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

Percentage Accurate: 64.8% → 99.8%

Time: 7.3s

Alternatives: 10

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 64.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -32500000000:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 310000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1 - x}{y} + \frac{x + -1}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -32500000000.0)
   (+ x (/ 1.0 y))
   (if (<= y 310000.0)
     (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
     (+ x (+ (/ (- 1.0 x) y) (/ (+ x -1.0) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -32500000000.0) {
		tmp = x + (1.0 / y);
	} else if (y <= 310000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + (((1.0 - x) / y) + ((x + -1.0) / (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-32500000000.0d0)) then
        tmp = x + (1.0d0 / y)
    else if (y <= 310000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = x + (((1.0d0 - x) / y) + ((x + (-1.0d0)) / (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -32500000000.0) {
		tmp = x + (1.0 / y);
	} else if (y <= 310000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + (((1.0 - x) / y) + ((x + -1.0) / (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -32500000000.0:
		tmp = x + (1.0 / y)
	elif y <= 310000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = x + (((1.0 - x) / y) + ((x + -1.0) / (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -32500000000.0)
		tmp = Float64(x + Float64(1.0 / y));
	elseif (y <= 310000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) / y) + Float64(Float64(x + -1.0) / Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -32500000000.0)
		tmp = x + (1.0 / y);
	elseif (y <= 310000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = x + (((1.0 - x) / y) + ((x + -1.0) / (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -32500000000.0], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 310000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.25e10

   1. Initial program 40.0%

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

      1. sub-neg 40.0%

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

      2. metadata-eval 40.0%

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

      3. associate--r- 40.0%

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

      4. neg-sub0 40.0%

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

      5. remove-double-neg 40.0%

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

      6. associate-/l* 53.3%

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

      7. +-commutative 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   if -3.25e10 < y < 3.1e5

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 3.1e5 < y

   1. Initial program 31.0%

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

      1. sub-neg 31.0%

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

      2. metadata-eval 31.0%

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

      3. associate--r- 31.0%

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

      4. neg-sub0 31.0%

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

      5. remove-double-neg 31.0%

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

      6. associate-/l* 51.8%

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

      7. +-commutative 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. sub-neg 100.0%

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

      11. div-sub 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

      15. unpow2 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -32500000000:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 310000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1 - x}{y} + \frac{x + -1}{y \cdot y}\right)\\ \end{array} \]

Alternative 2: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-118}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-127}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-98}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-59}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y))))
   (if (<= y -1.0)
     t_0
     (if (<= y -1.45e-118)
       (- 1.0 y)
       (if (<= y -8e-127)
         (* y x)
         (if (<= y 4e-98)
           1.0
           (if (<= y 1.8e-59) (* y x) (if (<= y 4.2e-7) (- 1.0 y) t_0))))))))

C

double code(double x, double y) {
	double t_0 = x + (1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= -1.45e-118) {
		tmp = 1.0 - y;
	} else if (y <= -8e-127) {
		tmp = y * x;
	} else if (y <= 4e-98) {
		tmp = 1.0;
	} else if (y <= 1.8e-59) {
		tmp = y * x;
	} else if (y <= 4.2e-7) {
		tmp = 1.0 - y;
	} 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 = x + (1.0d0 / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= (-1.45d-118)) then
        tmp = 1.0d0 - y
    else if (y <= (-8d-127)) then
        tmp = y * x
    else if (y <= 4d-98) then
        tmp = 1.0d0
    else if (y <= 1.8d-59) then
        tmp = y * x
    else if (y <= 4.2d-7) then
        tmp = 1.0d0 - y
    else
        tmp = 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 <= -1.0) {
		tmp = t_0;
	} else if (y <= -1.45e-118) {
		tmp = 1.0 - y;
	} else if (y <= -8e-127) {
		tmp = y * x;
	} else if (y <= 4e-98) {
		tmp = 1.0;
	} else if (y <= 1.8e-59) {
		tmp = y * x;
	} else if (y <= 4.2e-7) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (1.0 / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= -1.45e-118:
		tmp = 1.0 - y
	elif y <= -8e-127:
		tmp = y * x
	elif y <= 4e-98:
		tmp = 1.0
	elif y <= 1.8e-59:
		tmp = y * x
	elif y <= 4.2e-7:
		tmp = 1.0 - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= -1.45e-118)
		tmp = Float64(1.0 - y);
	elseif (y <= -8e-127)
		tmp = Float64(y * x);
	elseif (y <= 4e-98)
		tmp = 1.0;
	elseif (y <= 1.8e-59)
		tmp = Float64(y * x);
	elseif (y <= 4.2e-7)
		tmp = Float64(1.0 - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (1.0 / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= -1.45e-118)
		tmp = 1.0 - y;
	elseif (y <= -8e-127)
		tmp = y * x;
	elseif (y <= 4e-98)
		tmp = 1.0;
	elseif (y <= 1.8e-59)
		tmp = y * x;
	elseif (y <= 4.2e-7)
		tmp = 1.0 - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, -1.45e-118], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, -8e-127], N[(y * x), $MachinePrecision], If[LessEqual[y, 4e-98], 1.0, If[LessEqual[y, 1.8e-59], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.2e-7], N[(1.0 - y), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1 or 4.2e-7 < y

