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

Percentage Accurate: 66.4% → 99.8%

Time: 10.8s

Alternatives: 13

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: 66.4% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1400000000000:\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) + \frac{x + -1}{y \cdot y}\\ \mathbf{elif}\;y \leq 13000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\left(\frac{1 - x}{y \cdot y} + \left(\frac{x}{y} - x\right)\right) + \frac{x + -1}{{y}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1400000000000.0)
   (+ (+ x (/ (- 1.0 x) y)) (/ (+ x -1.0) (* y y)))
   (if (<= y 13000.0)
     (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
     (-
      (/ 1.0 y)
      (+
       (+ (/ (- 1.0 x) (* y y)) (- (/ x y) x))
       (/ (+ x -1.0) (pow y 3.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1400000000000.0) {
		tmp = (x + ((1.0 - x) / y)) + ((x + -1.0) / (y * y));
	} else if (y <= 13000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = (1.0 / y) - ((((1.0 - x) / (y * y)) + ((x / y) - x)) + ((x + -1.0) / 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) :: tmp
    if (y <= (-1400000000000.0d0)) then
        tmp = (x + ((1.0d0 - x) / y)) + ((x + (-1.0d0)) / (y * y))
    else if (y <= 13000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = (1.0d0 / y) - ((((1.0d0 - x) / (y * y)) + ((x / y) - x)) + ((x + (-1.0d0)) / (y ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1400000000000.0) {
		tmp = (x + ((1.0 - x) / y)) + ((x + -1.0) / (y * y));
	} else if (y <= 13000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = (1.0 / y) - ((((1.0 - x) / (y * y)) + ((x / y) - x)) + ((x + -1.0) / Math.pow(y, 3.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1400000000000.0:
		tmp = (x + ((1.0 - x) / y)) + ((x + -1.0) / (y * y))
	elif y <= 13000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = (1.0 / y) - ((((1.0 - x) / (y * y)) + ((x / y) - x)) + ((x + -1.0) / math.pow(y, 3.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1400000000000.0)
		tmp = Float64(Float64(x + Float64(Float64(1.0 - x) / y)) + Float64(Float64(x + -1.0) / Float64(y * y)));
	elseif (y <= 13000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(Float64(1.0 / y) - Float64(Float64(Float64(Float64(1.0 - x) / Float64(y * y)) + Float64(Float64(x / y) - x)) + Float64(Float64(x + -1.0) / (y ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1400000000000.0)
		tmp = (x + ((1.0 - x) / y)) + ((x + -1.0) / (y * y));
	elseif (y <= 13000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = (1.0 / y) - ((((1.0 - x) / (y * y)) + ((x / y) - x)) + ((x + -1.0) / (y ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1400000000000.0], N[(N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 13000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] - N[(N[(N[(N[(1.0 - x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4e12

   1. Initial program 23.7%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. mul-1-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. sub-neg 100.0%

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

      12. div-sub 100.0%

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

      13. sub-neg 100.0%

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

      14. metadata-eval 100.0%

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

      15. +-commutative 100.0%

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

      16. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if -1.4e12 < y < 13000

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      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%

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

   if 13000 < y

   1. Initial program 30.1%

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

      1. sub-neg 30.1%

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

      2. metadata-eval 30.1%

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

      3. associate--r- 30.1%

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

      4. neg-sub0 30.1%

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

      5. remove-double-neg 30.1%

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

      6. associate-/l* 51.0%

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

      7. +-commutative 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in y around 0 51.0%

