Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.5% → 99.9%

Time: 9.1s

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: 88.5% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

   1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   6. +-lowering-+.f64 99.8%

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

3. Simplified 99.8%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   7. /-lowering-/.f64 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \frac{x}{\frac{x + 1}{1 + \frac{x}{y}}} \]
8. Add Preprocessing

Alternative 2: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.0003:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -1.0)
     t_0
     (if (<= x -3.1e-63) (* x (/ x y)) (if (<= x 0.0003) x t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -3.1e-63) {
		tmp = x * (x / y);
	} else if (x <= 0.0003) {
		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 = 1.0d0 + (x / y)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-3.1d-63)) then
        tmp = x * (x / y)
    else if (x <= 0.0003d0) 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 = 1.0 + (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -3.1e-63) {
		tmp = x * (x / y);
	} else if (x <= 0.0003) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= -3.1e-63:
		tmp = x * (x / y)
	elif x <= 0.0003:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -3.1e-63)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 0.0003)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -3.1e-63)
		tmp = x * (x / y);
	elseif (x <= 0.0003)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -3.1e-63], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0003], x, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 2.99999999999999974e-4 < x

   1. Initial program 76.5%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. rgt-mult-inverse N/A

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

      13. +-commutative N/A

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

      14. neg-sub0 N/A

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

      15. associate-+l- N/A

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

      16. neg-sub0 N/A

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

      17. mul-1-neg N/A

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

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

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

   7. Simplified 97.2%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 96.5%

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

   if -1 < x < -3.09999999999999984e-63

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.0%

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

      1. /-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lft-identity N/A

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

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

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

      8. /-lowering-/.f64 76.0%

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

   7. Applied egg-rr 76.0%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 59.2%

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

   10. Simplified 59.2%

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

   if -3.09999999999999984e-63 < x < 2.99999999999999974e-4

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 82.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.0003:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + \left(1 + \frac{-1}{y}\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) (+ 1.0 (/ -1.0 y)))))
   (if (<= x -1.0) t_0 (if (<= x 1.26) (* x (+ 1.0 (/ x y))) t_0))))

C

double code(double x, double y) {
	double t_0 = (x / y) + (1.0 + (-1.0 / y));
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.26) {
		tmp = x * (1.0 + (x / 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 + ((-1.0d0) / y))
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.26d0) then
        tmp = x * (1.0d0 + (x / 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 + (-1.0 / y));
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.26) {
		tmp = x * (1.0 + (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + (1.0 + (-1.0 / y))
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 1.26:
		tmp = x * (1.0 + (x / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + Float64(1.0 + Float64(-1.0 / y)))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.26)
		tmp = Float64(x * Float64(1.0 + Float64(x / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + (1.0 + (-1.0 / y));
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.26)
		tmp = x * (1.0 + (x / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.26], N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.26000000000000001 < x

   1. Initial program 76.5%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. rgt-mult-inverse N/A

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

      13. +-commutative N/A

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

      14. neg-sub0 N/A

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

      15. associate-+l- N/A

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

      16. neg-sub0 N/A

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

      17. mul-1-neg N/A

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

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

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

   7. Simplified 97.2%

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

   if -1 < x < 1.26000000000000001

   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

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

   5. Simplified 96.8%

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

      1. /-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lft-identity N/A

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

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

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

      8. /-lowering-/.f64 96.8%

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

   7. Applied egg-rr 96.8%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y} + \left(1 + \frac{-1}{y}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(1 + \frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.0145:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x -4.3e-66) (* x (/ x y)) (if (<= x 0.0145) x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -4.3e-66) {
		tmp = x * (x / y);
	} else if (x <= 0.0145) {
		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 <= (-4.3d-66)) then
        tmp = x * (x / y)
    else if (x <= 0.0145d0) 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 <= -4.3e-66) {
		tmp = x * (x / y);
	} else if (x <= 0.0145) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= -4.3e-66:
		tmp = x * (x / y)
	elif x <= 0.0145:
		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 <= -4.3e-66)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 0.0145)
		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 <= -4.3e-66)
		tmp = x * (x / y);
	elseif (x <= 0.0145)
		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, -4.3e-66], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0145], x, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 0.0145000000000000007 < x

   1. Initial program 76.5%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 70.7%

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

   7. Simplified 70.7%

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

   if -1 < x < -4.30000000000000013e-66

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.0%

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

      1. /-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lft-identity N/A

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

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

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

      8. /-lowering-/.f64 76.0%

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

   7. Applied egg-rr 76.0%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 59.2%

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

   10. Simplified 59.2%

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

   if -4.30000000000000013e-66 < x < 0.0145000000000000007

   1. Initial program 99.9%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 82.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.0145:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (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 = 1.0 + (x / y);
	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 = 1.0d0 + (x / y)
    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 = 1.0 + (x / y);
	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 = 1.0 + (x / y)
	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(1.0 + Float64(x / y))
	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 = 1.0 + (x / y);
	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[(1.0 + N[(x / y), $MachinePrecision]), $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 76.5%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. rgt-mult-inverse N/A