   1. Initial program 36.1%

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

      1. sub-neg 36.1%

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

      2. metadata-eval 36.1%

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

      3. associate--r- 36.1%

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

      4. neg-sub0 36.1%

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

      5. remove-double-neg 36.1%

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

      6. associate-/l* 53.2%

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

      7. +-commutative 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in y around -inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. sub-neg 98.5%

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

      3. metadata-eval 98.5%

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

      4. distribute-neg-frac 98.5%

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

      5. +-commutative 98.5%

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

      6. distribute-neg-in 98.5%

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

      7. metadata-eval 98.5%

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

      8. sub-neg 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in x around 0 98.2%

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

   if -1 < y < -1.4499999999999999e-118 or 1.8e-59 < y < 4.2e-7

   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. metadata-eval 100.0%

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

      3. associate--r- 100.0%

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

      4. neg-sub0 100.0%

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

      5. remove-double-neg 100.0%

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

      6. associate-/l* 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 93.7%

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

      1. *-commutative 93.7%

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

      2. /-rgt-identity 93.7%

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

      3. associate-/r/ 93.6%

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

      4. div-sub 93.6%

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

      5. remove-double-div 93.6%

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

      6. associate-/r/ 93.7%

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

      7. /-rgt-identity 93.7%

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

      8. *-commutative 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in x around 0 82.4%

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

   if -1.4499999999999999e-118 < y < -8.0000000000000002e-127 or 3.99999999999999976e-98 < y < 1.8e-59

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

   6. Simplified 84.3%

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

   7. Taylor expanded in y around 0 84.3%

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

      1. *-commutative 84.3%

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

   9. Simplified 84.3%

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

   if -8.0000000000000002e-127 < y < 3.99999999999999976e-98

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 84.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-118}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-127}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-98}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-59}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 3: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-118}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-127}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-98}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-57}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y -1.45e-118)
     (- 1.0 y)
     (if (<= y -8e-127)
       (* y x)
       (if (<= y 4e-98)
         1.0
         (if (<= y 4.4e-57) (* y x) (if (<= y 4.2e-7) (- 1.0 y) x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= -1.45e-118) {
		tmp = 1.0 - y;
	} else if (y <= -8e-127) {
		tmp = y * x;
	} else if (y <= 4e-98) {
		tmp = 1.0;
	} else if (y <= 4.4e-57) {
		tmp = y * x;
	} else if (y <= 4.2e-7) {
		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 <= (-1.45d-118)) then
        tmp = 1.0d0 - y
    else if (y <= (-8d-127)) then
        tmp = y * x
    else if (y <= 4d-98) then
        tmp = 1.0d0
    else if (y <= 4.4d-57) then
        tmp = y * x
    else if (y <= 4.2d-7) 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 <= -1.45e-118) {
		tmp = 1.0 - y;
	} else if (y <= -8e-127) {
		tmp = y * x;
	} else if (y <= 4e-98) {
		tmp = 1.0;
	} else if (y <= 4.4e-57) {
		tmp = y * x;
	} else if (y <= 4.2e-7) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= -1.45e-118:
		tmp = 1.0 - y
	elif y <= -8e-127:
		tmp = y * x
	elif y <= 4e-98:
		tmp = 1.0
	elif y <= 4.4e-57:
		tmp = y * x
	elif y <= 4.2e-7:
		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 <= -1.45e-118)
		tmp = Float64(1.0 - y);
	elseif (y <= -8e-127)
		tmp = Float64(y * x);
	elseif (y <= 4e-98)
		tmp = 1.0;
	elseif (y <= 4.4e-57)
		tmp = Float64(y * x);
	elseif (y <= 4.2e-7)
		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 <= -1.45e-118)
		tmp = 1.0 - y;
	elseif (y <= -8e-127)
		tmp = y * x;
	elseif (y <= 4e-98)
		tmp = 1.0;
	elseif (y <= 4.4e-57)
		tmp = y * x;
	elseif (y <= 4.2e-7)
		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, -1.45e-118], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, -8e-127], N[(y * x), $MachinePrecision], If[LessEqual[y, 4e-98], 1.0, If[LessEqual[y, 4.4e-57], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.2e-7], N[(1.0 - y), $MachinePrecision], x]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1 or 4.2e-7 < y

