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

Percentage Accurate: 65.7% → 99.9%

Time: 9.1s

Alternatives: 11

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -22000 \lor \neg \left(y \leq 16000\right):\\ \;\;\;\;\left(\left(x + t_0\right) + \frac{1 - x}{{y}^{3}}\right) - \frac{t_0}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (or (<= y -22000.0) (not (<= y 16000.0)))
     (- (+ (+ x t_0) (/ (- 1.0 x) (pow y 3.0))) (/ t_0 y))
     (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if ((y <= -22000.0) || !(y <= 16000.0)) {
		tmp = ((x + t_0) + ((1.0 - x) / pow(y, 3.0))) - (t_0 / y);
	} else {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if ((y <= -22000.0) || !(y <= 16000.0))
		tmp = Float64(Float64(Float64(x + t_0) + Float64(Float64(1.0 - x) / (y ^ 3.0))) - Float64(t_0 / y));
	else
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[y, -22000.0], N[Not[LessEqual[y, 16000.0]], $MachinePrecision]], N[(N[(N[(x + t$95$0), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -22000 or 16000 < y

   1. Initial program 34.0%

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

      1. sub-neg 34.0%

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

      2. distribute-neg-frac 34.0%

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

      3. neg-mul-1 34.0%

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

      4. associate-*l/ 33.9%

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

      5. metadata-eval 33.9%

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

      6. associate-*l/ 33.9%

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

      7. associate-/r/ 33.9%

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

      8. metadata-eval 33.9%

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

      9. distribute-neg-frac 33.9%

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

      10. cancel-sign-sub-inv 33.9%

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

      11. associate-/r/ 33.9%

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

      12. associate-/r* 33.9%

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

      13. neg-mul-1 33.9%

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

      14. associate-/r/ 33.9%

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

      15. distribute-rgt-neg-in 33.9%

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

      16. associate-/r/ 33.9%

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

      17. distribute-neg-frac 33.9%

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

      18. metadata-eval 33.9%

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

      19. associate-/r/ 33.9%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in y around -inf 99.9%

      \[\leadsto \color{blue}{\left(\frac{x}{{y}^{2}} + \left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + x\right)\right)\right) - \frac{1}{{y}^{2}}} \]
   5. 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%

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

   6. Simplified 99.9%

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

   if -22000 < y < 16000

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-/l* 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. associate-/r/ 99.9%

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

      6. fma-def 99.9%

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

      7. neg-sub0 99.9%

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

      8. associate--r- 99.9%

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

      9. metadata-eval 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -22000 \lor \neg \left(y \leq 16000\right):\\ \;\;\;\;\left(\left(x + \frac{1 - x}{y}\right) + \frac{1 - x}{{y}^{3}}\right) - \frac{\frac{1 - x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -390000 \lor \neg \left(y \leq 340000\right):\\ \;\;\;\;\left(x + t_0\right) - \frac{t_0}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (or (<= y -390000.0) (not (<= y 340000.0)))
     (- (+ x t_0) (/ t_0 y))
     (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0))))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if ((y <= -390000.0) || !(y <= 340000.0)) {
		tmp = (x + t_0) - (t_0 / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - x) / y
    if ((y <= (-390000.0d0)) .or. (.not. (y <= 340000.0d0))) then
        tmp = (x + t_0) - (t_0 / y)
    else
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if ((y <= -390000.0) || !(y <= 340000.0)) {
		tmp = (x + t_0) - (t_0 / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 - x) / y
	tmp = 0
	if (y <= -390000.0) or not (y <= 340000.0):
		tmp = (x + t_0) - (t_0 / y)
	else:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if ((y <= -390000.0) || !(y <= 340000.0))
		tmp = Float64(Float64(x + t_0) - Float64(t_0 / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 - x) / y;
	tmp = 0.0;
	if ((y <= -390000.0) || ~((y <= 340000.0)))
		tmp = (x + t_0) - (t_0 / y);
	else
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[y, -390000.0], N[Not[LessEqual[y, 340000.0]], $MachinePrecision]], N[(N[(x + t$95$0), $MachinePrecision] - N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9e5 or 3.4e5 < y

