Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.3% → 99.8%

Time: 6.7s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -15500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 235000000:\\ \;\;\;\;\frac{\left(\frac{1}{\frac{y}{x}} + 1\right) \cdot x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -15500000000000.0)
     t_0
     (if (<= x 235000000.0) (/ (* (+ (/ 1.0 (/ y x)) 1.0) x) (+ 1.0 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -15500000000000.0) {
		tmp = t_0;
	} else if (x <= 235000000.0) {
		tmp = (((1.0 / (y / x)) + 1.0) * x) / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x - 1.0d0) / y) + 1.0d0
    if (x <= (-15500000000000.0d0)) then
        tmp = t_0
    else if (x <= 235000000.0d0) then
        tmp = (((1.0d0 / (y / x)) + 1.0d0) * x) / (1.0d0 + x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -15500000000000.0) {
		tmp = t_0;
	} else if (x <= 235000000.0) {
		tmp = (((1.0 / (y / x)) + 1.0) * x) / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - 1.0) / y) + 1.0
	tmp = 0
	if x <= -15500000000000.0:
		tmp = t_0
	elif x <= 235000000.0:
		tmp = (((1.0 / (y / x)) + 1.0) * x) / (1.0 + x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -15500000000000.0)
		tmp = t_0;
	elseif (x <= 235000000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(y / x)) + 1.0) * x) / Float64(1.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - 1.0) / y) + 1.0;
	tmp = 0.0;
	if (x <= -15500000000000.0)
		tmp = t_0;
	elseif (x <= 235000000.0)
		tmp = (((1.0 / (y / x)) + 1.0) * x) / (1.0 + x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -15500000000000.0], t$95$0, If[LessEqual[x, 235000000.0], N[(N[(N[(N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55e13 or 2.35e8 < x

   1. Initial program 73.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

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

       2. distribute-lft-out N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. associate-*r/ N/A

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

       6. *-rgt-identity N/A

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

       7. rgt-mult-inverse N/A

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

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

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

       9. associate-/r* N/A

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

       10. associate-*r/ N/A

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

       11. rgt-mult-inverse N/A

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

       12. +-commutative N/A

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

       13. associate-+r+ N/A

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

       14. sub-neg N/A

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

       15. div-sub N/A

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

       16. +-commutative N/A

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

       17. lower-+.f64 N/A

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

       18. lower-/.f64 N/A

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

       19. lower--.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -1.55e13 < x < 2.35e8

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15500000000000:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{elif}\;x \leq 235000000:\\ \;\;\;\;\frac{\left(\frac{1}{\frac{y}{x}} + 1\right) \cdot x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;t\_0 \leq 4000:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -500.0)
     (/ (- x 1.0) y)
     (if (<= t_0 4000.0) (/ x (+ 1.0 x)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 4000.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x / y) + 1.0d0) * x) / (1.0d0 + x)
    if (t_0 <= (-500.0d0)) then
        tmp = (x - 1.0d0) / y
    else if (t_0 <= 4000.0d0) then
        tmp = x / (1.0d0 + x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 4000.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x)
	tmp = 0
	if t_0 <= -500.0:
		tmp = (x - 1.0) / y
	elif t_0 <= 4000.0:
		tmp = x / (1.0 + x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = Float64(Float64(x - 1.0) / y);
	elseif (t_0 <= 4000.0)
		tmp = Float64(x / Float64(1.0 + x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	tmp = 0.0;
	if (t_0 <= -500.0)
		tmp = (x - 1.0) / y;
	elseif (t_0 <= 4000.0)
		tmp = x / (1.0 + x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 4000.0], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -500

   1. Initial program 77.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 94.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 91.5%