   1. Initial program 36.1%

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

      1. sub-neg 36.1%

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

      2. associate-*l/ 53.1%

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

      3. distribute-lft-neg-in 53.1%

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

      4. distribute-frac-neg 53.1%

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

      5. neg-sub0 53.1%

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

      6. associate--r- 53.1%

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

      7. metadata-eval 53.1%

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

      8. +-commutative 53.1%

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

      9. +-commutative 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in y around inf 76.2%

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

   if -1 < y < -1.4499999999999999e-118 or 4.39999999999999997e-57 < y < 4.2e-7

   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. metadata-eval 100.0%

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

      3. associate--r- 100.0%

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

      4. neg-sub0 100.0%

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

      5. remove-double-neg 100.0%

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

      6. associate-/l* 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 93.7%

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

      1. *-commutative 93.7%

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

      2. /-rgt-identity 93.7%

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

      3. associate-/r/ 93.6%

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

      4. div-sub 93.6%

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

      5. remove-double-div 93.6%

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

      6. associate-/r/ 93.7%

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

      7. /-rgt-identity 93.7%

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

      8. *-commutative 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in x around 0 82.4%

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

   if -1.4499999999999999e-118 < y < -8.0000000000000002e-127 or 3.99999999999999976e-98 < y < 4.39999999999999997e-57

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

   6. Simplified 84.3%

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

   7. Taylor expanded in y around 0 84.3%

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

      1. *-commutative 84.3%

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

   9. Simplified 84.3%

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

   if -8.0000000000000002e-127 < y < 3.99999999999999976e-98

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 84.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-118}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-127}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-98}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-57}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -55000000000:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 150000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -55000000000.0)
   (+ x (/ 1.0 y))
   (if (<= y 150000000.0)
     (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
     (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -55000000000.0) {
		tmp = x + (1.0 / y);
	} else if (y <= 150000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-55000000000.0d0)) then
        tmp = x + (1.0d0 / y)
    else if (y <= 150000000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = x + ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -55000000000.0) {
		tmp = x + (1.0 / y);
	} else if (y <= 150000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -55000000000.0:
		tmp = x + (1.0 / y)
	elif y <= 150000000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -55000000000.0)
		tmp = Float64(x + Float64(1.0 / y));
	elseif (y <= 150000000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -55000000000.0)
		tmp = x + (1.0 / y);
	elseif (y <= 150000000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -55000000000.0], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 150000000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.5e10

   1. Initial program 40.0%

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

      1. sub-neg 40.0%

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

      2. metadata-eval 40.0%

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

      3. associate--r- 40.0%

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

      4. neg-sub0 40.0%

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

      5. remove-double-neg 40.0%

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

      6. associate-/l* 53.3%

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

      7. +-commutative 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   if -5.5e10 < y < 1.5e8

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 1.5e8 < y

   1. Initial program 31.0%

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

      1. sub-neg 31.0%

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

      2. metadata-eval 31.0%

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

      3. associate--r- 31.0%

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

      4. neg-sub0 31.0%

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

      5. remove-double-neg 31.0%

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

      6. associate-/l* 51.8%

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

      7. +-commutative 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in y around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-neg-in 99.8%

