Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.7% → 99.8%

Time: 8.5s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\frac{x}{y} + 1}{x + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

   6. +-lowering-+.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y}\\ t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -6300000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 1400:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) y)) (t_1 (/ x (+ x 1.0))))
   (if (<= x -6300000.0)
     t_0
     (if (<= x 2.95e-95)
       t_1
       (if (<= x 2e-48) (/ x (/ y x)) (if (<= x 1400.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = (x + y) / y;
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -6300000.0) {
		tmp = t_0;
	} else if (x <= 2.95e-95) {
		tmp = t_1;
	} else if (x <= 2e-48) {
		tmp = x / (y / x);
	} else if (x <= 1400.0) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (x + y) / y
    t_1 = x / (x + 1.0d0)
    if (x <= (-6300000.0d0)) then
        tmp = t_0
    else if (x <= 2.95d-95) then
        tmp = t_1
    else if (x <= 2d-48) then
        tmp = x / (y / x)
    else if (x <= 1400.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / y;
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -6300000.0) {
		tmp = t_0;
	} else if (x <= 2.95e-95) {
		tmp = t_1;
	} else if (x <= 2e-48) {
		tmp = x / (y / x);
	} else if (x <= 1400.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / y
	t_1 = x / (x + 1.0)
	tmp = 0
	if x <= -6300000.0:
		tmp = t_0
	elif x <= 2.95e-95:
		tmp = t_1
	elif x <= 2e-48:
		tmp = x / (y / x)
	elif x <= 1400.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / y)
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -6300000.0)
		tmp = t_0;
	elseif (x <= 2.95e-95)
		tmp = t_1;
	elseif (x <= 2e-48)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 1400.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / y;
	t_1 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -6300000.0)
		tmp = t_0;
	elseif (x <= 2.95e-95)
		tmp = t_1;
	elseif (x <= 2e-48)
		tmp = x / (y / x);
	elseif (x <= 1400.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6300000.0], t$95$0, If[LessEqual[x, 2.95e-95], t$95$1, If[LessEqual[x, 2e-48], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1400.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.3e6 or 1400 < x

   1. Initial program 73.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right)\right), \color{blue}{x}\right) \]
   4. Step-by-step derivation

   5. Simplified 72.6%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + y\right), \color{blue}{y}\right) \]

      2. +-lowering-+.f64 99.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), y\right) \]

   8. Simplified 99.4%

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

   if -6.3e6 < x < 2.9499999999999999e-95 or 1.9999999999999999e-48 < x < 1400

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + x\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(x + \color{blue}{1}\right)\right) \]

      3. +-lowering-+.f64 77.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 77.5%

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

   if 2.9499999999999999e-95 < x < 1.9999999999999999e-48

   1. Initial program 99.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{x}{y} + 1\right) \cdot x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y} \cdot x + x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y} \cdot x\right), x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{\frac{y}{x}}\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{x}\right)\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

      6. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), x\right), \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 99.9%

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

      1. /-rgt-identity N/A

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

      2. associate-/r/ N/A

         \[\leadsto \frac{x}{y} \cdot x + x \]

      3. associate-*l/ N/A

         \[\leadsto \frac{x \cdot x}{y} + x \]

      4. remove-double-div N/A

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

      5. frac-add N/A

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

      6. associate-*l* N/A

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

      7. rgt-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. *-rgt-identity N/A

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

      10. div-inv N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x + y\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

      13. /-lowering-/.f64 92.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 92.5%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(y, x\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 83.9%

       \[\leadsto \frac{\color{blue}{x}}{\frac{y}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -1.75e+21)
     (/ x y)
     (if (<= x 3.6e-95)
       t_0
       (if (<= x 2.3e-46) (/ x (/ y x)) (if (<= x 2.45e+30) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1.75e+21) {
		tmp = x / y;
	} else if (x <= 3.6e-95) {
		tmp = t_0;
	} else if (x <= 2.3e-46) {
		tmp = x / (y / x);
	} else if (x <= 2.45e+30) {
		tmp = t_0;
	} else {
		tmp = x / 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 / (x + 1.0d0)
    if (x <= (-1.75d+21)) then
        tmp = x / y
    else if (x <= 3.6d-95) then
        tmp = t_0
    else if (x <= 2.3d-46) then
        tmp = x / (y / x)
    else if (x <= 2.45d+30) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1.75e+21) {
		tmp = x / y;
	} else if (x <= 3.6e-95) {
		tmp = t_0;
	} else if (x <= 2.3e-46) {
		tmp = x / (y / x);
	} else if (x <= 2.45e+30) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -1.75e+21:
		tmp = x / y
	elif x <= 3.6e-95:
		tmp = t_0
	elif x <= 2.3e-46:
		tmp = x / (y / x)
	elif x <= 2.45e+30:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.75e+21)
		tmp = Float64(x / y);
	elseif (x <= 3.6e-95)
		tmp = t_0;
	elseif (x <= 2.3e-46)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 2.45e+30)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -1.75e+21)
		tmp = x / y;
	elseif (x <= 3.6e-95)
		tmp = t_0;
	elseif (x <= 2.3e-46)
		tmp = x / (y / x);
	elseif (x <= 2.45e+30)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.75e+21], N[(x / y), $MachinePrecision], If[LessEqual[x, 3.6e-95], t$95$0, If[LessEqual[x, 2.3e-46], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.45e+30], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e21 or 2.44999999999999992e30 < x

