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

Percentage Accurate: 65.4% → 100.0%

Time: 12.9s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -12000:\\ \;\;\;\;x - \left(\frac{x + \frac{\left(t\_0 - -1\right) - x}{y}}{y} + \frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 12500:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + \left(-1 - t\_0\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) y)))
   (if (<= y -12000.0)
     (- x (+ (/ (+ x (/ (- (- t_0 -1.0) x) y)) y) (/ -1.0 y)))
     (if (<= y 12500.0)
       (fma y (/ (- 1.0 x) (- -1.0 y)) 1.0)
       (+ x (/ (+ (- 1.0 x) (/ (+ x (- -1.0 t_0)) y)) y))))))

C

double code(double x, double y) {
	double t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -12000.0) {
		tmp = x - (((x + (((t_0 - -1.0) - x) / y)) / y) + (-1.0 / y));
	} else if (y <= 12500.0) {
		tmp = fma(y, ((1.0 - x) / (-1.0 - y)), 1.0);
	} else {
		tmp = x + (((1.0 - x) + ((x + (-1.0 - t_0)) / y)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (y <= -12000.0)
		tmp = Float64(x - Float64(Float64(Float64(x + Float64(Float64(Float64(t_0 - -1.0) - x) / y)) / y) + Float64(-1.0 / y)));
	elseif (y <= 12500.0)
		tmp = fma(y, Float64(Float64(1.0 - x) / Float64(-1.0 - y)), 1.0);
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) + Float64(Float64(x + Float64(-1.0 - t_0)) / y)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -12000.0], N[(x - N[(N[(N[(x + N[(N[(N[(t$95$0 - -1.0), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 12500.0], N[(y * N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(x + N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -12000

   1. Initial program 31.6%

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

      1. sub-neg 31.6%

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

      2. +-commutative 31.6%

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

      3. *-commutative 31.6%

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

      4. associate-/l* 54.7%

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

      5. distribute-rgt-neg-in 54.7%

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

      6. fma-define 55.1%

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

      7. distribute-frac-neg2 55.1%

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

      8. +-commutative 55.1%

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

      9. distribute-neg-in 55.1%

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

      10. metadata-eval 55.1%

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

      11. unsub-neg 55.1%

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

   3. Simplified 55.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 54.7%

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

      2. associate-*r/ 31.6%

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

      3. *-commutative 31.6%

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

      4. sub-neg 31.6%

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

      5. metadata-eval 31.6%

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

      6. distribute-neg-in 31.6%

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

      7. distribute-neg-frac2 31.6%

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

      8. associate-*r/ 54.9%

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

      9. +-commutative 54.9%

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

      10. sub-neg 54.9%

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

      11. associate-*r/ 31.6%

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

      12. *-commutative 31.6%

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

      13. associate-/l* 54.7%

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

      14. +-commutative 54.7%

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

   6. Applied egg-rr 54.7%

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

   7. Taylor expanded in y around -inf 99.9%

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

   9. Simplified 99.9%

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

       1. associate-+l- 99.9%

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

       2. div-sub 100.0%

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

       3. +-commutative 100.0%

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

   11. Applied egg-rr 100.0%

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

   if -12000 < y < 12500

   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. *-commutative 99.9%

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

      4. associate-/l* 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   if 12500 < y

