Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 9.4s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. *-rgt-identity N/A

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

   4. associate-/l* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. associate-*r/ N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. associate-*r/ N/A

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

   9. *-rgt-identity N/A

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

   10. lower-/.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 100.0

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

8. Applied rewrites 100.0%

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

Alternative 2: 84.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{y + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+131}:\\ \;\;\;\;x - y \cdot x\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\left(1 - y\right) \cdot \left(y + x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (+ y 1.0))))
   (if (<= t_0 -1e+131)
     (- x (* y x))
     (if (<= t_0 -1e+67)
       (/ x y)
       (if (<= t_0 0.02)
         (* (- 1.0 y) (+ y x))
         (if (<= t_0 5e+34) 1.0 (+ 1.0 x)))))))

C

double code(double x, double y) {
	double t_0 = (y + x) / (y + 1.0);
	double tmp;
	if (t_0 <= -1e+131) {
		tmp = x - (y * x);
	} else if (t_0 <= -1e+67) {
		tmp = x / y;
	} else if (t_0 <= 0.02) {
		tmp = (1.0 - y) * (y + x);
	} else if (t_0 <= 5e+34) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) / (y + 1.0d0)
    if (t_0 <= (-1d+131)) then
        tmp = x - (y * x)
    else if (t_0 <= (-1d+67)) then
        tmp = x / y
    else if (t_0 <= 0.02d0) then
        tmp = (1.0d0 - y) * (y + x)
    else if (t_0 <= 5d+34) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) / (y + 1.0);
	double tmp;
	if (t_0 <= -1e+131) {
		tmp = x - (y * x);
	} else if (t_0 <= -1e+67) {
		tmp = x / y;
	} else if (t_0 <= 0.02) {
		tmp = (1.0 - y) * (y + x);
	} else if (t_0 <= 5e+34) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) / (y + 1.0)
	tmp = 0
	if t_0 <= -1e+131:
		tmp = x - (y * x)
	elif t_0 <= -1e+67:
		tmp = x / y
	elif t_0 <= 0.02:
		tmp = (1.0 - y) * (y + x)
	elif t_0 <= 5e+34:
		tmp = 1.0
	else:
		tmp = 1.0 + x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= -1e+131)
		tmp = Float64(x - Float64(y * x));
	elseif (t_0 <= -1e+67)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.02)
		tmp = Float64(Float64(1.0 - y) * Float64(y + x));
	elseif (t_0 <= 5e+34)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= -1e+131)
		tmp = x - (y * x);
	elseif (t_0 <= -1e+67)
		tmp = x / y;
	elseif (t_0 <= 0.02)
		tmp = (1.0 - y) * (y + x);
	elseif (t_0 <= 5e+34)
		tmp = 1.0;
	else
		tmp = 1.0 + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+131], N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e+67], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(N[(1.0 - y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+34], 1.0, N[(1.0 + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < -9.9999999999999991e130

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 85.3

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

   8. Applied rewrites 85.3%

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

   if -9.9999999999999991e130 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < -9.99999999999999983e66

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 84.6

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

   8. Applied rewrites 84.6%

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

   if -9.99999999999999983e66 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 93.3

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

   8. Applied rewrites 93.3%

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

   if 0.0200000000000000004 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 4.9999999999999998e34

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.1%

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

   if 4.9999999999999998e34 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 71.6

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

   14. Applied rewrites 71.6%

       \[\leadsto \color{blue}{1 + x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{y + 1} \leq -1 \cdot 10^{+131}:\\ \;\;\;\;x - y \cdot x\\ \mathbf{elif}\;\frac{y + x}{y + 1} \leq -1 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{y + x}{y + 1} \leq 0.02:\\ \;\;\;\;\left(1 - y\right) \cdot \left(y + x\right)\\ \mathbf{elif}\;\frac{y + x}{y + 1} \leq 5 \cdot 10^{+34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{y + 1}\\ \mathbf{if}\;t\_0 \leq 0.02:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (+ y 1.0))))
   (if (<= t_0 0.02) (+ y x) (if (<= t_0 5e+34) 1.0 (+ 1.0 x)))))

