Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.3% → 99.8%

Time: 6.8s

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: 88.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 99.9

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \frac{x - 1}{y}\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x - 1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (/ (- x 1.0) y)))
   (if (<= t_0 -50.0)
     t_1
     (if (<= t_0 5e-16)
       (* (- 1.0 x) x)
       (if (<= t_0 2.0) (/ (- x 1.0) x) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = (x - 1.0) / y;
	double tmp;
	if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-16) {
		tmp = (1.0 - x) * x;
	} else if (t_0 <= 2.0) {
		tmp = (x - 1.0) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    t_1 = (x - 1.0d0) / y
    if (t_0 <= (-50.0d0)) then
        tmp = t_1
    else if (t_0 <= 5d-16) then
        tmp = (1.0d0 - x) * x
    else if (t_0 <= 2.0d0) then
        tmp = (x - 1.0d0) / x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = (x - 1.0) / y;
	double tmp;
	if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-16) {
		tmp = (1.0 - x) * x;
	} else if (t_0 <= 2.0) {
		tmp = (x - 1.0) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	t_1 = (x - 1.0) / y
	tmp = 0
	if t_0 <= -50.0:
		tmp = t_1
	elif t_0 <= 5e-16:
		tmp = (1.0 - x) * x
	elif t_0 <= 2.0:
		tmp = (x - 1.0) / x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64(Float64(x - 1.0) / y)
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 5e-16)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(x - 1.0) / x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	t_1 = (x - 1.0) / y;
	tmp = 0.0;
	if (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 5e-16)
		tmp = (1.0 - x) * x;
	elseif (t_0 <= 2.0)
		tmp = (x - 1.0) / x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -50.0], t$95$1, If[LessEqual[t$95$0, 5e-16], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(x - 1.0), $MachinePrecision] / x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -50 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.3%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.6%

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

   if -50 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000004e-16

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{-1 \cdot {x}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.5%

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

   if 5.0000000000000004e-16 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{x - 1}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 83.5%

       \[\leadsto \frac{x - 1}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -50 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -50.0) (not (<= t_0 2.0)))
     (/ (- x 1.0) y)
     (/ x (+ 1.0 x)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -50.0) || !(t_0 <= 2.0)) {
		tmp = (x - 1.0) / y;
	} else {
		tmp = x / (1.0 + x);
	}
	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 / y) + 1.0d0)) / (x + 1.0d0)
    if ((t_0 <= (-50.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = (x - 1.0d0) / y
    else
        tmp = x / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -50.0) || !(t_0 <= 2.0)) {
		tmp = (x - 1.0) / y;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if (t_0 <= -50.0) or not (t_0 <= 2.0):
		tmp = (x - 1.0) / y
	else:
		tmp = x / (1.0 + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if ((t_0 <= -50.0) || !(t_0 <= 2.0))
		tmp = Float64(Float64(x - 1.0) / y);
	else
		tmp = Float64(x / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if ((t_0 <= -50.0) || ~((t_0 <= 2.0)))
		tmp = (x - 1.0) / y;
	else
		tmp = x / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -50.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -50 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.3%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.6%

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

   if -50 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 86.2

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

   5. Applied rewrites 86.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -50 \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2\right):\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -50000000000000:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)) -50000000000000.0)
   (* (- x) x)
   (* 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -50000000000000.0) {
		tmp = -x * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * ((x / y) + 1.0d0)) / (x + 1.0d0)) <= (-50000000000000.0d0)) then
        tmp = -x * x
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -50000000000000.0) {
		tmp = -x * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) + 1.0)) / (x + 1.0)) <= -50000000000000.0:
		tmp = -x * x
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0)) <= -50000000000000.0)
		tmp = Float64(Float64(-x) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) + 1.0)) / (x + 1.0)) <= -50000000000000.0)
		tmp = -x * x;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], -50000000000000.0], N[((-x) * x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5e13

   1. Initial program 66.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 24.7

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

   5. Applied rewrites 24.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{-1 \cdot {x}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 20.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot x\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 20.7%

       \[\leadsto \left(-x\right) \cdot x \]

   if -5e13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 92.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 50.3%

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

Alternative 5: 98.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.8%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.4%

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

   if -1 < x < 1

   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. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lft-mult-inverse N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. neg-mul-1 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 99.3%

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

4. Final simplification 98.4%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{1 + x} + \frac{{x}^{2}}{1 + x}}{y} \]

   3. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{1 + x}} + \frac{{x}^{2}}{1 + x}}{y} \]

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. +-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 87.9

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

5. Applied rewrites 87.9%

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.9%

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

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.8%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.4%

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

   if -1 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 98.3%

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

Alternative 8: 87.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -140000 \lor \neg \left(x \leq 280\right):\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -140000.0) (not (<= x 280.0)))
   (+ (/ (- x 1.0) y) 1.0)
   (/ x (+ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -140000.0) || !(x <= 280.0)) {
		tmp = ((x - 1.0) / y) + 1.0;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-140000.0d0)) .or. (.not. (x <= 280.0d0))) then
        tmp = ((x - 1.0d0) / y) + 1.0d0
    else
        tmp = x / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -140000.0) || !(x <= 280.0)) {
		tmp = ((x - 1.0) / y) + 1.0;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -140000.0) or not (x <= 280.0):
		tmp = ((x - 1.0) / y) + 1.0
	else:
		tmp = x / (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -140000.0) || !(x <= 280.0))
		tmp = Float64(Float64(Float64(x - 1.0) / y) + 1.0);
	else
		tmp = Float64(x / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -140000.0) || ~((x <= 280.0)))
		tmp = ((x - 1.0) / y) + 1.0;
	else
		tmp = x / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -140000.0], N[Not[LessEqual[x, 280.0]], $MachinePrecision]], N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e5 or 280 < x

   1. Initial program 72.9%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.7%

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

   if -1.4e5 < x < 280

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 75.9

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

   5. Applied rewrites 75.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -140000 \lor \neg \left(x \leq 280\right):\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.8%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.3%

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

   if -1 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{-1 \cdot {x}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.3%

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

4. Final simplification 74.3%

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

Alternative 10: 43.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. distribute-lft-out-- N/A

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

   6. associate-*r/ N/A

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

   7. *-rgt-identity N/A

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

   8. *-rgt-identity N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 56.3

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

5. Applied rewrites 56.3%

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

6. Taylor expanded in y around inf

   \[\leadsto x + \color{blue}{-1 \cdot {x}^{2}} \]
7. Step-by-step derivation

8. Applied rewrites 43.8%

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

Alternative 11: 38.7% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 99.9

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 41.3%

   \[\leadsto \color{blue}{1} \cdot x \]
8. 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 2024315 
(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)))