   1. Initial program 30.8%

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

      1. sub-neg 30.8%

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

      2. +-commutative 30.8%

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

      3. *-commutative 30.8%

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

      4. associate-/l* 53.8%

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

      5. distribute-rgt-neg-in 53.8%

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

      6. fma-define 53.6%

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

      7. distribute-frac-neg2 53.6%

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

      8. +-commutative 53.6%

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

      9. distribute-neg-in 53.6%

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

      10. metadata-eval 53.6%

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

      11. unsub-neg 53.6%

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

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 53.8%

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

      2. associate-*r/ 30.8%

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

      3. *-commutative 30.8%

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

      4. sub-neg 30.8%

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

      5. metadata-eval 30.8%

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

      6. distribute-neg-in 30.8%

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

      7. distribute-neg-frac2 30.8%

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

      8. associate-*r/ 53.8%

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

      9. +-commutative 53.8%

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

      10. sub-neg 53.8%

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

      11. associate-*r/ 30.8%

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

      12. *-commutative 30.8%

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

      13. associate-/l* 53.8%

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

      14. +-commutative 53.8%

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

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in y around -inf 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12000:\\ \;\;\;\;x - \left(\frac{x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}}{y} + \frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 12500:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + \left(-1 - \frac{x + -1}{y}\right)}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -12500 \lor \neg \left(y \leq 13000\right):\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + \left(-1 - \frac{x + -1}{y}\right)}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{1 - x}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -12500.0) (not (<= y 13000.0)))
   (+ x (/ (+ (- 1.0 x) (/ (+ x (- -1.0 (/ (+ x -1.0) y))) y)) y))
   (+ 1.0 (* y (/ (- 1.0 x) (- -1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -12500.0) || !(y <= 13000.0)) {
		tmp = x + (((1.0 - x) + ((x + (-1.0 - ((x + -1.0) / y))) / y)) / y);
	} else {
		tmp = 1.0 + (y * ((1.0 - 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 <= (-12500.0d0)) .or. (.not. (y <= 13000.0d0))) then
        tmp = x + (((1.0d0 - x) + ((x + ((-1.0d0) - ((x + (-1.0d0)) / y))) / y)) / y)
    else
        tmp = 1.0d0 + (y * ((1.0d0 - x) / ((-1.0d0) - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -12500.0) || !(y <= 13000.0)) {
		tmp = x + (((1.0 - x) + ((x + (-1.0 - ((x + -1.0) / y))) / y)) / y);
	} else {
		tmp = 1.0 + (y * ((1.0 - x) / (-1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -12500.0) or not (y <= 13000.0):
		tmp = x + (((1.0 - x) + ((x + (-1.0 - ((x + -1.0) / y))) / y)) / y)
	else:
		tmp = 1.0 + (y * ((1.0 - x) / (-1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -12500.0) || !(y <= 13000.0))
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) + Float64(Float64(x + Float64(-1.0 - Float64(Float64(x + -1.0) / y))) / y)) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(1.0 - x) / Float64(-1.0 - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -12500.0) || ~((y <= 13000.0)))
		tmp = x + (((1.0 - x) + ((x + (-1.0 - ((x + -1.0) / y))) / y)) / y);
	else
		tmp = 1.0 + (y * ((1.0 - x) / (-1.0 - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -12500.0], N[Not[LessEqual[y, 13000.0]], $MachinePrecision]], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(x + N[(-1.0 - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -12500 or 13000 < y

   1. Initial program 31.2%

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

      1. sub-neg 31.2%

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

      2. +-commutative 31.2%

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

      3. *-commutative 31.2%

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

      4. associate-/l* 54.2%

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

      5. distribute-rgt-neg-in 54.2%

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

      6. fma-define 54.3%

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

      7. distribute-frac-neg2 54.3%

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

      8. +-commutative 54.3%

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

      9. distribute-neg-in 54.3%

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

      10. metadata-eval 54.3%

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

      11. unsub-neg 54.3%

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

   3. Simplified 54.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 54.2%

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

      2. associate-*r/ 31.2%

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

      3. *-commutative 31.2%

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

      4. sub-neg 31.2%

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

      5. metadata-eval 31.2%

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

      6. distribute-neg-in 31.2%

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

      7. distribute-neg-frac2 31.2%

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

      8. associate-*r/ 54.3%

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

      9. +-commutative 54.3%

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

      10. sub-neg 54.3%

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

      11. associate-*r/ 31.2%

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

      12. *-commutative 31.2%

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

      13. associate-/l* 54.2%

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

      14. +-commutative 54.2%

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

   6. Applied egg-rr 54.2%

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

   7. Taylor expanded in y around -inf 100.0%

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

   9. Simplified 100.0%

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

   if -12500 < y < 13000

   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. *-commutative 99.9%

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

      4. associate-/l* 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. distribute-neg-frac2 99.9%