   1. Initial program 33.6%

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

      1. sub-neg 33.6%

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

      2. distribute-neg-frac 33.6%

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

      3. neg-mul-1 33.6%

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

      4. associate-*l/ 33.5%

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

      5. metadata-eval 33.5%

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

      6. associate-*l/ 33.5%

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

      7. associate-/r/ 33.5%

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

      8. metadata-eval 33.5%

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

      9. distribute-neg-frac 33.5%

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

      10. cancel-sign-sub-inv 33.5%

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

      11. associate-/r/ 33.5%

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

      12. associate-/r* 33.5%

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

      13. neg-mul-1 33.5%

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

      14. associate-/r/ 33.5%

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

      15. distribute-rgt-neg-in 33.5%

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

      16. associate-/r/ 33.5%

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

      17. distribute-neg-frac 33.5%

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

      18. metadata-eval 33.5%

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

      19. associate-/r/ 33.5%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\left(\frac{x}{{y}^{2}} + \left(-1 \cdot \frac{x - 1}{y} + x\right)\right) - \frac{1}{{y}^{2}}} \]
   5. 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. unsub-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. div-sub 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

      11. unpow2 100.0%

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

      12. associate-/r* 100.0%

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

   6. Simplified 100.0%

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

   if -3.9e5 < y < 3.4e5

   1. Initial program 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -390000 \lor \neg \left(y \leq 340000\right):\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) - \frac{\frac{1 - x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -155000000000 \lor \neg \left(y \leq 5000000000\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -155000000000.0) (not (<= y 5000000000.0)))
   (- x (/ -1.0 y))
   (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -155000000000.0) || !(y <= 5000000000.0)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-155000000000.0d0)) .or. (.not. (y <= 5000000000.0d0))) then
        tmp = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -155000000000.0) || !(y <= 5000000000.0)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -155000000000.0) or not (y <= 5000000000.0):
		tmp = x - (-1.0 / y)
	else:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -155000000000.0) || !(y <= 5000000000.0))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -155000000000.0) || ~((y <= 5000000000.0)))
		tmp = x - (-1.0 / y);
	else
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -155000000000.0], N[Not[LessEqual[y, 5000000000.0]], $MachinePrecision]], N[(x - N[(-1.0 / 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 < -1.55e11 or 5e9 < y

   1. Initial program 32.5%

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

      1. sub-neg 32.5%

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

      2. distribute-neg-frac 32.5%

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

      3. neg-mul-1 32.5%

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

      4. associate-*l/ 32.5%

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

      5. metadata-eval 32.5%

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

      6. associate-*l/ 32.5%

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

      7. associate-/r/ 32.5%

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

      8. metadata-eval 32.5%

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

      9. distribute-neg-frac 32.5%

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

      10. cancel-sign-sub-inv 32.5%

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

      11. associate-/r/ 32.5%

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

      12. associate-/r* 32.5%

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

      13. neg-mul-1 32.5%

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

      14. associate-/r/ 32.5%

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

      15. distribute-rgt-neg-in 32.5%

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

      16. associate-/r/ 32.5%

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

      17. distribute-neg-frac 32.5%

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

      18. metadata-eval 32.5%

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

      19. associate-/r/ 32.5%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. div-sub 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. metadata-eval 100.0%

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

      10. sub-neg 100.0%

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

      11. unsub-neg 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   if -1.55e11 < y < 5e9