   1. Initial program 72.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 78.2%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 78.2%

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

   if -1.75e21 < x < 3.6e-95 or 2.2999999999999999e-46 < x < 2.44999999999999992e30

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + x\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(x + \color{blue}{1}\right)\right) \]

      3. +-lowering-+.f64 76.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 76.8%

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

   if 3.6e-95 < x < 2.2999999999999999e-46

   1. Initial program 99.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{x}{y} + 1\right) \cdot x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y} \cdot x + x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y} \cdot x\right), x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{\frac{y}{x}}\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{x}\right)\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

      6. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), x\right), \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 99.9%

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

      1. /-rgt-identity N/A

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

      2. associate-/r/ N/A

         \[\leadsto \frac{x}{y} \cdot x + x \]

      3. associate-*l/ N/A

         \[\leadsto \frac{x \cdot x}{y} + x \]

      4. remove-double-div N/A

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

      5. frac-add N/A

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

      6. associate-*l* N/A

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

      7. rgt-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. *-rgt-identity N/A

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

      10. div-inv N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x + y\right), \color{blue}{\left(\frac{y}{x}\right)}\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\frac{\color{blue}{y}}{x}\right)\right) \]

      13. /-lowering-/.f64 92.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 92.5%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(y, x\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 83.9%

       \[\leadsto \frac{\color{blue}{x}}{\frac{y}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -5.5e+21)
     (/ x y)
     (if (<= x 3.6e-95)
       t_0
       (if (<= x 4.5e-47) (* x (/ x y)) (if (<= x 1.3e+30) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -5.5e+21) {
		tmp = x / y;
	} else if (x <= 3.6e-95) {
		tmp = t_0;
	} else if (x <= 4.5e-47) {
		tmp = x * (x / y);
	} else if (x <= 1.3e+30) {
		tmp = t_0;
	} else {
		tmp = x / 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 / (x + 1.0d0)
    if (x <= (-5.5d+21)) then
        tmp = x / y
    else if (x <= 3.6d-95) then
        tmp = t_0
    else if (x <= 4.5d-47) then
        tmp = x * (x / y)
    else if (x <= 1.3d+30) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -5.5e+21) {
		tmp = x / y;
	} else if (x <= 3.6e-95) {
		tmp = t_0;
	} else if (x <= 4.5e-47) {
		tmp = x * (x / y);
	} else if (x <= 1.3e+30) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -5.5e+21:
		tmp = x / y
	elif x <= 3.6e-95:
		tmp = t_0
	elif x <= 4.5e-47:
		tmp = x * (x / y)
	elif x <= 1.3e+30:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -5.5e+21)
		tmp = Float64(x / y);
	elseif (x <= 3.6e-95)
		tmp = t_0;
	elseif (x <= 4.5e-47)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 1.3e+30)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -5.5e+21)
		tmp = x / y;
	elseif (x <= 3.6e-95)
		tmp = t_0;
	elseif (x <= 4.5e-47)
		tmp = x * (x / y);
	elseif (x <= 1.3e+30)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e+21], N[(x / y), $MachinePrecision], If[LessEqual[x, 3.6e-95], t$95$0, If[LessEqual[x, 4.5e-47], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e+30], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5e21 or 1.29999999999999994e30 < x

   1. Initial program 72.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 78.2%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 78.2%

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

   if -5.5e21 < x < 3.6e-95 or 4.5e-47 < x < 1.29999999999999994e30

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + x\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(x + \color{blue}{1}\right)\right) \]

      3. +-lowering-+.f64 76.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 76.8%

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

   if 3.6e-95 < x < 4.5e-47

   1. Initial program 99.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{x}{y} + 1\right) \cdot x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y} \cdot x + x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y} \cdot x\right), x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{\frac{y}{x}}\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{x}\right)\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