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

   if -500 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4e3

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if 4e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 63.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -500:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 4000:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 4000:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -500.0) (/ x y) (if (<= t_0 4000.0) (/ x (+ 1.0 x)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = x / y;
	} else if (t_0 <= 4000.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x / y) + 1.0d0) * x) / (1.0d0 + x)
    if (t_0 <= (-500.0d0)) then
        tmp = x / y
    else if (t_0 <= 4000.0d0) then
        tmp = x / (1.0d0 + x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = x / y;
	} else if (t_0 <= 4000.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x)
	tmp = 0
	if t_0 <= -500.0:
		tmp = x / y
	elif t_0 <= 4000.0:
		tmp = x / (1.0 + x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 4000.0)
		tmp = Float64(x / Float64(1.0 + x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	tmp = 0.0;
	if (t_0 <= -500.0)
		tmp = x / y;
	elseif (t_0 <= 4000.0)
		tmp = x / (1.0 + x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 4000.0], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -500 or 4e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 69.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 87.6

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

   5. Applied rewrites 87.6%

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

   if -500 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4e3

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 86.6

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

   5. Applied rewrites 86.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -500:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 4000:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -500.0) (/ x y) (if (<= t_0 1e-5) (* (- 1.0 x) x) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = x / y;
	} else if (t_0 <= 1e-5) {
		tmp = (1.0 - x) * 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) :: t_0
    real(8) :: tmp
    t_0 = (((x / y) + 1.0d0) * x) / (1.0d0 + x)
    if (t_0 <= (-500.0d0)) then
        tmp = x / y
    else if (t_0 <= 1d-5) then
        tmp = (1.0d0 - x) * x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = x / y;
	} else if (t_0 <= 1e-5) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x)
	tmp = 0
	if t_0 <= -500.0:
		tmp = x / y
	elif t_0 <= 1e-5:
		tmp = (1.0 - x) * x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 1e-5)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	tmp = 0.0;
	if (t_0 <= -500.0)
		tmp = x / y;
	elseif (t_0 <= 1e-5)
		tmp = (1.0 - x) * x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1e-5], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -500 or 1.00000000000000008e-5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 76.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if -500 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000008e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot x, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 84.4%

      \[\leadsto \mathsf{fma}\left(-x, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 84.4%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 84.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -500:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 10^{-5}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -440000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -440000000000.0)
     t_0
     (if (<= x 5e+16) (/ (fma (/ x y) x x) (+ 1.0 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -440000000000.0) {
		tmp = t_0;
	} else if (x <= 5e+16) {
		tmp = fma((x / y), x, x) / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -440000000000.0)
		tmp = t_0;
	elseif (x <= 5e+16)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(1.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -440000000000.0], t$95$0, If[LessEqual[x, 5e+16], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4e11 or 5e16 < x

   1. Initial program 73.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

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

       2. distribute-lft-out N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. associate-*r/ N/A

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

       6. *-rgt-identity N/A

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

       7. rgt-mult-inverse N/A

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

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

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

       9. associate-/r* N/A

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

       10. associate-*r/ N/A

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

       11. rgt-mult-inverse N/A

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

       12. +-commutative N/A

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

       13. associate-+r+ N/A

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

       14. sub-neg N/A

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

       15. div-sub N/A

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

       16. +-commutative N/A

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

       17. lower-+.f64 N/A

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

       18. lower-/.f64 N/A

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

       19. lower--.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -4.4e11 < x < 5e16

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -440000000000:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 74.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

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

       2. distribute-lft-out N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. associate-*r/ N/A

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

       6. *-rgt-identity N/A

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

       7. rgt-mult-inverse N/A

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

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

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

       9. associate-/r* N/A

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

       10. associate-*r/ N/A

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

       11. rgt-mult-inverse N/A

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

       12. +-commutative N/A

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

       13. associate-+r+ N/A

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

       14. sub-neg N/A

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

       15. div-sub N/A

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

       16. +-commutative N/A

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

       17. lower-+.f64 N/A

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

       18. lower-/.f64 N/A

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

       19. lower--.f64 99.7

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

   11. Applied rewrites 99.7%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.9%

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

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 74.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

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

       2. distribute-lft-out N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. associate-*r/ N/A