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

      8. associate-*r/ 99.9%

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

      9. +-commutative 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-*r/ 99.9%

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

      12. *-commutative 99.9%

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

      13. associate-/l* 99.9%

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

      14. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12500 \lor \neg \left(y \leq 13000\right):\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + \left(-1 - \frac{x + -1}{y}\right)}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{1 - x}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -12500:\\ \;\;\;\;x - \left(\frac{x + \frac{\left(t\_0 - -1\right) - x}{y}}{y} + \frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 13000:\\ \;\;\;\;1 + y \cdot \frac{1 - x}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + \left(-1 - t\_0\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) y)))
   (if (<= y -12500.0)
     (- x (+ (/ (+ x (/ (- (- t_0 -1.0) x) y)) y) (/ -1.0 y)))
     (if (<= y 13000.0)
       (+ 1.0 (* y (/ (- 1.0 x) (- -1.0 y))))
       (+ x (/ (+ (- 1.0 x) (/ (+ x (- -1.0 t_0)) y)) y))))))

C

double code(double x, double y) {
	double t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -12500.0) {
		tmp = x - (((x + (((t_0 - -1.0) - x) / y)) / y) + (-1.0 / y));
	} else if (y <= 13000.0) {
		tmp = 1.0 + (y * ((1.0 - x) / (-1.0 - y)));
	} else {
		tmp = x + (((1.0 - x) + ((x + (-1.0 - t_0)) / y)) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + (-1.0d0)) / y
    if (y <= (-12500.0d0)) then
        tmp = x - (((x + (((t_0 - (-1.0d0)) - x) / y)) / y) + ((-1.0d0) / y))
    else if (y <= 13000.0d0) then
        tmp = 1.0d0 + (y * ((1.0d0 - x) / ((-1.0d0) - y)))
    else
        tmp = x + (((1.0d0 - x) + ((x + ((-1.0d0) - t_0)) / y)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -12500.0) {
		tmp = x - (((x + (((t_0 - -1.0) - x) / y)) / y) + (-1.0 / y));
	} else if (y <= 13000.0) {
		tmp = 1.0 + (y * ((1.0 - x) / (-1.0 - y)));
	} else {
		tmp = x + (((1.0 - x) + ((x + (-1.0 - t_0)) / y)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + -1.0) / y
	tmp = 0
	if y <= -12500.0:
		tmp = x - (((x + (((t_0 - -1.0) - x) / y)) / y) + (-1.0 / y))
	elif y <= 13000.0:
		tmp = 1.0 + (y * ((1.0 - x) / (-1.0 - y)))
	else:
		tmp = x + (((1.0 - x) + ((x + (-1.0 - t_0)) / y)) / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (y <= -12500.0)
		tmp = Float64(x - Float64(Float64(Float64(x + Float64(Float64(Float64(t_0 - -1.0) - x) / y)) / y) + Float64(-1.0 / y)));
	elseif (y <= 13000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(1.0 - x) / Float64(-1.0 - y))));
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) + Float64(Float64(x + Float64(-1.0 - t_0)) / y)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + -1.0) / y;
	tmp = 0.0;
	if (y <= -12500.0)
		tmp = x - (((x + (((t_0 - -1.0) - x) / y)) / y) + (-1.0 / y));
	elseif (y <= 13000.0)
		tmp = 1.0 + (y * ((1.0 - x) / (-1.0 - y)));
	else
		tmp = x + (((1.0 - x) + ((x + (-1.0 - t_0)) / y)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -12500.0], N[(x - N[(N[(N[(x + N[(N[(N[(t$95$0 - -1.0), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 13000.0], N[(1.0 + N[(y * N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(x + N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -12500