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

   5. Taylor expanded in y around -inf 99.9%

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

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1400000000000:\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) + \frac{x + -1}{y \cdot y}\\ \mathbf{elif}\;y \leq 13000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\left(\frac{1 - x}{y \cdot y} + \left(\frac{x}{y} - x\right)\right) + \frac{x + -1}{{y}^{3}}\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ t_1 := \frac{x + -1}{y \cdot y}\\ \mathbf{if}\;y \leq -1400000000000:\\ \;\;\;\;\left(x + t_0\right) + t_1\\ \mathbf{elif}\;y \leq 12000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(t_0 + \frac{1 - x}{{y}^{3}}\right)\right) + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)) (t_1 (/ (+ x -1.0) (* y y))))
   (if (<= y -1400000000000.0)
     (+ (+ x t_0) t_1)
     (if (<= y 12000.0)
       (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
       (+ (+ x (+ t_0 (/ (- 1.0 x) (pow y 3.0)))) t_1)))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double t_1 = (x + -1.0) / (y * y);
	double tmp;
	if (y <= -1400000000000.0) {
		tmp = (x + t_0) + t_1;
	} else if (y <= 12000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = (x + (t_0 + ((1.0 - x) / pow(y, 3.0)))) + t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 - x) / y
    t_1 = (x + (-1.0d0)) / (y * y)
    if (y <= (-1400000000000.0d0)) then
        tmp = (x + t_0) + t_1
    else if (y <= 12000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = (x + (t_0 + ((1.0d0 - x) / (y ** 3.0d0)))) + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double t_1 = (x + -1.0) / (y * y);
	double tmp;
	if (y <= -1400000000000.0) {
		tmp = (x + t_0) + t_1;
	} else if (y <= 12000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = (x + (t_0 + ((1.0 - x) / Math.pow(y, 3.0)))) + t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 - x) / y
	t_1 = (x + -1.0) / (y * y)
	tmp = 0
	if y <= -1400000000000.0:
		tmp = (x + t_0) + t_1
	elif y <= 12000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = (x + (t_0 + ((1.0 - x) / math.pow(y, 3.0)))) + t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	t_1 = Float64(Float64(x + -1.0) / Float64(y * y))
	tmp = 0.0
	if (y <= -1400000000000.0)
		tmp = Float64(Float64(x + t_0) + t_1);
	elseif (y <= 12000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x + Float64(t_0 + Float64(Float64(1.0 - x) / (y ^ 3.0)))) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 - x) / y;
	t_1 = (x + -1.0) / (y * y);
	tmp = 0.0;
	if (y <= -1400000000000.0)
		tmp = (x + t_0) + t_1;
	elseif (y <= 12000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = (x + (t_0 + ((1.0 - x) / (y ^ 3.0)))) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1400000000000.0], N[(N[(x + t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[y, 12000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(t$95$0 + N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4e12

   1. Initial program 23.7%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. mul-1-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. sub-neg 100.0%

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

      12. div-sub 100.0%

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

      13. sub-neg 100.0%

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

      14. metadata-eval 100.0%

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

      15. +-commutative 100.0%

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

      16. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if -1.4e12 < y < 12000

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      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%

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

   if 12000 < y

   1. Initial program 30.1%

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

   2. Taylor expanded in y around -inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate--l+ 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1400000000000:\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) + \frac{x + -1}{y \cdot y}\\ \mathbf{elif}\;y \leq 12000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(\frac{1 - x}{y} + \frac{1 - x}{{y}^{3}}\right)\right) + \frac{x + -1}{y \cdot y}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1400000000000 \lor \neg \left(y \leq 230000\right):\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) + \frac{x + -1}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1400000000000.0) (not (<= y 230000.0)))
   (+ (+ x (/ (- 1.0 x) y)) (/ (+ x -1.0) (* y y)))
   (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1400000000000.0) || !(y <= 230000.0)) {
		tmp = (x + ((1.0 - x) / y)) + ((x + -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) :: tmp
    if ((y <= (-1400000000000.0d0)) .or. (.not. (y <= 230000.0d0))) then
        tmp = (x + ((1.0d0 - x) / y)) + ((x + (-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 tmp;
	if ((y <= -1400000000000.0) || !(y <= 230000.0)) {
		tmp = (x + ((1.0 - x) / y)) + ((x + -1.0) / (y * y));
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1400000000000.0) or not (y <= 230000.0):
		tmp = (x + ((1.0 - x) / y)) + ((x + -1.0) / (y * y))
	else:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1400000000000.0) || !(y <= 230000.0))
		tmp = Float64(Float64(x + Float64(Float64(1.0 - x) / y)) + Float64(Float64(x + -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)
	tmp = 0.0;
	if ((y <= -1400000000000.0) || ~((y <= 230000.0)))
		tmp = (x + ((1.0 - x) / y)) + ((x + -1.0) / (y * 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, -1400000000000.0], N[Not[LessEqual[y, 230000.0]], $MachinePrecision]], N[(N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / N[(y * 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 < -1.4e12 or 2.3e5 < y