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. distribute-neg-frac 99.2%

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

      3. neg-mul-1 99.2%

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

      4. associate-*l/ 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-*l/ 99.2%

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

      7. associate-/r/ 99.2%

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

      8. metadata-eval 99.2%

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

      9. distribute-neg-frac 99.2%

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

      10. cancel-sign-sub-inv 99.2%

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

      11. associate-/r/ 99.2%

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

      12. associate-/r* 99.2%

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

      13. neg-mul-1 99.2%

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

      14. associate-/r/ 99.2%

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

      15. distribute-rgt-neg-in 99.2%

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

      16. associate-/r/ 99.2%

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

      17. distribute-neg-frac 99.2%

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

      18. metadata-eval 99.2%

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

      19. associate-/r/ 99.2%

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

   3. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -155000000000 \lor \neg \left(y \leq 5000000000\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -33000000000 \lor \neg \left(y \leq 9500000000\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -33000000000.0) (not (<= y 9500000000.0)))
   (- x (/ -1.0 y))
   (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -33000000000.0) || !(y <= 9500000000.0)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-33000000000.0d0)) .or. (.not. (y <= 9500000000.0d0))) then
        tmp = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -33000000000.0) || !(y <= 9500000000.0)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -33000000000.0) or not (y <= 9500000000.0):
		tmp = x - (-1.0 / y)
	else:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -33000000000.0) || !(y <= 9500000000.0))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -33000000000.0) || ~((y <= 9500000000.0)))
		tmp = x - (-1.0 / y);
	else
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -33000000000.0], N[Not[LessEqual[y, 9500000000.0]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3e10 or 9.5e9 < y

   1. Initial program 32.5%

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

      1. sub-neg 32.5%

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

      2. distribute-neg-frac 32.5%

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

      3. neg-mul-1 32.5%

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

      4. associate-*l/ 32.5%

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

      5. metadata-eval 32.5%

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

      6. associate-*l/ 32.5%

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

      7. associate-/r/ 32.5%

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

      8. metadata-eval 32.5%

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

      9. distribute-neg-frac 32.5%

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

      10. cancel-sign-sub-inv 32.5%

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

      11. associate-/r/ 32.5%

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

      12. associate-/r* 32.5%

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

      13. neg-mul-1 32.5%

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

      14. associate-/r/ 32.5%

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

      15. distribute-rgt-neg-in 32.5%

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

      16. associate-/r/ 32.5%

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

      17. distribute-neg-frac 32.5%

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

      18. metadata-eval 32.5%

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

      19. associate-/r/ 32.5%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. div-sub 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. metadata-eval 100.0%

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

      10. sub-neg 100.0%

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

      11. unsub-neg 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   if -3.3e10 < y < 9.5e9

   1. Initial program 99.2%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -33000000000 \lor \neg \left(y \leq 9500000000\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]

Alternative 5: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

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 - (y * (1.0 - 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 <= 1.0d0))) then
        tmp = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 - (y * (1.0d0 - x))
    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 - (y * (1.0 - x));
	}
	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 - (y * (1.0 - x))
	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(y * Float64(1.0 - x)));
	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 - (y * (1.0 - x));
	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[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 34.9%

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

      1. sub-neg 34.9%

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

      2. distribute-neg-frac 34.9%

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

      3. neg-mul-1 34.9%

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

      4. associate-*l/ 34.9%

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

      5. metadata-eval 34.9%

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

      6. associate-*l/ 34.9%

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

      7. associate-/r/ 34.9%

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

      8. metadata-eval 34.9%

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

      9. distribute-neg-frac 34.9%

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

      10. cancel-sign-sub-inv 34.9%

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

      11. associate-/r/ 34.9%

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

      12. associate-/r* 34.9%

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

      13. neg-mul-1 34.9%

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

      14. associate-/r/ 34.9%

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

      15. distribute-rgt-neg-in 34.9%

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

      16. associate-/r/ 34.9%

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

      17. distribute-neg-frac 34.9%

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

      18. metadata-eval 34.9%

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

      19. associate-/r/ 34.9%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in y around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. associate--l+ 97.6%

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

      3. div-sub 97.6%

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

      4. sub-neg 97.6%

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

      5. +-commutative 97.6%

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

      6. metadata-eval 97.6%

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

      7. distribute-neg-in 97.6%

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

      8. distribute-neg-frac 97.6%

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

      9. metadata-eval 97.6%

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

      10. sub-neg 97.6%

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

      11. unsub-neg 97.6%

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

      12. sub-neg 97.6%

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

      13. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in x around 0 97.3%

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

   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. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-*l/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. associate-/r/ 99.9%

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

      12. associate-/r* 99.9%

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

      13. neg-mul-1 99.9%

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

      14. associate-/r/ 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. associate-/r/ 99.9%