      6. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), x\right), \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 99.9%

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

      1. /-rgt-identity N/A

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

      2. associate-/r/ N/A

         \[\leadsto \frac{x}{y} \cdot x + x \]

      3. distribute-lft1-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{x}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{x}{y}\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{y}\right)\right), x\right) \]

      7. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, y\right)\right), x\right) \]

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, x\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 83.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), x\right) \]

   12. Simplified 83.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 x))))
   (if (<= x -1.0)
     (/ x y)
     (if (<= x 3.6e-95)
       t_0
       (if (<= x 1.55e-47) (* x (/ x y)) (if (<= x 0.32) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - x);
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 3.6e-95) {
		tmp = t_0;
	} else if (x <= 1.55e-47) {
		tmp = x * (x / y);
	} else if (x <= 0.32) {
		tmp = t_0;
	} else {
		tmp = x / 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 - x)
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 3.6d-95) then
        tmp = t_0
    else if (x <= 1.55d-47) then
        tmp = x * (x / y)
    else if (x <= 0.32d0) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - x);
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 3.6e-95) {
		tmp = t_0;
	} else if (x <= 1.55e-47) {
		tmp = x * (x / y);
	} else if (x <= 0.32) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - x)
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 3.6e-95:
		tmp = t_0
	elif x <= 1.55e-47:
		tmp = x * (x / y)
	elif x <= 0.32:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 3.6e-95)
		tmp = t_0;
	elseif (x <= 1.55e-47)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 0.32)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - x);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 3.6e-95)
		tmp = t_0;
	elseif (x <= 1.55e-47)
		tmp = x * (x / y);
	elseif (x <= 0.32)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 3.6e-95], t$95$0, If[LessEqual[x, 1.55e-47], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.32], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 0.320000000000000007 < x

   1. Initial program 74.2%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 73.7%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 73.7%

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

   if -1 < x < 3.6e-95 or 1.5499999999999999e-47 < x < 0.320000000000000007

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{1 + x}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + x\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(x + \color{blue}{1}\right)\right)\right) \]

      3. +-lowering-+.f64 78.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 78.1%

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

   8. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{x}\right)\right) \]

      4. --lowering--.f64 77.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right) \]

   10. Simplified 77.5%

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

   if 3.6e-95 < x < 1.5499999999999999e-47

   1. Initial program 99.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{x}{y} + 1\right) \cdot x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y} \cdot x + x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y} \cdot x\right), x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{\frac{y}{x}}\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{x}\right)\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

      6. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), x\right), \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 99.9%

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

      1. /-rgt-identity N/A

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

      2. associate-/r/ N/A

         \[\leadsto \frac{x}{y} \cdot x + x \]

      3. distribute-lft1-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{x}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{x}{y}\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{y}\right)\right), x\right) \]

      7. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, y\right)\right), x\right) \]

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, x\right) \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 83.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, y\right), x\right) \]

   12. Simplified 83.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-95}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;x \cdot \left(\left(\frac{x}{y} + 1\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) y)))
   (if (<= x -1.0) t_0 (if (<= x 0.85) (* x (- (+ (/ x y) 1.0) x)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.85) {
		tmp = x * (((x / y) + 1.0) - x);
	} 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 + y) / y
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 0.85d0) then
        tmp = x * (((x / y) + 1.0d0) - x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.85) {
		tmp = x * (((x / y) + 1.0) - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / y
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 0.85:
		tmp = x * (((x / y) + 1.0) - x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.85)
		tmp = Float64(x * Float64(Float64(Float64(x / y) + 1.0) - x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / y;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.85)
		tmp = x * (((x / y) + 1.0) - x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.85], N[(x * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.849999999999999978 < x

   1. Initial program 74.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right)\right), \color{blue}{x}\right) \]
   4. Step-by-step derivation

   5. Simplified 71.3%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + y\right), \color{blue}{y}\right) \]

      2. +-lowering-+.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), y\right) \]

   8. Simplified 97.0%

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

   if -1 < x < 0.849999999999999978

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(\frac{1}{y} - 1\right)\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + x \cdot \left(\frac{1}{y} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + x \cdot \left(\frac{1}{y} + -1\right)\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\frac{1}{y} \cdot x + \color{blue}{-1 \cdot x}\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\frac{1 \cdot x}{y} + \color{blue}{-1} \cdot x\right)\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\frac{x}{y} + -1 \cdot x\right)\right)\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(1 + \frac{x}{y}\right) + \color{blue}{-1 \cdot x}\right)\right) \]