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

      13. +-commutative N/A

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

      14. neg-sub0 N/A

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

      15. associate-+l- N/A

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

      16. neg-sub0 N/A

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

      17. mul-1-neg N/A

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

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

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

   7. Simplified 97.2%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 96.5%

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

   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

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

   5. Simplified 96.8%

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

      1. /-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lft-identity N/A

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

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

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

      8. /-lowering-/.f64 96.8%

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

   7. Applied egg-rr 96.8%

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

4. Final simplification 96.6%

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

Alternative 6: 86.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -31:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 12000:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -31.0) t_0 (if (<= x 12000.0) (* x (/ 1.0 (+ x 1.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -31.0) {
		tmp = t_0;
	} else if (x <= 12000.0) {
		tmp = x * (1.0 / (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 = 1.0d0 + (x / y)
    if (x <= (-31.0d0)) then
        tmp = t_0
    else if (x <= 12000.0d0) then
        tmp = x * (1.0d0 / (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 = 1.0 + (x / y);
	double tmp;
	if (x <= -31.0) {
		tmp = t_0;
	} else if (x <= 12000.0) {
		tmp = x * (1.0 / (x + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -31.0:
		tmp = t_0
	elif x <= 12000.0:
		tmp = x * (1.0 / (x + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -31.0)
		tmp = t_0;
	elseif (x <= 12000.0)
		tmp = Float64(x * Float64(1.0 / Float64(x + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -31.0)
		tmp = t_0;
	elseif (x <= 12000.0)
		tmp = x * (1.0 / (x + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -31.0], t$95$0, If[LessEqual[x, 12000.0], N[(x * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -31 or 12000 < x

   1. Initial program 76.0%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. rgt-mult-inverse N/A

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

      13. +-commutative N/A

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

      14. neg-sub0 N/A

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

      15. associate-+l- N/A

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

      16. neg-sub0 N/A

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

      17. mul-1-neg N/A

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

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

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

   7. Simplified 98.6%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 98.0%

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

   if -31 < x < 12000

   1. Initial program 99.8%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 76.0%

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

   7. Simplified 76.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -31:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 12000:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -31:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 11000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -31.0) t_0 (if (<= x 11000.0) (/ x (+ x 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -31.0) {
		tmp = t_0;
	} else if (x <= 11000.0) {
		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 = 1.0d0 + (x / y)
    if (x <= (-31.0d0)) then
        tmp = t_0
    else if (x <= 11000.0d0) 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 = 1.0 + (x / y);
	double tmp;
	if (x <= -31.0) {
		tmp = t_0;
	} else if (x <= 11000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -31.0:
		tmp = t_0
	elif x <= 11000.0:
		tmp = x / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -31.0)
		tmp = t_0;
	elseif (x <= 11000.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -31.0)
		tmp = t_0;
	elseif (x <= 11000.0)
		tmp = x / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -31.0], t$95$0, If[LessEqual[x, 11000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -31 or 11000 < x

   1. Initial program 76.0%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. rgt-mult-inverse N/A

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

      13. +-commutative N/A

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

      14. neg-sub0 N/A

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

      15. associate-+l- N/A

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

      16. neg-sub0 N/A

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

      17. mul-1-neg N/A

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

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

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

   7. Simplified 98.6%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 98.0%

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

   if -31 < x < 11000

   1. Initial program 99.8%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 75.9%

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

   7. Simplified 75.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -31:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 11000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 0.38:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ x y) (if (<= x 0.38) x (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 0.38) {
		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 <= 0.38d0) 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 <= 0.38) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 0.38:
		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 <= 0.38)
		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 <= 0.38)
		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, 0.38], x, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.38 < x

   1. Initial program 76.5%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 70.7%

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

   7. Simplified 70.7%

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

   if -1 < x < 0.38

   1. Initial program 99.8%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 74.9%

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

Alternative 9: 50.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -205000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -205000000.0) 1.0 (if (<= x 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -205000000.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-205000000.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -205000000.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -205000000.0:
		tmp = 1.0
	elif x <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -205000000.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -205000000.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -205000000.0], 1.0, If[LessEqual[x, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -2.05e8 or 1 < x

   1. Initial program 76.0%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 28.8%

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

   7. Simplified 28.8%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 27.8%

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

   if -2.05e8 < x < 1

   1. Initial program 99.8%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      6. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 73.2%

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

Alternative 10: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return x * ((1.0 + (x / y)) / (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)) / (x + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

   1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   6. +-lowering-+.f64 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

   \[\leadsto x \cdot \frac{1 + \frac{x}{y}}{x + 1} \]
6. Add Preprocessing

Alternative 11: 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 87.9%

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

   1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   6. +-lowering-+.f64 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around inf

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

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

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

   2. +-commutative N/A

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

   3. +-lowering-+.f64 52.0%

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

7. Simplified 52.0%

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

8. Taylor expanded in x around inf

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

10. Simplified 15.9%

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

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024161 
(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)))