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

      7. metadata-eval 99.8%

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

      8. sub-neg 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -55000000000:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 150000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 5: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-118}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.55 \cdot 10^{-127}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-98}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-59}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y -1.45e-118)
     1.0
     (if (<= y -3.55e-127)
       (* y x)
       (if (<= y 4e-98)
         1.0
         (if (<= y 1.85e-59) (* y x) (if (<= y 4.2e-7) 1.0 x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= -1.45e-118) {
		tmp = 1.0;
	} else if (y <= -3.55e-127) {
		tmp = y * x;
	} else if (y <= 4e-98) {
		tmp = 1.0;
	} else if (y <= 1.85e-59) {
		tmp = y * x;
	} else if (y <= 4.2e-7) {
		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 <= (-1.45d-118)) then
        tmp = 1.0d0
    else if (y <= (-3.55d-127)) then
        tmp = y * x
    else if (y <= 4d-98) then
        tmp = 1.0d0
    else if (y <= 1.85d-59) then
        tmp = y * x
    else if (y <= 4.2d-7) 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 <= -1.45e-118) {
		tmp = 1.0;
	} else if (y <= -3.55e-127) {
		tmp = y * x;
	} else if (y <= 4e-98) {
		tmp = 1.0;
	} else if (y <= 1.85e-59) {
		tmp = y * x;
	} else if (y <= 4.2e-7) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= -1.45e-118:
		tmp = 1.0
	elif y <= -3.55e-127:
		tmp = y * x
	elif y <= 4e-98:
		tmp = 1.0
	elif y <= 1.85e-59:
		tmp = y * x
	elif y <= 4.2e-7:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= -1.45e-118)
		tmp = 1.0;
	elseif (y <= -3.55e-127)
		tmp = Float64(y * x);
	elseif (y <= 4e-98)
		tmp = 1.0;
	elseif (y <= 1.85e-59)
		tmp = Float64(y * x);
	elseif (y <= 4.2e-7)
		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 <= -1.45e-118)
		tmp = 1.0;
	elseif (y <= -3.55e-127)
		tmp = y * x;
	elseif (y <= 4e-98)
		tmp = 1.0;
	elseif (y <= 1.85e-59)
		tmp = y * x;
	elseif (y <= 4.2e-7)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, -1.45e-118], 1.0, If[LessEqual[y, -3.55e-127], N[(y * x), $MachinePrecision], If[LessEqual[y, 4e-98], 1.0, If[LessEqual[y, 1.85e-59], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.2e-7], 1.0, x]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 4.2e-7 < y

   1. Initial program 36.1%

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

      1. sub-neg 36.1%

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

      2. associate-*l/ 53.1%

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

      3. distribute-lft-neg-in 53.1%

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

      4. distribute-frac-neg 53.1%

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

      5. neg-sub0 53.1%

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

      6. associate--r- 53.1%

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

      7. metadata-eval 53.1%

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

      8. +-commutative 53.1%

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

      9. +-commutative 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in y around inf 76.2%

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

   if -1 < y < -1.4499999999999999e-118 or -3.5500000000000001e-127 < y < 3.99999999999999976e-98 or 1.85e-59 < y < 4.2e-7

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 83.4%

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

   if -1.4499999999999999e-118 < y < -3.5500000000000001e-127 or 3.99999999999999976e-98 < y < 1.85e-59

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

   6. Simplified 84.3%

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

   7. Taylor expanded in y around 0 84.3%

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

      1. *-commutative 84.3%

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

   9. Simplified 84.3%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-118}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.55 \cdot 10^{-127}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-98}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-59}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.21999999999999997 < y

   1. Initial program 35.6%

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

      1. sub-neg 35.6%

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

      2. metadata-eval 35.6%

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

      3. associate--r- 35.6%

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

      4. neg-sub0 35.6%

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

      5. remove-double-neg 35.6%

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

      6. associate-/l* 52.9%

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

      7. +-commutative 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in y around -inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. distribute-neg-frac 99.3%