   1. Initial program 31.6%

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

      1. sub-neg 31.6%

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

      2. +-commutative 31.6%

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

      3. *-commutative 31.6%

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

      4. associate-/l* 54.7%

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

      5. distribute-rgt-neg-in 54.7%

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

      6. fma-define 55.1%

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

      7. distribute-frac-neg2 55.1%

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

      8. +-commutative 55.1%

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

      9. distribute-neg-in 55.1%

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

      10. metadata-eval 55.1%

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

      11. unsub-neg 55.1%

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

   3. Simplified 55.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 54.7%

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

      2. associate-*r/ 31.6%

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

      3. *-commutative 31.6%

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

      4. sub-neg 31.6%

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

      5. metadata-eval 31.6%

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

      6. distribute-neg-in 31.6%

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

      7. distribute-neg-frac2 31.6%

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

      8. associate-*r/ 54.9%

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

      9. +-commutative 54.9%

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

      10. sub-neg 54.9%

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

      11. associate-*r/ 31.6%

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

      12. *-commutative 31.6%

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

      13. associate-/l* 54.7%

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

      14. +-commutative 54.7%

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

   6. Applied egg-rr 54.7%

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

   7. Taylor expanded in y around -inf 99.9%

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

   9. Simplified 99.9%

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

       1. associate-+l- 99.9%

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

       2. div-sub 100.0%

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

       3. +-commutative 100.0%

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

   11. Applied egg-rr 100.0%

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

   if -12500 < y < 13000

   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. *-commutative 99.9%

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

      4. associate-/l* 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. distribute-neg-frac2 99.9%

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

      8. associate-*r/ 99.9%

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

      9. +-commutative 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-*r/ 99.9%

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

      12. *-commutative 99.9%

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

      13. associate-/l* 99.9%

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

      14. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   if 13000 < y

   1. Initial program 30.8%

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

      1. sub-neg 30.8%

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

      2. +-commutative 30.8%

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

      3. *-commutative 30.8%

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

      4. associate-/l* 53.8%

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

      5. distribute-rgt-neg-in 53.8%

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

      6. fma-define 53.6%

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

      7. distribute-frac-neg2 53.6%

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

      8. +-commutative 53.6%

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

      9. distribute-neg-in 53.6%

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

      10. metadata-eval 53.6%

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

      11. unsub-neg 53.6%

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

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 53.8%

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

      2. associate-*r/ 30.8%

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

      3. *-commutative 30.8%

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

      4. sub-neg 30.8%

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

      5. metadata-eval 30.8%

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

      6. distribute-neg-in 30.8%

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

      7. distribute-neg-frac2 30.8%

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

      8. associate-*r/ 53.8%

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

      9. +-commutative 53.8%

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

      10. sub-neg 53.8%

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

      11. associate-*r/ 30.8%

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

      12. *-commutative 30.8%

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

      13. associate-/l* 53.8%

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

      14. +-commutative 53.8%

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

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in y around -inf 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12500:\\ \;\;\;\;x - \left(\frac{x + \frac{\left(\frac{x + -1}{y} - -1\right) - x}{y}}{y} + \frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 13000:\\ \;\;\;\;1 + y \cdot \frac{1 - x}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + \left(-1 - \frac{x + -1}{y}\right)}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -135000000 \lor \neg \left(y \leq 240000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -135000000.0) (not (<= y 240000000.0)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -135000000.0) || !(y <= 240000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - 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 <= (-135000000.0d0)) .or. (.not. (y <= 240000000.0d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    else
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -135000000.0) || !(y <= 240000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -135000000.0) or not (y <= 240000000.0):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -135000000.0) || !(y <= 240000000.0))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -135000000.0) || ~((y <= 240000000.0)))
		tmp = x - ((x + -1.0) / y);
	else
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -135000000.0], N[Not[LessEqual[y, 240000000.0]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e8 or 2.4e8 < y

   1. Initial program 29.6%

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

      1. associate-/l* 53.3%

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

      2. +-commutative 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in y around -inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. mul-1-neg 99.7%

