

      3. remove-double-neg 9.5%

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

      4. +-commutative 9.5%

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

   3. Simplified 9.5%

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

   5. Taylor expanded in y around -inf 100.0%

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

      1. associate-+r+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.9 \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.1\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) + \left(\frac{x + -1}{{y}^{2}} + \frac{1 - x}{{y}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t\_0 \leq 0.999998 \lor \neg \left(t\_0 \leq 1.1\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) \cdot \left(\frac{-1}{y} - -1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (or (<= t_0 0.999998) (not (<= t_0 1.1)))
     (- 1.0 (* (- 1.0 x) (/ y (+ 1.0 y))))
     (+ x (/ (* (- 1.0 x) (- (/ -1.0 y) -1.0)) y)))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if ((t_0 <= 0.999998) || !(t_0 <= 1.1)) {
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	} else {
		tmp = x + (((1.0 - x) * ((-1.0 / y) - -1.0)) / 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 = ((1.0d0 - x) * y) / (1.0d0 + y)
    if ((t_0 <= 0.999998d0) .or. (.not. (t_0 <= 1.1d0))) then
        tmp = 1.0d0 - ((1.0d0 - x) * (y / (1.0d0 + y)))
    else
        tmp = x + (((1.0d0 - x) * (((-1.0d0) / y) - (-1.0d0))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if ((t_0 <= 0.999998) || !(t_0 <= 1.1)) {
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	} else {
		tmp = x + (((1.0 - x) * ((-1.0 / y) - -1.0)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (1.0 + y)
	tmp = 0
	if (t_0 <= 0.999998) or not (t_0 <= 1.1):
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)))
	else:
		tmp = x + (((1.0 - x) * ((-1.0 / y) - -1.0)) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	tmp = 0.0
	if ((t_0 <= 0.999998) || !(t_0 <= 1.1))
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) * Float64(y / Float64(1.0 + y))));
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) * Float64(Float64(-1.0 / y) - -1.0)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (1.0 + y);
	tmp = 0.0;
	if ((t_0 <= 0.999998) || ~((t_0 <= 1.1)))
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	else
		tmp = x + (((1.0 - x) * ((-1.0 / y) - -1.0)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.999998], N[Not[LessEqual[t$95$0, 1.1]], $MachinePrecision]], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] * N[(N[(-1.0 / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 0.999998000000000054 or 1.1000000000000001 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 84.5%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. remove-double-neg 99.9%

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

      4. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 0.999998000000000054 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.1000000000000001

   1. Initial program 8.4%

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

      1. associate-/l* 8.5%

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

      2. remove-double-neg 8.5%

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

      3. remove-double-neg 8.5%

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

      4. +-commutative 8.5%

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

   3. Simplified 8.5%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. neg-mul-1 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--l+ 100.0%

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

      5. div-sub 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. neg-mul-1 100.0%

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

      9. metadata-eval 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. associate-*r/ 100.0%

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

      14. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in y around -inf 100.0%

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

       1. associate--l+ 100.0%

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

       2. div-sub 100.0%

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

       3. sub-neg 100.0%

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

       4. metadata-eval 100.0%

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

       5. +-commutative 100.0%

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

       6. metadata-eval 100.0%

          \[\leadsto x + \left(-1 \cdot \frac{x - 1}{y} + \frac{\color{blue}{\left(0 - 1\right)} + x}{{y}^{2}}\right) \]

       7. associate--r- 100.0%

          \[\leadsto x + \left(-1 \cdot \frac{x - 1}{y} + \frac{\color{blue}{0 - \left(1 - x\right)}}{{y}^{2}}\right) \]