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

      17. distribute-neg-frac 99.9%

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

      18. metadata-eval 99.9%

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

      19. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.8%

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

4. Final simplification 97.5%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 32.8%

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

      1. sub-neg 32.8%

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

      2. distribute-neg-frac 32.8%

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

      3. neg-mul-1 32.8%

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

      4. associate-*l/ 32.5%

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

      5. metadata-eval 32.5%

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

      6. associate-*l/ 32.5%

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

      7. associate-/r/ 32.5%

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

      8. metadata-eval 32.5%

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

      9. distribute-neg-frac 32.5%

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

      10. cancel-sign-sub-inv 32.5%

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

      11. associate-/r/ 32.8%

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

      12. associate-/r* 32.8%

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

      13. neg-mul-1 32.8%

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

      14. associate-/r/ 32.5%

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

      15. distribute-rgt-neg-in 32.5%

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

      16. associate-/r/ 32.8%

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

      17. distribute-neg-frac 32.8%

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

      18. metadata-eval 32.8%

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

      19. associate-/r/ 32.5%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in y around inf 96.8%

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

      1. +-commutative 96.8%

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

      2. associate--l+ 96.8%

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

      3. div-sub 96.8%

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

      4. sub-neg 96.8%

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

      5. +-commutative 96.8%

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

      6. metadata-eval 96.8%

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

      7. distribute-neg-in 96.8%

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

      8. distribute-neg-frac 96.8%

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

      9. metadata-eval 96.8%

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

      10. sub-neg 96.8%

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

      11. unsub-neg 96.8%

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

      12. sub-neg 96.8%

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

      13. metadata-eval 96.8%

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

   6. Simplified 96.8%

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

   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. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-*l/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. associate-/r/ 99.9%

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

      12. associate-/r* 99.9%

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

      13. neg-mul-1 99.9%

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

      14. associate-/r/ 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. associate-/r/ 99.9%

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

      17. distribute-neg-frac 99.9%

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

      18. metadata-eval 99.9%

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

      19. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.8%

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

   if 1 < y

   1. Initial program 37.1%

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

      1. sub-neg 37.1%

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

      2. distribute-neg-frac 37.1%

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

      3. neg-mul-1 37.1%

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

      4. associate-*l/ 37.2%

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

      5. metadata-eval 37.2%

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

      6. associate-*l/ 37.2%

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

      7. associate-/r/ 37.2%

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

      8. metadata-eval 37.2%

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

      9. distribute-neg-frac 37.2%

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

      10. cancel-sign-sub-inv 37.2%

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

      11. associate-/r/ 36.9%

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

      12. associate-/r* 36.9%

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

      13. neg-mul-1 36.9%

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

      14. associate-/r/ 37.2%

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

      15. distribute-rgt-neg-in 37.2%

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

      16. associate-/r/ 36.9%

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

      17. distribute-neg-frac 36.9%

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

      18. metadata-eval 36.9%

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

      19. associate-/r/ 37.2%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in y around inf 98.4%

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

      1. +-commutative 98.4%

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

      2. associate--l+ 98.4%

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

      3. div-sub 98.4%

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

      4. sub-neg 98.4%

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

      5. +-commutative 98.4%

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

      6. metadata-eval 98.4%

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

      7. distribute-neg-in 98.4%

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

      8. distribute-neg-frac 98.4%

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

      9. metadata-eval 98.4%

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

      10. sub-neg 98.4%

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

      11. unsub-neg 98.4%

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

      12. sub-neg 98.4%

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

      13. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in x around 0 98.4%

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

4. Final simplification 97.7%

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

Alternative 7: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

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 + (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 <= 1.0d0))) then
        tmp = x - ((-1.0d0) / 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 <= 1.0)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * x);
	}
	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 + (y * x)
	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(y * x));
	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 + (y * x);
	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[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 34.9%

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

      1. sub-neg 34.9%

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

      2. distribute-neg-frac 34.9%

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

      3. neg-mul-1 34.9%

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

      4. associate-*l/ 34.9%

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

      5. metadata-eval 34.9%

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

      6. associate-*l/ 34.9%

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

      7. associate-/r/ 34.9%

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

      8. metadata-eval 34.9%

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

      9. distribute-neg-frac 34.9%

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

      10. cancel-sign-sub-inv 34.9%

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

      11. associate-/r/ 34.9%

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

      12. associate-/r* 34.9%

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

      13. neg-mul-1 34.9%

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

      14. associate-/r/ 34.9%

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

      15. distribute-rgt-neg-in 34.9%

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

      16. associate-/r/ 34.9%

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

      17. distribute-neg-frac 34.9%

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

      18. metadata-eval 34.9%

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

      19. associate-/r/ 34.9%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in y around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. associate--l+ 97.6%

