Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 8

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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, 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ t_1 := \frac{x}{y - -1}\\ \mathbf{if}\;t\_0 \leq -10000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))) (t_1 (/ x (- y -1.0))))
   (if (<= t_0 -10000000.0)
     t_1
     (if (<= t_0 2e-12)
       (fma 1.0 y x)
       (if (<= t_0 2.0) (/ y (- y -1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + 1.0);
	double t_1 = x / (y - -1.0);
	double tmp;
	if (t_0 <= -10000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-12) {
		tmp = fma(1.0, y, x);
	} else if (t_0 <= 2.0) {
		tmp = y / (y - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + 1.0))
	t_1 = Float64(x / Float64(y - -1.0))
	tmp = 0.0
	if (t_0 <= -10000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-12)
		tmp = fma(1.0, y, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10000000.0], t$95$1, If[LessEqual[t$95$0, 2e-12], N[(1.0 * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(y - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < -1e7 or 2 < (/.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 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. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. *-rgt-identity N/A

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

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

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -1e7 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 1.99999999999999996e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.0%

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

   if 1.99999999999999996e-12 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. *-rgt-identity N/A

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

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

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 3: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ t_1 := \frac{x}{y - -1}\\ \mathbf{if}\;t\_0 \leq -10000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))) (t_1 (/ x (- y -1.0))))
   (if (<= t_0 -10000000.0)
     t_1
     (if (<= t_0 2e-12)
       (fma 1.0 y x)
       (if (<= t_0 2.0) (- 1.0 (/ 1.0 y)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + 1.0);
	double t_1 = x / (y - -1.0);
	double tmp;
	if (t_0 <= -10000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-12) {
		tmp = fma(1.0, y, x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (1.0 / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + 1.0))
	t_1 = Float64(x / Float64(y - -1.0))
	tmp = 0.0
	if (t_0 <= -10000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-12)
		tmp = fma(1.0, y, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(1.0 / y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10000000.0], t$95$1, If[LessEqual[t$95$0, 2e-12], N[(1.0 * y + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < -1e7 or 2 < (/.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 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. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. *-rgt-identity N/A

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

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

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -1e7 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 1.99999999999999996e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.0%

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

   if 1.99999999999999996e-12 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 2

   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. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

      20. distribute-lft-neg-out N/A

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

      21. distribute-rgt-neg-in N/A

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

      22. metadata-eval N/A

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

      23. *-rgt-identity N/A

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

      24. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.8%

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

Alternative 4: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      \[\leadsto \color{blue}{\left(1 + \frac{x}{y}\right) - \frac{1}{y}} \]
   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. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

      20. distribute-lft-neg-out N/A

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

      21. distribute-rgt-neg-in N/A

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

      22. metadata-eval N/A

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

      23. *-rgt-identity N/A

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

      24. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

4. Final simplification 98.9%

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

Alternative 5: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.85\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.85)))
   (- 1.0 (/ (- x) y))
   (fma (- 1.0 x) y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.85)) {
		tmp = 1.0 - (-x / y);
	} else {
		tmp = fma((1.0 - x), y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.85))
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	else
		tmp = fma(Float64(1.0 - x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.85]], $MachinePrecision]], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.849999999999999978 < 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. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

      20. distribute-lft-neg-out N/A

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

      21. distribute-rgt-neg-in N/A

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

      22. metadata-eval N/A

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

      23. *-rgt-identity N/A

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

      24. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.1%

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

   if -1 < y < 0.849999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.85\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      \[\leadsto \color{blue}{\left(1 + \frac{x}{y}\right) - \frac{1}{y}} \]
   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. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

      20. distribute-lft-neg-out N/A

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

      21. distribute-rgt-neg-in N/A

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

      22. metadata-eval N/A

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

      23. *-rgt-identity N/A

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

      24. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.2%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

4. Final simplification 87.5%

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

Alternative 7: 60.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.7 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.7e+16))) (/ x y) (fma (- 1.0 x) y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.7e+16)) {
		tmp = x / y;
	} else {
		tmp = fma((1.0 - x), y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.7e+16))
		tmp = Float64(x / y);
	else
		tmp = fma(Float64(1.0 - x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.7e+16]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.7e16 < 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. remove-double-neg N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. *-commutative N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

      20. distribute-lft-neg-out N/A

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

      21. distribute-rgt-neg-in N/A

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

      22. metadata-eval N/A

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

      23. *-rgt-identity N/A

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

      24. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 23.8%

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

   if -1 < y < 1.7e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.7 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, y, x\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x, y)
	return fma(1.0, y, x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 47.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 47.8%

   \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))