       8. neg-sub0 100.0%

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

       9. neg-mul-1 100.0%

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

       10. associate-*r/ 100.0%

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

       11. +-commutative 100.0%

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

       12. distribute-lft-in 100.0%

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

       13. mul-1-neg 100.0%

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

       14. unpow2 100.0%

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

       15. associate-/l/ 100.0%

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

       16. *-rgt-identity 100.0%

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

       17. associate-*r/ 100.0%

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

       18. sub-neg 100.0%

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

       19. metadata-eval 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.999998 \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.1\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) \cdot \left(\frac{-1}{y} - -1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t\_0 \leq 0.999998 \lor \neg \left(t\_0 \leq 1.1\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (or (<= t_0 0.999998) (not (<= t_0 1.1)))
     (- 1.0 (* (- 1.0 x) (/ y (+ 1.0 y))))
     (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if ((t_0 <= 0.999998) || !(t_0 <= 1.1)) {
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	} else {
		tmp = x + (1.0 / 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 = ((1.0d0 - x) * y) / (1.0d0 + y)
    if ((t_0 <= 0.999998d0) .or. (.not. (t_0 <= 1.1d0))) then
        tmp = 1.0d0 - ((1.0d0 - x) * (y / (1.0d0 + y)))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if ((t_0 <= 0.999998) || !(t_0 <= 1.1)) {
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (1.0 + y)
	tmp = 0
	if (t_0 <= 0.999998) or not (t_0 <= 1.1):
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	tmp = 0.0
	if ((t_0 <= 0.999998) || !(t_0 <= 1.1))
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) * Float64(y / Float64(1.0 + y))));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (1.0 + y);
	tmp = 0.0;
	if ((t_0 <= 0.999998) || ~((t_0 <= 1.1)))
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.999998], N[Not[LessEqual[t$95$0, 1.1]], $MachinePrecision]], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 0.999998000000000054 or 1.1000000000000001 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 84.5%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. remove-double-neg 99.9%

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

      4. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 0.999998000000000054 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.1000000000000001

   1. Initial program 8.4%

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

      1. associate-/l* 8.5%

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

      2. remove-double-neg 8.5%

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

      3. remove-double-neg 8.5%

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

      4. +-commutative 8.5%

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

   3. Simplified 8.5%

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

   5. Taylor expanded in y around inf 99.5%

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

      1. associate--l+ 99.5%

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

      2. div-sub 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. neg-mul-1 99.5%

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

      6. metadata-eval 99.5%

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

      7. distribute-lft-in 99.5%

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

      8. metadata-eval 99.5%

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

      9. sub-neg 99.5%

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

      10. associate-*r/ 99.5%

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

      11. mul-1-neg 99.5%

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

      12. unsub-neg 99.5%

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

      13. sub-neg 99.5%

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

      14. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.999998 \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.1\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -200000:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 75000:\\ \;\;\;\;1 + x \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -200000.0)
   (+ x (/ 1.0 y))
   (if (<= y 75000.0) (+ 1.0 (* x (/ y (+ 1.0 y)))) (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -200000.0) {
		tmp = x + (1.0 / y);
	} else if (y <= 75000.0) {
		tmp = 1.0 + (x * (y / (1.0 + y)));
	} else {
		tmp = x + ((1.0 - 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 <= (-200000.0d0)) then
        tmp = x + (1.0d0 / y)
    else if (y <= 75000.0d0) then
        tmp = 1.0d0 + (x * (y / (1.0d0 + y)))
    else
        tmp = x + ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -200000.0) {
		tmp = x + (1.0 / y);
	} else if (y <= 75000.0) {
		tmp = 1.0 + (x * (y / (1.0 + y)));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -200000.0:
		tmp = x + (1.0 / y)
	elif y <= 75000.0:
		tmp = 1.0 + (x * (y / (1.0 + y)))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -200000.0)
		tmp = Float64(x + Float64(1.0 / y));
	elseif (y <= 75000.0)
		tmp = Float64(1.0 + Float64(x * Float64(y / Float64(1.0 + y))));
	else
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -200000.0)
		tmp = x + (1.0 / y);
	elseif (y <= 75000.0)
		tmp = 1.0 + (x * (y / (1.0 + y)));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -200000.0], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 75000.0], N[(1.0 + N[(x * N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e5

   1. Initial program 21.9%

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

      1. associate-/l* 42.8%

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

      2. remove-double-neg 42.8%

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

      3. remove-double-neg 42.8%

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

      4. +-commutative 42.8%

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

   3. Simplified 42.8%

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

   5. Taylor expanded in y around inf 99.5%

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

      1. associate--l+ 99.5%

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

      2. div-sub 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. neg-mul-1 99.5%

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

      6. metadata-eval 99.5%

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

      7. distribute-lft-in 99.5%

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

      8. metadata-eval 99.5%

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

      9. sub-neg 99.5%

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

      10. associate-*r/ 99.5%

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

      11. mul-1-neg 99.5%

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

      12. unsub-neg 99.5%

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

      13. sub-neg 99.5%

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

      14. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 99.5%

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

   if -2e5 < y < 75000

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-/l* 98.6%

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

      3. associate-*l* 98.6%

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

      4. *-commutative 98.6%

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

      5. mul-1-neg 98.6%

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

      6. distribute-neg-frac2 98.6%

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

      7. distribute-neg-in 98.6%

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

      8. metadata-eval 98.6%

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

      9. sub-neg 98.6%

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

   7. Simplified 98.6%

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

   if 75000 < y

   1. Initial program 35.5%

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

      1. associate-/l* 61.9%

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

      2. remove-double-neg 61.9%

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

      3. remove-double-neg 61.9%

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

      4. +-commutative 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in y around inf 99.1%