   1. Initial program 26.6%

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

   2. Taylor expanded in y around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. mul-1-neg 99.8%

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

      5. sub-neg 99.8%

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

      6. metadata-eval 99.8%

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

      7. distribute-neg-frac 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. metadata-eval 99.8%

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

      11. sub-neg 99.8%

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

      12. div-sub 99.8%

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

      13. sub-neg 99.8%

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

      14. metadata-eval 99.8%

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

      15. +-commutative 99.8%

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

      16. unpow2 99.8%

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

   4. Simplified 99.8%

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

   if -1.4e12 < y < 2.3e5

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      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%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1400000000000 \lor \neg \left(y \leq 230000\right):\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) + \frac{x + -1}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1400000000000 \lor \neg \left(y \leq 155000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1400000000000.0) (not (<= y 155000000.0)))
   (+ 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 <= -1400000000000.0) || !(y <= 155000000.0)) {
		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 <= (-1400000000000.0d0)) .or. (.not. (y <= 155000000.0d0))) 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 <= -1400000000000.0) || !(y <= 155000000.0)) {
		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 <= -1400000000000.0) or not (y <= 155000000.0):
		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 <= -1400000000000.0) || !(y <= 155000000.0))
		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 <= -1400000000000.0) || ~((y <= 155000000.0)))
		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, -1400000000000.0], N[Not[LessEqual[y, 155000000.0]], $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 < -1.4e12 or 1.55e8 < y

   1. Initial program 25.9%

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

   2. Taylor expanded in y around -inf 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. metadata-eval 99.9%

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

      9. sub-neg 99.9%

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

   4. Simplified 99.9%

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

   if -1.4e12 < y < 1.55e8

   1. Initial program 98.9%

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

      1. sub-neg 98.9%

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

      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 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1400000000000 \lor \neg \left(y \leq 155000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 5: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8200000:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e+130)
   x
   (if (<= y -5.4e+103)
     (/ 1.0 y)
     (if (<= y -1.0) x (if (<= y 8200000.0) (+ 1.0 (* y x)) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e+130) {
		tmp = x;
	} else if (y <= -5.4e+103) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = x;
	} else if (y <= 8200000.0) {
		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 <= (-5d+130)) then
        tmp = x
    else if (y <= (-5.4d+103)) then
        tmp = 1.0d0 / y
    else if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 8200000.0d0) 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 <= -5e+130) {
		tmp = x;
	} else if (y <= -5.4e+103) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = x;
	} else if (y <= 8200000.0) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e+130:
		tmp = x
	elif y <= -5.4e+103:
		tmp = 1.0 / y
	elif y <= -1.0:
		tmp = x
	elif y <= 8200000.0:
		tmp = 1.0 + (y * x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e+130)
		tmp = x;
	elseif (y <= -5.4e+103)
		tmp = Float64(1.0 / y);
	elseif (y <= -1.0)
		tmp = x;
	elseif (y <= 8200000.0)
		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 <= -5e+130)
		tmp = x;
	elseif (y <= -5.4e+103)
		tmp = 1.0 / y;
	elseif (y <= -1.0)
		tmp = x;
	elseif (y <= 8200000.0)
		tmp = 1.0 + (y * x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e+130], x, If[LessEqual[y, -5.4e+103], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, -1.0], x, If[LessEqual[y, 8200000.0], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.9999999999999996e130 or -5.39999999999999985e103 < y < -1 or 8.2e6 < y

   1. Initial program 29.6%

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

      1. sub-neg 29.6%

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

      2. associate-*l/ 54.8%

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

      3. distribute-lft-neg-in 54.8%

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

      4. distribute-frac-neg 54.8%

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

      5. neg-sub0 54.8%

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

      6. associate--r- 54.8%

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

      7. metadata-eval 54.8%

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

      8. +-commutative 54.8%

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

      9. +-commutative 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in y around inf 75.8%