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

      3. neg-sub0 99.7%

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

      4. associate-+l- 99.7%

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

      5. neg-sub0 99.7%

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

      6. +-commutative 99.7%

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

      7. sub-neg 99.7%

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

   7. Simplified 99.7%

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

   if -1.35e8 < y < 2.4e8

   1. Initial program 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -135000000 \lor \neg \left(y \leq 240000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -135000000 \lor \neg \left(y \leq 240000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -135000000.0) (not (<= y 240000000.0)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (* (- 1.0 x) (/ y (- -1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -135000000.0) || !(y <= 240000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-135000000.0d0)) .or. (.not. (y <= 240000000.0d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    else
        tmp = 1.0d0 + ((1.0d0 - x) * (y / ((-1.0d0) - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -135000000.0) || !(y <= 240000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -135000000.0) or not (y <= 240000000.0):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -135000000.0) || !(y <= 240000000.0))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) * Float64(y / Float64(-1.0 - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -135000000.0) || ~((y <= 240000000.0)))
		tmp = x - ((x + -1.0) / y);
	else
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -135000000.0], N[Not[LessEqual[y, 240000000.0]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e8 or 2.4e8 < y

   1. Initial program 29.6%

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

      1. associate-/l* 53.3%

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

      2. +-commutative 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in y around -inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. mul-1-neg 99.7%

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

      3. neg-sub0 99.7%

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

      4. associate-+l- 99.7%

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

      5. neg-sub0 99.7%

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

      6. +-commutative 99.7%

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

      7. sub-neg 99.7%

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

   7. Simplified 99.7%

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

   if -1.35e8 < y < 2.4e8

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -135000000 \lor \neg \left(y \leq 240000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -102000000 \lor \neg \left(y \leq 200000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{1 - x}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -102000000.0) (not (<= y 200000000.0)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (* y (/ (- 1.0 x) (- -1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -102000000.0) || !(y <= 200000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * ((1.0 - 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 <= (-102000000.0d0)) .or. (.not. (y <= 200000000.0d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    else
        tmp = 1.0d0 + (y * ((1.0d0 - x) / ((-1.0d0) - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -102000000.0) || !(y <= 200000000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * ((1.0 - x) / (-1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -102000000.0) or not (y <= 200000000.0):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + (y * ((1.0 - x) / (-1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -102000000.0) || !(y <= 200000000.0))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(1.0 - x) / Float64(-1.0 - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -102000000.0) || ~((y <= 200000000.0)))
		tmp = x - ((x + -1.0) / y);
	else
		tmp = 1.0 + (y * ((1.0 - x) / (-1.0 - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -102000000.0], N[Not[LessEqual[y, 200000000.0]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.02e8 or 2e8 < y

   1. Initial program 29.6%

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

      1. associate-/l* 53.3%

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

      2. +-commutative 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in y around -inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. mul-1-neg 99.7%

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

      3. neg-sub0 99.7%

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

      4. associate-+l- 99.7%

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

      5. neg-sub0 99.7%

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

      6. +-commutative 99.7%

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

      7. sub-neg 99.7%

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

   7. Simplified 99.7%

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

   if -1.02e8 < y < 2e8

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/l* 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      6. fma-define 99.8%

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

      7. distribute-frac-neg2 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. metadata-eval 99.8%

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

      11. unsub-neg 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-commutative 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

      6. distribute-neg-in 99.7%

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

      7. distribute-neg-frac2 99.7%

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

      8. associate-*r/ 99.7%

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

      9. +-commutative 99.7%

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

      10. sub-neg 99.7%

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

      11. associate-*r/ 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-/l* 99.7%

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

      14. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -102000000 \lor \neg \left(y \leq 200000000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{1 - x}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -280000:\\ \;\;\;\;x + \frac{\left(1 - x\right) + t\_0}{y}\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) y)))
   (if (<= y -280000.0)
     (+ x (/ (+ (- 1.0 x) t_0) y))
     (if (<= y 250000000.0)
       (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))
       (- x t_0)))))

