Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, C

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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) / (1.0d0 - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) / (1.0d0 - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) / (1.0d0 - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 73.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-200}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;-y\\ \mathbf{elif}\;t\_0 \leq 50:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 -5e-200)
     (fma y x x)
     (if (<= t_0 0.02) (- y) (if (<= t_0 50.0) 1.0 (fma y x x))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= -5e-200) {
		tmp = fma(y, x, x);
	} else if (t_0 <= 0.02) {
		tmp = -y;
	} else if (t_0 <= 50.0) {
		tmp = 1.0;
	} else {
		tmp = fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= -5e-200)
		tmp = fma(y, x, x);
	elseif (t_0 <= 0.02)
		tmp = Float64(-y);
	elseif (t_0 <= 50.0)
		tmp = 1.0;
	else
		tmp = fma(y, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-200], N[(y * x + x), $MachinePrecision], If[LessEqual[t$95$0, 0.02], (-y), If[LessEqual[t$95$0, 50.0], 1.0, N[(y * x + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -4.99999999999999991e-200 or 50 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   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. lower--.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.1%

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

   if -4.99999999999999991e-200 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.9%

      \[\leadsto -y \]

   if 0.0200000000000000004 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 50

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 95.4%

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

Alternative 3: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 0.02 \lor \neg \left(t\_0 \leq 50\right):\\ \;\;\;\;x - \mathsf{fma}\left(y, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (or (<= t_0 0.02) (not (<= t_0 50.0))) (- x (fma y x y)) 1.0)))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if ((t_0 <= 0.02) || !(t_0 <= 50.0)) {
		tmp = x - fma(y, x, y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if ((t_0 <= 0.02) || !(t_0 <= 50.0))
		tmp = Float64(x - fma(y, x, y));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.02], N[Not[LessEqual[t$95$0, 50.0]], $MachinePrecision]], N[(x - N[(y * x + y), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.0200000000000000004 or 50 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 82.0%

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

   if 0.0200000000000000004 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 50

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 95.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.02 \lor \neg \left(\frac{x - y}{1 - y} \leq 50\right):\\ \;\;\;\;x - \mathsf{fma}\left(y, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 0.02 \lor \neg \left(t\_0 \leq 50\right):\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (or (<= t_0 0.02) (not (<= t_0 50.0))) (fma -1.0 y x) 1.0)))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if ((t_0 <= 0.02) || !(t_0 <= 50.0)) {
		tmp = fma(-1.0, y, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if ((t_0 <= 0.02) || !(t_0 <= 50.0))
		tmp = fma(-1.0, y, x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.02], N[Not[LessEqual[t$95$0, 50.0]], $MachinePrecision]], N[(-1.0 * y + x), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.0200000000000000004 or 50 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.8%

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

   if 0.0200000000000000004 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 50

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 95.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.02 \lor \neg \left(\frac{x - y}{1 - y} \leq 50\right):\\ \;\;\;\;\mathsf{fma}\left(-1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (- 1.0 (/ (- x 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 - ((x - 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(Float64(x - 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[(N[(x - 1.0), $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%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 97.6%

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

Alternative 6: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.84 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.84) (not (<= y 1.0))) (- 1.0 (/ x y)) (fma (+ -1.0 x) y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.84) || !(y <= 1.0)) {
		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 <= -0.84) || !(y <= 1.0))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = fma(Float64(-1.0 + x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.84], N[Not[LessEqual[y, 1.0]], $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 < -0.839999999999999969 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.3%

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

   if -0.839999999999999969 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.84 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.02:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.02) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.02) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((x - y) / (1.0d0 - y)) <= 0.02d0) then
        tmp = -y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.02) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.02:
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.02)
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x - y) / (1.0 - y)) <= 0.02)
		tmp = -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.02], (-y), 1.0]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 85.8

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

   5. Applied rewrites 85.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 26.5%

      \[\leadsto -y \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 80.0

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 79.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 71.8%

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

Alternative 8: 86.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.3%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 79.8%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 9: 38.3% accurate, 18.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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 = 1.0d0
end function

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

   1. associate-+r+ N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. associate-+l- N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 52.8

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 53.0%

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

9. Taylor expanded in y around inf

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

11. Applied rewrites 43.7%

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

Reproduce

herbie shell --seed 2024350 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))