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

      5. +-commutative 99.3%

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

      6. distribute-neg-in 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

   6. Simplified 99.3%

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

   if -1 < y < 1.21999999999999997

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate--r- 100.0%

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

      4. neg-sub0 100.0%

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

      5. remove-double-neg 100.0%

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

      6. associate-/l* 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. /-rgt-identity 98.1%

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

      3. associate-/r/ 98.0%

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

      4. div-sub 98.0%

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

      5. remove-double-div 98.0%

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

      6. associate-/r/ 98.1%

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

      7. /-rgt-identity 98.1%

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

      8. *-commutative 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in x around inf 97.6%

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

      1. associate-*r* 97.6%

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

      2. neg-mul-1 97.6%

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

      3. *-commutative 97.6%

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

   9. Simplified 97.6%

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

4. Final simplification 98.4%

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

Alternative 7: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 35.6%

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

      1. sub-neg 35.6%

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

      2. metadata-eval 35.6%

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

      3. associate--r- 35.6%

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

      4. neg-sub0 35.6%

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

      5. remove-double-neg 35.6%

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

      6. associate-/l* 52.9%

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

      7. +-commutative 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in y around -inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. distribute-neg-frac 99.3%

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

      5. +-commutative 99.3%

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

      6. distribute-neg-in 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

   6. Simplified 99.3%

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

   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. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate--r- 100.0%

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

      4. neg-sub0 100.0%

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

      5. remove-double-neg 100.0%

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

      6. associate-/l* 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. /-rgt-identity 98.1%

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

      3. associate-/r/ 98.0%

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

      4. div-sub 98.0%

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

      5. remove-double-div 98.0%

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

      6. associate-/r/ 98.1%

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

      7. /-rgt-identity 98.1%

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

      8. *-commutative 98.1%

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

   6. Simplified 98.1%

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

4. Final simplification 98.7%

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

Alternative 8: 98.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 35.6%

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

      1. sub-neg 35.6%

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

      2. metadata-eval 35.6%

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

      3. associate--r- 35.6%

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

      4. neg-sub0 35.6%

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

      5. remove-double-neg 35.6%

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

      6. associate-/l* 52.9%

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

      7. +-commutative 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in y around -inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. distribute-neg-frac 99.3%

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

      5. +-commutative 99.3%

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

      6. distribute-neg-in 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around 0 98.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate--r- 100.0%

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

      4. neg-sub0 100.0%

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

      5. remove-double-neg 100.0%

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

      6. associate-/l* 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. /-rgt-identity 98.1%

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

      3. associate-/r/ 98.0%

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

      4. div-sub 98.0%

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

      5. remove-double-div 98.0%

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

      6. associate-/r/ 98.1%

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

      7. /-rgt-identity 98.1%

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

      8. *-commutative 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in x around inf 97.6%

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

      1. associate-*r* 97.6%

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

      2. neg-mul-1 97.6%

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

      3. *-commutative 97.6%

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

   9. Simplified 97.6%

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

4. Final simplification 98.2%

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

Alternative 9: 74.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 4.2e-7) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 4.2e-7) {
		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 <= 4.2d-7) 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 <= 4.2e-7) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 4.2e-7:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 4.2e-7)
		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 <= 4.2e-7)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 4.2e-7], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 4.2e-7 < y

   1. Initial program 36.1%

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

      1. sub-neg 36.1%

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

      2. associate-*l/ 53.1%

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

      3. distribute-lft-neg-in 53.1%

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

      4. distribute-frac-neg 53.1%

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

      5. neg-sub0 53.1%

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

      6. associate--r- 53.1%

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

      7. metadata-eval 53.1%

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

      8. +-commutative 53.1%

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

      9. +-commutative 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in y around inf 76.2%

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

   if -1 < y < 4.2e-7

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 76.3%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 38.4% 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 68.8%

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

   1. sub-neg 68.8%

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

   2. associate-*l/ 77.1%

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

   3. distribute-lft-neg-in 77.1%

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

   4. distribute-frac-neg 77.1%

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

   5. neg-sub0 77.1%

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

   6. associate--r- 77.1%

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

   7. metadata-eval 77.1%

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

   8. +-commutative 77.1%

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

   9. +-commutative 77.1%

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

3. Simplified 77.1%

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

4. Taylor expanded in y around 0 40.8%

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

5. Final simplification 40.8%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023271 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

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

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