C

double code(double x, double y) {
	double t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -280000.0) {
		tmp = x + (((1.0 - x) + t_0) / y);
	} else if (y <= 250000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x - 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 = (x + (-1.0d0)) / y
    if (y <= (-280000.0d0)) then
        tmp = x + (((1.0d0 - x) + t_0) / y)
    else if (y <= 250000000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    else
        tmp = x - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -280000.0) {
		tmp = x + (((1.0 - x) + t_0) / y);
	} else if (y <= 250000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x - t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + -1.0) / y
	tmp = 0
	if y <= -280000.0:
		tmp = x + (((1.0 - x) + t_0) / y)
	elif y <= 250000000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = x - t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (y <= -280000.0)
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) + t_0) / y));
	elseif (y <= 250000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = Float64(x - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + -1.0) / y;
	tmp = 0.0;
	if (y <= -280000.0)
		tmp = x + (((1.0 - x) + t_0) / y);
	elseif (y <= 250000000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = x - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -280000.0], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] + t$95$0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 250000000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8e5

   1. Initial program 31.6%

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

      1. associate-/l* 54.9%

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

      2. +-commutative 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in y around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. distribute-lft-out-- 99.5%

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

   7. Simplified 99.5%

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

   if -2.8e5 < y < 2.5e8

   1. Initial program 99.9%

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

   if 2.5e8 < y

   1. Initial program 29.8%

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

      1. associate-/l* 53.1%

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

      2. +-commutative 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in y around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. sub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -280000:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \lor \neg \left(y \leq 230000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.0) (not (<= y 230000.0)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (* y (/ x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.0) || !(y <= 230000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * (x / (y + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-9.0d0)) .or. (.not. (y <= 230000.0d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    else
        tmp = 1.0d0 + (y * (x / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9.0) || !(y <= 230000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (y * (x / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.0) or not (y <= 230000.0):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + (y * (x / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.0) || !(y <= 230000.0))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9.0) || ~((y <= 230000.0)))
		tmp = x - ((x + -1.0) / y);
	else
		tmp = 1.0 + (y * (x / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.0], N[Not[LessEqual[y, 230000.0]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9 or 2.3e5 < y

   1. Initial program 31.1%

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

      1. associate-/l* 54.3%

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

      2. +-commutative 54.3%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in y around -inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. mul-1-neg 98.6%

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

      3. neg-sub0 98.6%

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

      4. associate-+l- 98.6%

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

      5. neg-sub0 98.6%

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

      6. +-commutative 98.6%

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

      7. sub-neg 98.6%

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

   7. Simplified 98.6%

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

   if -9 < y < 2.3e5

   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. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac2 100.0%

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

      8. associate-*r/ 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. associate-*r/ 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-/l* 100.0%

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

      14. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 98.1%

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

      1. neg-mul-1 98.1%

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

      2. distribute-neg-frac2 98.1%

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

      3. distribute-neg-in 98.1%

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

      4. metadata-eval 98.1%

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

      5. unsub-neg 98.1%

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

   9. Simplified 98.1%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \lor \neg \left(y \leq 230000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \lor \neg \left(y \leq 220000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.2) (not (<= y 220000.0)))
   (- x (/ (+ x -1.0) y))
   (+ 1.0 (* x (/ y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.2) || !(y <= 220000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (x * (y / (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 <= (-5.2d0)) .or. (.not. (y <= 220000.0d0))) then
        tmp = x - ((x + (-1.0d0)) / y)
    else
        tmp = 1.0d0 + (x * (y / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5.2) || !(y <= 220000.0)) {
		tmp = x - ((x + -1.0) / y);
	} else {
		tmp = 1.0 + (x * (y / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.2) or not (y <= 220000.0):
		tmp = x - ((x + -1.0) / y)
	else:
		tmp = 1.0 + (x * (y / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.2) || !(y <= 220000.0))
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(1.0 + Float64(x * Float64(y / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.2) || ~((y <= 220000.0)))
		tmp = x - ((x + -1.0) / y);
	else
		tmp = 1.0 + (x * (y / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.2], N[Not[LessEqual[y, 220000.0]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.20000000000000018 or 2.2e5 < y