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

   if -4.9999999999999996e130 < y < -5.39999999999999985e103

   1. Initial program 4.5%

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

   2. Taylor expanded in x around 0 4.0%

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

   3. Taylor expanded in y around inf 72.1%

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

   if -1 < y < 8.2e6

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0 95.2%

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

      1. sub-neg 95.2%

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

      2. distribute-lft-in 95.2%

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

      3. distribute-rgt-neg-out 95.2%

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

      4. unsub-neg 95.2%

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

      5. *-rgt-identity 95.2%

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

   4. Simplified 95.2%

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

   5. Taylor expanded in x around inf 94.9%

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

      1. mul-1-neg 94.9%

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

      2. distribute-rgt-neg-in 94.9%

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

   7. Simplified 94.9%

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

      1. *-commutative 94.9%

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

      2. cancel-sign-sub 94.9%

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

      3. *-commutative 94.9%

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

      4. +-commutative 94.9%

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

   9. Applied egg-rr 94.9%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8200000:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq 8200000:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e+130)
   x
   (if (<= y -6.5e+103)
     (/ 1.0 y)
     (if (<= y -1.0) (- x (/ x y)) (if (<= y 8200000.0) (+ 1.0 (* y x)) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e+130) {
		tmp = x;
	} else if (y <= -6.5e+103) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = x - (x / y);
	} else if (y <= 8200000.0) {
		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 <= (-5d+130)) then
        tmp = x
    else if (y <= (-6.5d+103)) then
        tmp = 1.0d0 / y
    else if (y <= (-1.0d0)) then
        tmp = x - (x / y)
    else if (y <= 8200000.0d0) 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 <= -5e+130) {
		tmp = x;
	} else if (y <= -6.5e+103) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = x - (x / y);
	} else if (y <= 8200000.0) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e+130:
		tmp = x
	elif y <= -6.5e+103:
		tmp = 1.0 / y
	elif y <= -1.0:
		tmp = x - (x / y)
	elif y <= 8200000.0:
		tmp = 1.0 + (y * x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e+130)
		tmp = x;
	elseif (y <= -6.5e+103)
		tmp = Float64(1.0 / y);
	elseif (y <= -1.0)
		tmp = Float64(x - Float64(x / y));
	elseif (y <= 8200000.0)
		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 <= -5e+130)
		tmp = x;
	elseif (y <= -6.5e+103)
		tmp = 1.0 / y;
	elseif (y <= -1.0)
		tmp = x - (x / y);
	elseif (y <= 8200000.0)
		tmp = 1.0 + (y * x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e+130], x, If[LessEqual[y, -6.5e+103], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, -1.0], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8200000.0], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.9999999999999996e130 or 8.2e6 < y

   1. Initial program 23.9%

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

      1. sub-neg 23.9%

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

      2. associate-*l/ 53.3%

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

      3. distribute-lft-neg-in 53.3%

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

      4. distribute-frac-neg 53.3%

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

      5. neg-sub0 53.3%

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

      6. associate--r- 53.3%

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

      7. metadata-eval 53.3%

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

      8. +-commutative 53.3%

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

      9. +-commutative 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in y around inf 79.5%