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

      1. associate--l+ 99.1%

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

      2. div-sub 99.1%

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

      3. sub-neg 99.1%

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

      4. +-commutative 99.1%

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

      5. neg-mul-1 99.1%

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

      6. metadata-eval 99.1%

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

      7. distribute-lft-in 99.1%

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

      8. metadata-eval 99.1%

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

      9. sub-neg 99.1%

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

      10. associate-*r/ 99.1%

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

      11. mul-1-neg 99.1%

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

      12. unsub-neg 99.1%

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

      13. sub-neg 99.1%

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

      14. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -200000:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 75000:\\ \;\;\;\;1 + x \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 0.42:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e+95) x (if (<= y -1.0) (/ 1.0 y) (if (<= y 0.42) (- 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e+95) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 0.42) {
		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 <= (-9d+95)) then
        tmp = x
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0 / y
    else if (y <= 0.42d0) 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 <= -9e+95) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 0.42) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9e+95:
		tmp = x
	elif y <= -1.0:
		tmp = 1.0 / y
	elif y <= 0.42:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e+95)
		tmp = x;
	elseif (y <= -1.0)
		tmp = Float64(1.0 / y);
	elseif (y <= 0.42)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9e+95)
		tmp = x;
	elseif (y <= -1.0)
		tmp = 1.0 / y;
	elseif (y <= 0.42)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e+95], x, If[LessEqual[y, -1.0], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, 0.42], N[(1.0 - y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.00000000000000033e95 or 0.419999999999999984 < y

   1. Initial program 29.0%

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

      1. associate-/l* 55.4%

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

      2. remove-double-neg 55.4%

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

      3. remove-double-neg 55.4%

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

      4. +-commutative 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in y around inf 81.7%

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

   if -9.00000000000000033e95 < y < -1

   1. Initial program 36.8%

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

      1. associate-/l* 41.0%

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

      2. remove-double-neg 41.0%

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

      3. remove-double-neg 41.0%

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

      4. +-commutative 41.0%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in y around inf 92.9%

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

      1. associate--l+ 92.9%

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

      2. div-sub 92.9%

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

      3. sub-neg 92.9%

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

      4. +-commutative 92.9%

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

      5. neg-mul-1 92.9%

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

      6. metadata-eval 92.9%

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

      7. distribute-lft-in 92.9%

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

      8. metadata-eval 92.9%

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

      9. sub-neg 92.9%

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

      10. associate-*r/ 92.9%

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

      11. mul-1-neg 92.9%

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

      12. unsub-neg 92.9%

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

      13. sub-neg 92.9%

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

      14. metadata-eval 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in x around 0 64.3%

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

   if -1 < y < 0.419999999999999984

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 72.9%

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

   6. Taylor expanded in y around 0 72.9%

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

      1. neg-mul-1 72.9%

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

      2. unsub-neg 72.9%

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

   8. Simplified 72.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 0.42:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.82)))
   (+ x (/ 1.0 y))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.82)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 + (y * (x + -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 <= (-1.0d0)) .or. (.not. (y <= 0.82d0))) then
        tmp = x + (1.0d0 / y)
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.82)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.82):
		tmp = x + (1.0 / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.82))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.82)))
		tmp = x + (1.0 / y);
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.82]], $MachinePrecision]], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.819999999999999951 < y

   1. Initial program 29.8%

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

      1. associate-/l* 52.6%

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

      2. remove-double-neg 52.6%

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

      3. remove-double-neg 52.6%

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

      4. +-commutative 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. associate--l+ 98.1%

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

      2. div-sub 98.1%

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

      3. sub-neg 98.1%

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

      4. +-commutative 98.1%

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

      5. neg-mul-1 98.1%

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

      6. metadata-eval 98.1%

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

      7. distribute-lft-in 98.1%

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

      8. metadata-eval 98.1%

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

      9. sub-neg 98.1%

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

      10. associate-*r/ 98.1%

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

      11. mul-1-neg 98.1%

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

      12. unsub-neg 98.1%

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

      13. sub-neg 98.1%

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

      14. metadata-eval 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in x around 0 97.5%

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

   if -1 < y < 0.819999999999999951

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.1%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

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 + -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 <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    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 + -1.0));
	}
	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 + -1.0))
	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(y * Float64(x + -1.0)));
	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 + -1.0));
	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[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 29.8%

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

      1. associate-/l* 52.6%

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

      2. remove-double-neg 52.6%

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

      3. remove-double-neg 52.6%

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

      4. +-commutative 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. associate--l+ 98.1%