   1. Initial program 31.1%

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

      1. associate-/l* 54.3%

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

      2. +-commutative 54.3%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in y around -inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. mul-1-neg 98.6%

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

      3. neg-sub0 98.6%

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

      4. associate-+l- 98.6%

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

      5. neg-sub0 98.6%

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

      6. +-commutative 98.6%

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

      7. sub-neg 98.6%

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

   7. Simplified 98.6%

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

   if -5.20000000000000018 < y < 2.2e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. associate-/l* 98.0%

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

      3. +-commutative 98.0%

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

      4. distribute-rgt-neg-in 98.0%

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

      5. distribute-neg-frac2 98.0%

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

      6. +-commutative 98.0%

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

      7. distribute-neg-in 98.0%

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

      8. metadata-eval 98.0%

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

      9. sub-neg 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \lor \neg \left(y \leq 220000\right):\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \frac{y}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 31.7%

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

      1. associate-/l* 54.6%

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

      2. +-commutative 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in y around -inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. mul-1-neg 98.4%

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

      3. neg-sub0 98.4%

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

      4. associate-+l- 98.4%

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

      5. neg-sub0 98.4%

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

      6. +-commutative 98.4%

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

      7. sub-neg 98.4%

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

   7. Simplified 98.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 97.2%

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

4. Final simplification 97.8%

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

Alternative 11: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.2)) {
		tmp = x - ((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 <= 1.2d0))) then
        tmp = x - ((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 <= 1.2)) {
		tmp = x - ((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 <= 1.2):
		tmp = x - ((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 <= 1.2))
		tmp = Float64(x - Float64(Float64(x + -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 <= 1.2)))
		tmp = x - ((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, 1.2]], $MachinePrecision]], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.19999999999999996 < y

   1. Initial program 31.7%

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

      1. associate-/l* 54.6%

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

      2. +-commutative 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in y around -inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. mul-1-neg 98.4%

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

      3. neg-sub0 98.4%

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

      4. associate-+l- 98.4%

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

      5. neg-sub0 98.4%

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

      6. +-commutative 98.4%

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

      7. sub-neg 98.4%

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

   7. Simplified 98.4%

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

   if -1 < y < 1.19999999999999996

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. 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. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac2 100.0%

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

      8. associate-*r/ 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. associate-*r/ 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-/l* 100.0%

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

      14. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 98.0%

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

      1. neg-mul-1 98.0%

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

      2. distribute-neg-frac2 98.0%

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

      3. distribute-neg-in 98.0%

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

      4. metadata-eval 98.0%

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

      5. unsub-neg 98.0%

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

   9. Simplified 98.0%

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

   10. Taylor expanded in y around 0 96.4%

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

       1. associate-*r* 96.4%

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

       2. *-commutative 96.4%

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

       3. neg-mul-1 96.4%

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

   12. Simplified 96.4%

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

4. Final simplification 97.3%

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

Alternative 12: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.001 \lor \neg \left(y \leq 4.5 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.001) (not (<= y 4.5e-8)))
   (* x (/ y (+ y 1.0)))
   (+ 1.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.001) || !(y <= 4.5e-8)) {
		tmp = x * (y / (y + 1.0));
	} 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 <= (-0.001d0)) .or. (.not. (y <= 4.5d-8))) then
        tmp = x * (y / (y + 1.0d0))
    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 <= -0.001) || !(y <= 4.5e-8)) {
		tmp = x * (y / (y + 1.0));
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.001) or not (y <= 4.5e-8):
		tmp = x * (y / (y + 1.0))
	else:
		tmp = 1.0 + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.001) || !(y <= 4.5e-8))
		tmp = Float64(x * Float64(y / Float64(y + 1.0)));
	else
		tmp = Float64(1.0 + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.001) || ~((y <= 4.5e-8)))
		tmp = x * (y / (y + 1.0));
	else
		tmp = 1.0 + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.001], N[Not[LessEqual[y, 4.5e-8]], $MachinePrecision]], N[(x * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e-3 or 4.49999999999999993e-8 < y