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

       6. *-rgt-identity N/A

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

       7. rgt-mult-inverse N/A

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

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

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

       9. associate-/r* N/A

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

       10. associate-*r/ N/A

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

       11. rgt-mult-inverse N/A

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

       12. +-commutative N/A

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

       13. associate-+r+ N/A

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

       14. sub-neg N/A

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

       15. div-sub N/A

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

       16. +-commutative N/A

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

       17. lower-+.f64 N/A

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

       18. lower-/.f64 N/A

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

       19. lower--.f64 99.7

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

   11. Applied rewrites 99.7%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 8: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.22:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -1.0) t_0 (if (<= x 1.22) (fma (/ x y) x x) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.22) {
		tmp = fma((x / y), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.22)
		tmp = fma(Float64(x / y), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.22], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.21999999999999997 < x

   1. Initial program 74.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

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

       2. distribute-lft-out N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. associate-*r/ N/A

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

       6. *-rgt-identity N/A

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

       7. rgt-mult-inverse N/A

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

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

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

       9. associate-/r* N/A

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

       10. associate-*r/ N/A

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

       11. rgt-mult-inverse N/A

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

       12. +-commutative N/A

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

       13. associate-+r+ N/A

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

       14. sub-neg N/A

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

       15. div-sub N/A

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

       16. +-commutative N/A

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

       17. lower-+.f64 N/A

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

       18. lower-/.f64 N/A

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

       19. lower--.f64 99.7

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

   11. Applied rewrites 99.7%

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

   if -1 < x < 1.21999999999999997

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.1%

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

Alternative 9: 86.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -220000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2400:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -220000000.0) t_0 (if (<= x 2400.0) (/ x (+ 1.0 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -220000000.0) {
		tmp = t_0;
	} else if (x <= 2400.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x - 1.0d0) / y) + 1.0d0
    if (x <= (-220000000.0d0)) then
        tmp = t_0
    else if (x <= 2400.0d0) then
        tmp = x / (1.0d0 + x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -220000000.0) {
		tmp = t_0;
	} else if (x <= 2400.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - 1.0) / y) + 1.0
	tmp = 0
	if x <= -220000000.0:
		tmp = t_0
	elif x <= 2400.0:
		tmp = x / (1.0 + x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -220000000.0)
		tmp = t_0;
	elseif (x <= 2400.0)
		tmp = Float64(x / Float64(1.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - 1.0) / y) + 1.0;
	tmp = 0.0;
	if (x <= -220000000.0)
		tmp = t_0;
	elseif (x <= 2400.0)
		tmp = x / (1.0 + x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -220000000.0], t$95$0, If[LessEqual[x, 2400.0], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2e8 or 2400 < x

   1. Initial program 74.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. sub-neg N/A

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

       2. distribute-lft-out N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. associate-*r/ N/A

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

       6. *-rgt-identity N/A

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

       7. rgt-mult-inverse N/A

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

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

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

       9. associate-/r* N/A

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

       10. associate-*r/ N/A

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

       11. rgt-mult-inverse N/A

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

       12. +-commutative N/A

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

       13. associate-+r+ N/A

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

       14. sub-neg N/A

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

       15. div-sub N/A

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

       16. +-commutative N/A

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

       17. lower-+.f64 N/A

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

       18. lower-/.f64 N/A

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

       19. lower--.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -2.2e8 < x < 2400

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 75.3

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

   5. Applied rewrites 75.3%

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

Alternative 10: 43.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f64 N/A

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

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

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 53.0

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

5. Applied rewrites 53.0%

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

6. Taylor expanded in y around inf

   \[\leadsto \mathsf{fma}\left(-1 \cdot x, x, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 39.5%

   \[\leadsto \mathsf{fma}\left(-x, x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 39.5%

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

11. Taylor expanded in y around inf

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

13. Applied rewrites 39.5%

    \[\leadsto \left(1 - x\right) \cdot x \]
14. Add Preprocessing

Alternative 11: 39.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f64 N/A

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

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

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 53.0

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

5. Applied rewrites 53.0%

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

6. Taylor expanded in y around inf

   \[\leadsto \mathsf{fma}\left(-1 \cdot x, x, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 39.5%

   \[\leadsto \mathsf{fma}\left(-x, x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 39.5%

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

11. Taylor expanded in x around 0

    \[\leadsto 1 \cdot x \]
12. Step-by-step derivation

13. Applied rewrites 36.3%

    \[\leadsto 1 \cdot x \]
14. 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 2024255 
(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)))