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

      2. div-sub 98.1%

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

      3. sub-neg 98.1%

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

      4. +-commutative 98.1%

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

      5. neg-mul-1 98.1%

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

      6. metadata-eval 98.1%

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

      7. distribute-lft-in 98.1%

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

      8. metadata-eval 98.1%

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

      9. sub-neg 98.1%

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

      10. associate-*r/ 98.1%

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

      11. mul-1-neg 98.1%

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

      12. unsub-neg 98.1%

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

      13. sub-neg 98.1%

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

      14. metadata-eval 98.1%

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

   7. Simplified 98.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.1%

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

4. Final simplification 98.6%

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

Alternative 8: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.045\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.045))) (+ x (/ 1.0 y)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.045)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 - 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 <= 0.045d0))) then
        tmp = x + (1.0d0 / y)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.045)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.045):
		tmp = x + (1.0 / y)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.045))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.045)))
		tmp = x + (1.0 / y);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.045]], $MachinePrecision]], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.044999999999999998 < y

   1. Initial program 30.3%

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

      1. associate-/l* 53.0%

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

      2. remove-double-neg 53.0%

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

      3. remove-double-neg 53.0%

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

      4. +-commutative 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in y around inf 97.4%

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

      1. associate--l+ 97.4%

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

      2. div-sub 97.4%

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

      3. sub-neg 97.4%

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

      4. +-commutative 97.4%

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

      5. neg-mul-1 97.4%

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

      6. metadata-eval 97.4%

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

      7. distribute-lft-in 97.4%

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

      8. metadata-eval 97.4%

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

      9. sub-neg 97.4%

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

      10. associate-*r/ 97.4%

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

      11. mul-1-neg 97.4%

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

      12. unsub-neg 97.4%

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

      13. sub-neg 97.4%

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

      14. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in x around 0 96.8%

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

   if -1 < y < 0.044999999999999998

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 72.9%

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

   6. Taylor expanded in y around 0 72.9%

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

      1. neg-mul-1 72.9%

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

      2. unsub-neg 72.9%

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

   8. Simplified 72.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.045\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = 1.0 + (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 / y)
    else
        tmp = 1.0d0 + (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 / y);
	} else {
		tmp = 1.0 + (x * y);
	}
	return tmp;
}

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(1.0 + Float64(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 / y);
	else
		tmp = 1.0 + (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[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 29.8%

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

      1. associate-/l* 52.6%

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

      2. remove-double-neg 52.6%

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

      3. remove-double-neg 52.6%

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

      4. +-commutative 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. associate--l+ 98.1%

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

      2. div-sub 98.1%

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

      3. sub-neg 98.1%

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

      4. +-commutative 98.1%

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

      5. neg-mul-1 98.1%

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

      6. metadata-eval 98.1%

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

      7. distribute-lft-in 98.1%

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

      8. metadata-eval 98.1%

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

      9. sub-neg 98.1%

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

      10. associate-*r/ 98.1%

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

      11. mul-1-neg 98.1%

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

      12. unsub-neg 98.1%

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

      13. sub-neg 98.1%

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

      14. metadata-eval 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in x around 0 97.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/l* 99.3%

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

      3. associate-*l* 99.3%

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

      4. *-commutative 99.3%

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

      5. mul-1-neg 99.3%

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

      6. distribute-neg-frac2 99.3%

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

      7. distribute-neg-in 99.3%

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

      8. metadata-eval 99.3%

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

      9. sub-neg 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in y around 0 98.4%

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

      1. neg-mul-1 98.4%

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

      2. distribute-lft-neg-in 98.4%

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

   10. Simplified 98.4%

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

4. Final simplification 97.9%

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

Alternative 10: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.24:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.24) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.24) {
		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 <= 0.24d0) 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 <= 0.24) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.24:
		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 <= 0.24)
		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 <= 0.24)
		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, 0.24], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.23999999999999999 < y

   1. Initial program 30.3%

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

      1. associate-/l* 53.0%

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

      2. remove-double-neg 53.0%

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

      3. remove-double-neg 53.0%

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

      4. +-commutative 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in y around inf 73.1%

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

   if -1 < y < 0.23999999999999999

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 72.9%

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

   6. Taylor expanded in y around 0 72.9%

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

      1. neg-mul-1 72.9%

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

      2. unsub-neg 72.9%

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

   8. Simplified 72.9%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.24:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 0.035) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.035) {
		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 <= 0.035d0) 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 <= 0.035) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.035:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.035)
		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 <= 0.035)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.035], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.035000000000000003 < y

   1. Initial program 30.3%

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

      1. associate-/l* 53.0%

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

      2. remove-double-neg 53.0%

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

      3. remove-double-neg 53.0%

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

      4. +-commutative 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in y around inf 73.1%

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

   if -1 < y < 0.035000000000000003

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 72.2%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.8% 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 64.9%

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

   1. associate-/l* 76.3%

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

   2. remove-double-neg 76.3%

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

   3. remove-double-neg 76.3%

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

   4. +-commutative 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in y around 0 37.6%

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

6. Final simplification 37.6%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 99.6% accurate, 0.5× 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 2024043 
(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))))