   1. Initial program 33.9%

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

      1. sub-neg 33.9%

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

      2. +-commutative 33.9%

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

      3. *-commutative 33.9%

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

      4. associate-/l* 56.0%

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

      5. distribute-rgt-neg-in 56.0%

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

      6. fma-define 56.1%

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

      7. distribute-frac-neg2 56.1%

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

      8. +-commutative 56.1%

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

      9. distribute-neg-in 56.1%

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

      10. metadata-eval 56.1%

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

      11. unsub-neg 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in x around inf 55.0%

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

      1. associate-/l* 77.2%

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

   7. Simplified 77.2%

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

   if -1e-3 < y < 4.49999999999999993e-8

   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. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac2 100.0%

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

      8. associate-*r/ 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. associate-*r/ 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-/l* 100.0%

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

      14. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 98.4%

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

      1. neg-mul-1 98.4%

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

      2. distribute-neg-frac2 98.4%

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

      3. distribute-neg-in 98.4%

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

      4. metadata-eval 98.4%

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

      5. unsub-neg 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in y around 0 98.0%

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

       1. associate-*r* 98.0%

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

       2. *-commutative 98.0%

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

       3. neg-mul-1 98.0%

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

   12. Simplified 98.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.001 \lor \neg \left(y \leq 4.5 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 32:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 32.0) (+ 1.0 (* y x)) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 32.0) {
		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 <= 32.0d0) 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 <= 32.0) {
		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 <= 32.0:
		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 <= 32.0)
		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 <= 32.0)
		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, 32.0], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 32 < y

   1. Initial program 31.7%

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

      1. associate-/l* 54.6%

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

      2. +-commutative 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in y around inf 75.8%

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

   if -1 < y < 32

   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. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac2 100.0%

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

      8. associate-*r/ 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. associate-*r/ 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-/l* 100.0%

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

      14. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 98.0%

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

      1. neg-mul-1 98.0%

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

      2. distribute-neg-frac2 98.0%

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

      3. distribute-neg-in 98.0%

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

      4. metadata-eval 98.0%

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

      5. unsub-neg 98.0%

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

   9. Simplified 98.0%

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

   10. Taylor expanded in y around 0 96.4%

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

       1. associate-*r* 96.4%

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

       2. *-commutative 96.4%

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

       3. neg-mul-1 96.4%

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

   12. Simplified 96.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 32:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-5}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 3e-5) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 3e-5) {
		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 <= 3d-5) 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 <= 3e-5) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 3e-5:
		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 <= 3e-5)
		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 <= 3e-5)
		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, 3e-5], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 3.00000000000000008e-5 < y

   1. Initial program 32.8%

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

      1. associate-/l* 55.4%

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

      2. +-commutative 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in y around inf 74.8%

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

   if -1 < y < 3.00000000000000008e-5

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.2%

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

   6. Taylor expanded in x around 0 79.0%

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

Alternative 15: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 3e-5) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 3e-5) {
		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 <= 3d-5) 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 <= 3e-5) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 3e-5:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 3e-5)
		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 <= 3e-5)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 3e-5], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 3.00000000000000008e-5 < y

   1. Initial program 32.8%

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

      1. associate-/l* 55.4%

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

      2. +-commutative 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in y around inf 74.8%

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

   if -1 < y < 3.00000000000000008e-5

   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. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 78.1%

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

Alternative 16: 38.7% 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 68.3%

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

   1. sub-neg 68.3%

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

   2. +-commutative 68.3%

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

   3. *-commutative 68.3%

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

   4. associate-/l* 78.9%

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

   5. distribute-rgt-neg-in 78.9%

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

   6. fma-define 78.9%

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

   7. distribute-frac-neg2 78.9%

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

   8. +-commutative 78.9%

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

   9. distribute-neg-in 78.9%

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

   10. metadata-eval 78.9%

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

   11. unsub-neg 78.9%

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

3. Simplified 78.9%

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

5. Taylor expanded in y around 0 43.0%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer target: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024111 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :alt
  (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))))