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

      3. div-sub 97.6%

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

      4. sub-neg 97.6%

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

      5. +-commutative 97.6%

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

      6. metadata-eval 97.6%

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

      7. distribute-neg-in 97.6%

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

      8. distribute-neg-frac 97.6%

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

      9. metadata-eval 97.6%

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

      10. sub-neg 97.6%

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

      11. unsub-neg 97.6%

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

      12. sub-neg 97.6%

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

      13. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in x around 0 97.3%

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

   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. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-*l/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. associate-/r/ 99.9%

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

      12. associate-/r* 99.9%

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

      13. neg-mul-1 99.9%

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

      14. associate-/r/ 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. associate-/r/ 99.9%

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

      17. distribute-neg-frac 99.9%

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

      18. metadata-eval 99.9%

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

      19. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.8%

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

   5. Taylor expanded in x around inf 97.2%

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

      1. mul-1-neg 97.2%

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

      2. distribute-rgt-neg-in 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in y around 0 97.2%

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

4. Final simplification 97.2%

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

Alternative 8: 86.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 170000000:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 170000000.0) (+ 1.0 (* y x)) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 170000000.0) {
		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 <= (-1.0d0)) then
        tmp = x
    else if (y <= 170000000.0d0) 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 <= -1.0) {
		tmp = x;
	} else if (y <= 170000000.0) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 170000000.0:
		tmp = 1.0 + (y * x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 170000000.0)
		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 <= -1.0)
		tmp = x;
	elseif (y <= 170000000.0)
		tmp = 1.0 + (y * x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 170000000.0], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.7e8 < y

   1. Initial program 34.3%

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

      1. sub-neg 34.3%

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

      2. distribute-neg-frac 34.3%

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

      3. neg-mul-1 34.3%

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

      4. associate-*l/ 34.3%

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

      5. metadata-eval 34.3%

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

      6. associate-*l/ 34.3%

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

      7. associate-/r/ 34.3%

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

      8. metadata-eval 34.3%

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

      9. distribute-neg-frac 34.3%

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

      10. cancel-sign-sub-inv 34.3%

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

      11. associate-/r/ 34.3%

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

      12. associate-/r* 34.3%

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

      13. neg-mul-1 34.3%

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

      14. associate-/r/ 34.3%

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

      15. distribute-rgt-neg-in 34.3%

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

      16. associate-/r/ 34.3%

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

      17. distribute-neg-frac 34.3%

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

      18. metadata-eval 34.3%

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

      19. associate-/r/ 34.3%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in y around inf 78.8%

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

   if -1 < y < 1.7e8

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-neg-frac 99.6%

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

      3. neg-mul-1 99.6%

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

      4. associate-*l/ 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*l/ 99.5%

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

      7. associate-/r/ 99.5%

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

      8. metadata-eval 99.5%

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

      9. distribute-neg-frac 99.5%

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

      10. cancel-sign-sub-inv 99.5%

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

      11. associate-/r/ 99.5%

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

      12. associate-/r* 99.5%

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

      13. neg-mul-1 99.5%

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

      14. associate-/r/ 99.5%

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

      15. distribute-rgt-neg-in 99.5%

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

      16. associate-/r/ 99.5%

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

      17. distribute-neg-frac 99.5%

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

      18. metadata-eval 99.5%

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

      19. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 96.3%

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

   5. Taylor expanded in x around inf 96.0%

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

      1. mul-1-neg 96.0%

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

      2. distribute-rgt-neg-in 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in y around 0 96.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 170000000:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.8:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.8) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.8) {
		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 <= 0.8d0) 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 <= 0.8) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.8:
		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 <= 0.8)
		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 <= 0.8)
		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, 0.8], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.80000000000000004 < y