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

   if -4.9999999999999996e130 < y < -6.50000000000000001e103

   1. Initial program 4.5%

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

   2. Taylor expanded in x around 0 4.0%

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

   3. Taylor expanded in y around inf 72.1%

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

   if -6.50000000000000001e103 < y < -1

   1. Initial program 53.0%

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

      1. sub-neg 53.0%

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

      2. associate-*l/ 61.3%

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

      3. distribute-lft-neg-in 61.3%

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

      4. distribute-frac-neg 61.3%

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

      5. neg-sub0 61.3%

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

      6. associate--r- 61.3%

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

      7. metadata-eval 61.3%

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

      8. +-commutative 61.3%

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

      9. +-commutative 61.3%

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

   3. Simplified 61.3%

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

   4. Taylor expanded in x around inf 61.1%

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

   5. Taylor expanded in y around inf 62.6%

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

      1. mul-1-neg 62.6%

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

      2. +-commutative 62.6%

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

      3. sub-neg 62.6%

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

   7. Simplified 62.6%

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

   if -1 < y < 8.2e6

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0 95.2%

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

      1. sub-neg 95.2%

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

      2. distribute-lft-in 95.2%

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

      3. distribute-rgt-neg-out 95.2%

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

      4. unsub-neg 95.2%

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

      5. *-rgt-identity 95.2%

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

   4. Simplified 95.2%

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

   5. Taylor expanded in x around inf 94.9%

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

      1. mul-1-neg 94.9%

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

      2. distribute-rgt-neg-in 94.9%

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

   7. Simplified 94.9%

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

      1. *-commutative 94.9%

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

      2. cancel-sign-sub 94.9%

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

      3. *-commutative 94.9%

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

      4. +-commutative 94.9%

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

   9. Applied egg-rr 94.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq 8200000:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -140000000000 \lor \neg \left(y \leq 1920\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -140000000000.0) (not (<= y 1920.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* y (/ x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -140000000000.0) || !(y <= 1920.0)) {
		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 <= (-140000000000.0d0)) .or. (.not. (y <= 1920.0d0))) 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 <= -140000000000.0) || !(y <= 1920.0)) {
		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 <= -140000000000.0) or not (y <= 1920.0):
		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 <= -140000000000.0) || !(y <= 1920.0))
		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 <= -140000000000.0) || ~((y <= 1920.0)))
		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, -140000000000.0], N[Not[LessEqual[y, 1920.0]], $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 < -1.4e11 or 1920 < y

   1. Initial program 27.2%

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

   2. Taylor expanded in y around -inf 99.2%

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

      1. +-commutative 99.2%

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

      2. mul-1-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

      5. distribute-neg-frac 99.2%

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

      6. +-commutative 99.2%

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

      7. distribute-neg-in 99.2%

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

      8. metadata-eval 99.2%

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

      9. sub-neg 99.2%

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

   4. Simplified 99.2%

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

   if -1.4e11 < y < 1920

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      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 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -140000000000 \lor \neg \left(y \leq 1920\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 8: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00095:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e+130)
   x
   (if (<= y -6.5e+103)
     (/ 1.0 y)
     (if (<= y -1.0) x (if (<= y 0.00095) (- 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e+130) {
		tmp = x;
	} else if (y <= -6.5e+103) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.00095) {
		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 <= (-5d+130)) then
        tmp = x
    else if (y <= (-6.5d+103)) then
        tmp = 1.0d0 / y
    else if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.00095d0) 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 <= -5e+130) {
		tmp = x;
	} else if (y <= -6.5e+103) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.00095) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e+130:
		tmp = x
	elif y <= -6.5e+103:
		tmp = 1.0 / y
	elif y <= -1.0:
		tmp = x
	elif y <= 0.00095:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e+130)
		tmp = x;
	elseif (y <= -6.5e+103)
		tmp = Float64(1.0 / y);
	elseif (y <= -1.0)
		tmp = x;
	elseif (y <= 0.00095)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5e+130)
		tmp = x;
	elseif (y <= -6.5e+103)
		tmp = 1.0 / y;
	elseif (y <= -1.0)
		tmp = x;
	elseif (y <= 0.00095)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e+130], x, If[LessEqual[y, -6.5e+103], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.00095], N[(1.0 - y), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.9999999999999996e130 or -6.50000000000000001e103 < y < -1 or 9.49999999999999998e-4 < y

   1. Initial program 31.4%

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

      1. sub-neg 31.4%

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

      2. associate-*l/ 55.8%

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

      3. distribute-lft-neg-in 55.8%

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

      4. distribute-frac-neg 55.8%

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

      5. neg-sub0 55.8%

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

      6. associate--r- 55.8%

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

      7. metadata-eval 55.8%

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

      8. +-commutative 55.8%

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

      9. +-commutative 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in y around inf 73.6%

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

   if -4.9999999999999996e130 < y < -6.50000000000000001e103

   1. Initial program 4.5%

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

   2. Taylor expanded in x around 0 4.0%

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

   3. Taylor expanded in y around inf 72.1%

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

   if -1 < y < 9.49999999999999998e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.7%

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

      1. sub-neg 97.7%

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

      2. distribute-lft-in 97.7%

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

      3. distribute-rgt-neg-out 97.7%

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

      4. unsub-neg 97.7%

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

      5. *-rgt-identity 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in x around 0 78.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00095:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.2\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.2))) (+ x (/ (- 1.0 x) y)) (+ 1.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.2)) {
		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 <= 1.2d0))) 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 <= 1.2)) {
		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 <= 1.2):
		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 <= 1.2))
		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 <= 1.2)))
		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, 1.2]], $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 1.19999999999999996 < y