C

double code(double x, double y) {
	double t_0 = (y + x) / (y + 1.0);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = y + x;
	} else if (t_0 <= 5e+34) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) / (y + 1.0d0)
    if (t_0 <= 0.02d0) then
        tmp = y + x
    else if (t_0 <= 5d+34) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) / (y + 1.0);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = y + x;
	} else if (t_0 <= 5e+34) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) / (y + 1.0)
	tmp = 0
	if t_0 <= 0.02:
		tmp = y + x
	elif t_0 <= 5e+34:
		tmp = 1.0
	else:
		tmp = 1.0 + x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= 0.02)
		tmp = Float64(y + x);
	elseif (t_0 <= 5e+34)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= 0.02)
		tmp = y + x;
	elseif (t_0 <= 5e+34)
		tmp = 1.0;
	else
		tmp = 1.0 + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.02], N[(y + x), $MachinePrecision], If[LessEqual[t$95$0, 5e+34], 1.0, N[(1.0 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 82.2%

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

      1. lift-+.f64 N/A

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

      2. *-lft-identity 82.2

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

   10. Applied rewrites 82.2%

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

   if 0.0200000000000000004 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 4.9999999999999998e34

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.1%

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

   if 4.9999999999999998e34 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 71.6

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

   14. Applied rewrites 71.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{y + 1} \leq 0.02:\\ \;\;\;\;y + x\\ \mathbf{elif}\;\frac{y + x}{y + 1} \leq 5 \cdot 10^{+34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y + 1}\\ \mathbf{if}\;y \leq -0.0008:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.0051:\\ \;\;\;\;\left(1 - y\right) \cdot \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x (+ y 1.0)))))
   (if (<= y -0.0008) t_0 (if (<= y 0.0051) (* (- 1.0 y) (+ y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / (y + 1.0));
	double tmp;
	if (y <= -0.0008) {
		tmp = t_0;
	} else if (y <= 0.0051) {
		tmp = (1.0 - y) * (y + 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 + 1.0d0))
    if (y <= (-0.0008d0)) then
        tmp = t_0
    else if (y <= 0.0051d0) then
        tmp = (1.0d0 - y) * (y + 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 + 1.0));
	double tmp;
	if (y <= -0.0008) {
		tmp = t_0;
	} else if (y <= 0.0051) {
		tmp = (1.0 - y) * (y + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / (y + 1.0))
	tmp = 0
	if y <= -0.0008:
		tmp = t_0
	elif y <= 0.0051:
		tmp = (1.0 - y) * (y + x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / Float64(y + 1.0)))
	tmp = 0.0
	if (y <= -0.0008)
		tmp = t_0;
	elseif (y <= 0.0051)
		tmp = Float64(Float64(1.0 - y) * Float64(y + x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / (y + 1.0));
	tmp = 0.0;
	if (y <= -0.0008)
		tmp = t_0;
	elseif (y <= 0.0051)
		tmp = (1.0 - y) * (y + x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0008], t$95$0, If[LessEqual[y, 0.0051], N[(N[(1.0 - y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000038e-4 or 0.0051000000000000004 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 99.4%

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

   if -8.00000000000000038e-4 < y < 0.0051000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 99.4

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

   8. Applied rewrites 99.4%

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

Alternative 5: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\left(1 - y\right) \cdot \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= y -1.0) t_0 (if (<= y 0.75) (* (- 1.0 y) (+ y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.75) {
		tmp = (1.0 - y) * (y + 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 (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 0.75d0) then
        tmp = (1.0d0 - y) * (y + 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 (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.75) {
		tmp = (1.0 - y) * (y + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 0.75:
		tmp = (1.0 - y) * (y + x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.75)
		tmp = Float64(Float64(1.0 - y) * Float64(y + 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 (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.75)
		tmp = (1.0 - y) * (y + 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[y, -1.0], t$95$0, If[LessEqual[y, 0.75], N[(N[(1.0 - y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.75 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 98.8

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

   8. Applied rewrites 98.8%

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

   if -1 < y < 0.75

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 98.8

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

   8. Applied rewrites 98.8%

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

Alternative 6: 86.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 75.0%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 98.8

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

   8. Applied rewrites 98.8%

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

Alternative 7: 86.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 75.0%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

Alternative 8: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 9: 62.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+49}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.75e+49) (+ 1.0 x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.75e+49) {
		tmp = 1.0 + 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 (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.75d+49) then
        tmp = 1.0d0 + x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.75e+49) {
		tmp = 1.0 + x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 1.75e+49:
		tmp = 1.0 + x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.75e+49)
		tmp = Float64(1.0 + x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 1.75e+49)
		tmp = 1.0 + x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 1.75e+49], N[(1.0 + x), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.74999999999999987e49 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 78.0%

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

   if -1 < y < 1.74999999999999987e49

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 49.3%

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

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 43.8

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

   14. Applied rewrites 43.8%

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

Alternative 10: 39.2% accurate, 18.0× 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 100.0%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 41.1%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))