   1. Initial program 35.5%

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

      1. sub-neg 35.5%

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

      2. distribute-neg-frac 35.5%

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

      3. neg-mul-1 35.5%

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

      4. associate-*l/ 35.4%

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

      5. metadata-eval 35.4%

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

      6. associate-*l/ 35.4%

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

      7. associate-/r/ 35.4%

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

      8. metadata-eval 35.4%

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

      9. distribute-neg-frac 35.4%

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

      10. cancel-sign-sub-inv 35.4%

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

      11. associate-/r/ 35.4%

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

      12. associate-/r* 35.4%

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

      13. neg-mul-1 35.4%

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

      14. associate-/r/ 35.4%

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

      15. distribute-rgt-neg-in 35.4%

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

      16. associate-/r/ 35.4%

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

      17. distribute-neg-frac 35.4%

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

      18. metadata-eval 35.4%

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

      19. associate-/r/ 35.4%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in y around inf 77.1%

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

   if -1 < y < 0.80000000000000004

   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. distribute-neg-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-*l/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. associate-/r/ 99.9%

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

      12. associate-/r* 99.9%

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

      13. neg-mul-1 99.9%

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

      14. associate-/r/ 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. associate-/r/ 99.9%

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

      17. distribute-neg-frac 99.9%

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

      18. metadata-eval 99.9%

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

      19. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.3%

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

   5. Taylor expanded in x around 0 78.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.8:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 74.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 170000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 170000000.0) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 170000000.0) {
		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 <= 170000000.0d0) 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 <= 170000000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 170000000.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 170000000.0)
		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 <= 170000000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 170000000.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.7e8 < y

   1. Initial program 34.3%

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

      1. sub-neg 34.3%

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

      2. distribute-neg-frac 34.3%

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

      3. neg-mul-1 34.3%

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

      4. associate-*l/ 34.3%

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

      5. metadata-eval 34.3%

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

      6. associate-*l/ 34.3%

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

      7. associate-/r/ 34.3%

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

      8. metadata-eval 34.3%

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

      9. distribute-neg-frac 34.3%

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

      10. cancel-sign-sub-inv 34.3%

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

      11. associate-/r/ 34.3%

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

      12. associate-/r* 34.3%

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

      13. neg-mul-1 34.3%

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

      14. associate-/r/ 34.3%

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

      15. distribute-rgt-neg-in 34.3%

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

      16. associate-/r/ 34.3%

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

      17. distribute-neg-frac 34.3%

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

      18. metadata-eval 34.3%

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

      19. associate-/r/ 34.3%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in y around inf 78.8%

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

   if -1 < y < 1.7e8

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-neg-frac 99.6%

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

      3. neg-mul-1 99.6%

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

      4. associate-*l/ 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*l/ 99.5%

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

      7. associate-/r/ 99.5%

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

      8. metadata-eval 99.5%

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

      9. distribute-neg-frac 99.5%

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

      10. cancel-sign-sub-inv 99.5%

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

      11. associate-/r/ 99.5%

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

      12. associate-/r* 99.5%

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

      13. neg-mul-1 99.5%

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

      14. associate-/r/ 99.5%

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

      15. distribute-rgt-neg-in 99.5%

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

      16. associate-/r/ 99.5%

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

      17. distribute-neg-frac 99.5%

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

      18. metadata-eval 99.5%

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

      19. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 76.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 170000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 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 69.0%

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

   1. sub-neg 69.0%

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

   2. distribute-neg-frac 69.0%

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

   3. neg-mul-1 69.0%

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

   4. associate-*l/ 69.0%

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

   5. metadata-eval 69.0%

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

   6. associate-*l/ 69.0%

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

   7. associate-/r/ 69.0%

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

   8. metadata-eval 69.0%

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

   9. distribute-neg-frac 69.0%

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

   10. cancel-sign-sub-inv 69.0%

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

   11. associate-/r/ 68.9%

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

   12. associate-/r* 68.9%

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

   13. neg-mul-1 68.9%

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

   14. associate-/r/ 69.0%

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

   15. distribute-rgt-neg-in 69.0%

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

   16. associate-/r/ 68.9%

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

   17. distribute-neg-frac 68.9%

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

   18. metadata-eval 68.9%

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

   19. associate-/r/ 69.0%

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

3. Simplified 78.8%

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

4. Taylor expanded in y around 0 42.3%

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

5. Final simplification 42.3%

   \[\leadsto 1 \]

Developer target: 99.7% 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 2023214 
(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))))