   1. Initial program 28.7%

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

   2. Taylor expanded in y around -inf 97.4%

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

      1. +-commutative 97.4%

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

      2. mul-1-neg 97.4%

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

      3. sub-neg 97.4%

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

      4. metadata-eval 97.4%

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

      5. distribute-neg-frac 97.4%

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

      6. +-commutative 97.4%

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

      7. distribute-neg-in 97.4%

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

      8. metadata-eval 97.4%

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

      9. sub-neg 97.4%

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

   4. Simplified 97.4%

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

   if -1 < y < 1.19999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.2%

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

      1. sub-neg 97.2%

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

      2. distribute-lft-in 97.2%

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

      3. distribute-rgt-neg-out 97.2%

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

      4. unsub-neg 97.2%

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

      5. *-rgt-identity 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in x around inf 96.8%

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

      1. mul-1-neg 96.8%

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

      2. distribute-rgt-neg-in 96.8%

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

   7. Simplified 96.8%

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

      1. *-commutative 96.8%

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

      2. cancel-sign-sub 96.8%

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

      3. *-commutative 96.8%

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

      4. +-commutative 96.8%

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

   9. Applied egg-rr 96.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.2\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]

Alternative 10: 98.3% 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 28.7%

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

   2. Taylor expanded in y around -inf 97.4%

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

      1. +-commutative 97.4%

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

      2. mul-1-neg 97.4%

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

      3. sub-neg 97.4%

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

      4. metadata-eval 97.4%

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

      5. distribute-neg-frac 97.4%

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

      6. +-commutative 97.4%

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

      7. distribute-neg-in 97.4%

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

      8. metadata-eval 97.4%

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

      9. sub-neg 97.4%

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

   4. Simplified 97.4%

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

   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 97.2%

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

      1. sub-neg 97.2%

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

      2. distribute-lft-in 97.2%

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

      3. distribute-rgt-neg-out 97.2%

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

      4. unsub-neg 97.2%

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

      5. *-rgt-identity 97.2%

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

   4. Simplified 97.2%

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

4. Final simplification 97.3%

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

Alternative 11: 73.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00095:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.00095) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.00095) {
		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 <= 0.00095d0) 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 <= 0.00095) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.00095:
		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 <= 0.00095)
		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 <= 0.00095)
		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, 0.00095], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 9.49999999999999998e-4 < y

   1. Initial program 29.3%

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

      1. sub-neg 29.3%

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

      2. associate-*l/ 52.5%

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

      3. distribute-lft-neg-in 52.5%

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

      4. distribute-frac-neg 52.5%

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

      5. neg-sub0 52.5%

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

      6. associate--r- 52.5%

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

      7. metadata-eval 52.5%

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

      8. +-commutative 52.5%

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

      9. +-commutative 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in y around inf 69.6%

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

   if -1 < y < 9.49999999999999998e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.7%

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

      1. sub-neg 97.7%

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

      2. distribute-lft-in 97.7%

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

      3. distribute-rgt-neg-out 97.7%

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

      4. unsub-neg 97.7%

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

      5. *-rgt-identity 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in x around 0 78.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00095:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 73.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00095:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 0.00095) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.00095) {
		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 <= 0.00095d0) 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 <= 0.00095) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.00095:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.00095)
		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 <= 0.00095)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.00095], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 9.49999999999999998e-4 < y

   1. Initial program 29.3%

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

      1. sub-neg 29.3%

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

      2. associate-*l/ 52.5%

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

      3. distribute-lft-neg-in 52.5%

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

      4. distribute-frac-neg 52.5%

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

      5. neg-sub0 52.5%

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

      6. associate--r- 52.5%

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

      7. metadata-eval 52.5%

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

      8. +-commutative 52.5%

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

      9. +-commutative 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in y around inf 69.6%

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

   if -1 < y < 9.49999999999999998e-4

   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 77.6%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.00095:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13: 39.1% 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 64.6%

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

   1. sub-neg 64.6%

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

   2. associate-*l/ 76.3%

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

   3. distribute-lft-neg-in 76.3%

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

   4. distribute-frac-neg 76.3%

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

   5. neg-sub0 76.3%

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

   6. associate--r- 76.3%

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

   7. metadata-eval 76.3%

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

   8. +-commutative 76.3%

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

   9. +-commutative 76.3%

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

3. Simplified 76.3%

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

4. Taylor expanded in y around 0 40.6%

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

5. Final simplification 40.6%

   \[\leadsto 1 \]

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 2023279 
(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))))