      8. lft-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{x} \cdot x + \frac{x}{y}\right) + -1 \cdot x\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{x} \cdot x + \frac{1 \cdot x}{y}\right) + -1 \cdot x\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{x} \cdot x + \frac{1}{y} \cdot x\right) + -1 \cdot x\right)\right) \]

      11. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{x} + \frac{1}{y}\right) + \color{blue}{-1} \cdot x\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{x} + \frac{1}{y}\right) + \left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{x} + \frac{1}{y}\right) - \color{blue}{x}\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(x \cdot \left(\frac{1}{x} + \frac{1}{y}\right)\right), \color{blue}{x}\right)\right) \]

      15. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{1}{x} \cdot x + \frac{1}{y} \cdot x\right), x\right)\right) \]

      16. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(1 + \frac{1}{y} \cdot x\right), x\right)\right) \]

      17. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(1 + \frac{1 \cdot x}{y}\right), x\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(1 + \frac{x}{y}\right), x\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{y}\right)\right), x\right)\right) \]

      20. /-lowering-/.f64 98.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, y\right)\right), x\right)\right) \]

   7. Simplified 98.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;x \cdot \left(\left(\frac{x}{y} + 1\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 74.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right)\right), \color{blue}{x}\right) \]
   4. Step-by-step derivation

   5. Simplified 71.3%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + y\right), \color{blue}{y}\right) \]

      2. +-lowering-+.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), y\right) \]

   8. Simplified 97.0%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{x}{y} + 1\right) \cdot x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y} \cdot x + x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y} \cdot x\right), x\right), \mathsf{+.f64}\left(\color{blue}{x}, 1\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{\frac{y}{x}}\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{x}\right)\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

      6. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), x\right), \mathsf{+.f64}\left(x, 1\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, x\right)\right), x\right), \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.9%

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

      1. /-rgt-identity N/A

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

      2. associate-/r/ N/A

         \[\leadsto \frac{x}{y} \cdot x + x \]

      3. distribute-lft1-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{x}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{x}{y}\right), x\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{y}\right)\right), x\right) \]

      7. /-lowering-/.f64 97.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, y\right)\right), x\right) \]

   9. Applied egg-rr 97.8%

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

4. Final simplification 97.4%

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

Alternative 8: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 0.78:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ x y) (if (<= x 0.78) (* x (- 1.0 x)) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 0.78) {
		tmp = x * (1.0 - x);
	} else {
		tmp = 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 (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 0.78d0) then
        tmp = x * (1.0d0 - x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 0.78) {
		tmp = x * (1.0 - x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 0.78:
		tmp = x * (1.0 - x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 0.78)
		tmp = Float64(x * Float64(1.0 - x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 0.78)
		tmp = x * (1.0 - x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 0.78], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.78000000000000003 < x

   1. Initial program 74.2%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 73.7%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 73.7%

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

   if -1 < x < 0.78000000000000003

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{1 + x}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + x\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(x + \color{blue}{1}\right)\right)\right) \]

      3. +-lowering-+.f64 72.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 72.2%

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

   8. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + -1 \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(1 - \color{blue}{x}\right)\right) \]

      4. --lowering--.f64 71.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right) \]

   10. Simplified 71.6%

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

Alternative 9: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 160:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ x y) (if (<= x 160.0) x (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 160.0) {
		tmp = x;
	} else {
		tmp = 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 (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 160.0d0) then
        tmp = x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 160.0) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 160.0:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 160.0)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 160.0)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 160.0], x, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 160 < x

   1. Initial program 74.0%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 74.3%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 74.3%

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

   if -1 < x < 160

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   6. Step-by-step derivation

   7. Simplified 70.6%

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

Alternative 10: 49.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 74.2%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{1 + x}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + x\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(x + \color{blue}{1}\right)\right)\right) \]

      3. +-lowering-+.f64 27.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 27.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 25.7%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

      6. +-lowering-+.f64 99.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   6. Step-by-step derivation

   7. Simplified 71.0%

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

Alternative 11: 14.6% 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 87.5%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{x}{y} + 1}{x + 1}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y} + 1\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{x}{y}\right), 1\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \left(x + 1\right)\right)\right) \]

   6. +-lowering-+.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(x, y\right), 1\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
4. Add Preprocessing

5. Taylor expanded in y around inf

   \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{1 + x}\right)}\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(1 + x\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \left(x + \color{blue}{1}\right)\right)\right) \]

   3. +-lowering-+.f64 50.3%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

7. Simplified 50.3%

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

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 14.5%

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

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1))))

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