Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.3% → 99.9%

Time: 7.6s

Alternatives: 8

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.9%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. +-lowering-+.f64 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 97.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 78.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 96.9%

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

      1. *-rgt-identity N/A

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

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

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

      3. /-lowering-/.f64 96.9%

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

   9. Applied egg-rr 96.9%

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

   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. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 96.4%

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

   8. Taylor expanded in y around 0

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

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

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

      2. +-lowering-+.f64 96.4%

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

   10. Simplified 96.4%

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

4. Final simplification 96.6%

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

Alternative 3: 97.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 78.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 96.9%

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

      1. *-rgt-identity N/A

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

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

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

      3. /-lowering-/.f64 96.9%

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

   9. Applied egg-rr 96.9%

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

   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. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 96.4%

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

4. Final simplification 96.6%

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

Alternative 4: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ \mathbf{if}\;x \leq -6200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)))
   (if (<= x -6200000.0) t_0 (if (<= x 0.00165) (/ x (+ x 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if (x <= -6200000.0) {
		tmp = t_0;
	} else if (x <= 0.00165) {
		tmp = x / (x + 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) + 1.0d0
    if (x <= (-6200000.0d0)) then
        tmp = t_0
    else if (x <= 0.00165d0) then
        tmp = x / (x + 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) + 1.0;
	double tmp;
	if (x <= -6200000.0) {
		tmp = t_0;
	} else if (x <= 0.00165) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	tmp = 0
	if x <= -6200000.0:
		tmp = t_0
	elif x <= 0.00165:
		tmp = x / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	tmp = 0.0
	if (x <= -6200000.0)
		tmp = t_0;
	elseif (x <= 0.00165)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + 1.0;
	tmp = 0.0;
	if (x <= -6200000.0)
		tmp = t_0;
	elseif (x <= 0.00165)
		tmp = x / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -6200000.0], t$95$0, If[LessEqual[x, 0.00165], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2e6 or 0.00165 < x

   1. Initial program 77.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 98.9%

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

      1. *-rgt-identity N/A

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

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

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

      3. /-lowering-/.f64 98.9%

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

   9. Applied egg-rr 98.9%

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

   if -6.2e6 < x < 0.00165

   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

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

   5. Simplified 71.2%

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

Alternative 5: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.00165:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0))) (if (<= x -1.0) t_0 (if (<= x 0.00165) x t_0))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.00165) {
		tmp = 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) + 1.0d0
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 0.00165d0) then
        tmp = 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) + 1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.00165) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 0.00165:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.00165)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + 1.0;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.00165)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.00165], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.00165 < x

   1. Initial program 78.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 96.2%

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

      1. *-rgt-identity N/A

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

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

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

      3. /-lowering-/.f64 96.2%

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

   9. Applied egg-rr 96.2%

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

   if -1 < x < 0.00165

   1. Initial program 99.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 71.1%

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

Alternative 6: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 0.00135:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ x y) (if (<= x 0.00135) x (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 0.00135) {
		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 <= 0.00135d0) 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 <= 0.00135) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 0.00135:
		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 <= 0.00135)
		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 <= 0.00135)
		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, 0.00135], x, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.0013500000000000001 < x

   1. Initial program 78.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 72.1%

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

   7. Simplified 72.1%

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

   if -1 < x < 0.0013500000000000001

   1. Initial program 99.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 71.1%

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

Alternative 7: 48.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.8e-6) 1.0 (if (<= x 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.8e-6) {
		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 <= (-3.8d-6)) 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 <= -3.8e-6) {
		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 <= -3.8e-6:
		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 <= -3.8e-6)
		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 <= -3.8e-6)
		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, -3.8e-6], 1.0, If[LessEqual[x, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e-6 or 1 < x

   1. Initial program 78.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 27.5%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 25.9%

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

   if -3.8e-6 < x < 1

   1. Initial program 99.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 72.7%

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

Alternative 8: 14.7% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 88.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 49.5%

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

6. Taylor expanded in x around inf

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

8. Simplified 15.2%

   \[\leadsto \color{blue}{1} \]
9. 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 2024138 
(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)))