Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.7% → 99.9%

Time: 11.0s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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 * ((x / y) + 1.0d0)) / (x + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 * ((x / y) + 1.0d0)) / (x + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.9

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 99.9

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.999998:\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))))
   (if (<= t_0 (- INFINITY))
     (/ x y)
     (if (<= t_0 0.999998)
       (* (/ (+ y x) (fma y x y)) x)
       (if (<= t_0 2.0)
         (/ x (- x -1.0))
         (if (<= t_0 4e+165) (/ (* (/ x y) x) (- x -1.0)) (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= 0.999998) {
		tmp = ((y + x) / fma(y, x, y)) * x;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_0 <= 4e+165) {
		tmp = ((x / y) * x) / (x - -1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_0 <= 0.999998)
		tmp = Float64(Float64(Float64(y + x) / fma(y, x, y)) * x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_0 <= 4e+165)
		tmp = Float64(Float64(Float64(x / y) * x) / Float64(x - -1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.999998], N[(N[(N[(y + x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+165], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 3.9999999999999996e165 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 44.6%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. associate-/l* N/A

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

      10. frac-2neg-rev N/A

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

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

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

      12. sqrt-prod N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

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

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 9.0

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot x}}{y}\right) \]

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*r/ N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.999998000000000054

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. div-add-rev N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 83.4

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

   7. Applied rewrites 83.4%

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

   9. Applied rewrites 98.5%

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

   if 0.999998000000000054 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-rgt-identity N/A

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

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

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999996e165

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 0.999998:\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))))
   (if (<= t_0 -1e+15)
     (/ x y)
     (if (<= t_0 0.5)
       (fma (- (/ x y) x) x x)
       (if (<= t_0 2.0)
         (/ x (- x -1.0))
         (if (<= t_0 4e+165) (* (/ x (fma y x y)) x) (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -1e+15) {
		tmp = x / y;
	} else if (t_0 <= 0.5) {
		tmp = fma(((x / y) - x), x, x);
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else if (t_0 <= 4e+165) {
		tmp = (x / fma(y, x, y)) * x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_0 <= -1e+15)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.5)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	elseif (t_0 <= 4e+165)
		tmp = Float64(Float64(x / fma(y, x, y)) * x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+15], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+165], N[(N[(x / N[(y * x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e15 or 3.9999999999999996e165 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 59.7%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. associate-/l* N/A

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

      10. frac-2neg-rev N/A

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

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

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

      12. sqrt-prod N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

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

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 6.7

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

   4. Applied rewrites 6.7%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot x}}{y}\right) \]

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*r/ N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 93.7

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

   7. Applied rewrites 93.7%

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

   if -1e15 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. difference-of-squares-rev N/A

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

      15. difference-of-sqr--1-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 N/A

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

      20. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.3%

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

   if 0.5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-rgt-identity N/A

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

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

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999996e165

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 81.0

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

   5. Applied rewrites 81.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 4 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))))
   (if (<= t_0 -1e+15)
     (/ x y)
     (if (<= t_0 0.5)
       (fma (- (/ x y) x) x x)
       (if (<= t_0 2.0) (/ x (- x -1.0)) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -1e+15) {
		tmp = x / y;
	} else if (t_0 <= 0.5) {
		tmp = fma(((x / y) - x), x, x);
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_0 <= -1e+15)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.5)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+15], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e15 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. associate-/l* N/A

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

      10. frac-2neg-rev N/A

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

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

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

      12. sqrt-prod N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

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

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 10.2

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

   4. Applied rewrites 10.2%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot x}}{y}\right) \]

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*r/ N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 86.5

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

   7. Applied rewrites 86.5%

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

   if -1e15 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. difference-of-squares-rev N/A

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

      15. difference-of-sqr--1-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 N/A

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

      20. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.3%

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

   if 0.5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-rgt-identity N/A

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

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

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 4 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 4e+165)))
     (/ x y)
     (/ (fma (/ x y) x x) (- x -1.0)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 4e+165)) {
		tmp = x / y;
	} else {
		tmp = fma((x / y), x, x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 4e+165))
		tmp = Float64(x / y);
	else
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 4e+165]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -inf.0 or 3.9999999999999996e165 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 44.6%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. associate-/l* N/A

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

      10. frac-2neg-rev N/A

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

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

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

      12. sqrt-prod N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

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

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 9.0

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot x}}{y}\right) \]

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*r/ N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 3.9999999999999996e165

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -\infty \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 4 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+15} \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))))
   (if (or (<= t_0 -1e+15) (not (<= t_0 2.0))) (/ x y) (/ x (- x -1.0)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if ((t_0 <= -1e+15) || !(t_0 <= 2.0)) {
		tmp = x / y;
	} else {
		tmp = x / (x - -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) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x - (-1.0d0))
    if ((t_0 <= (-1d+15)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = x / y
    else
        tmp = x / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if ((t_0 <= -1e+15) || !(t_0 <= 2.0)) {
		tmp = x / y;
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0)
	tmp = 0
	if (t_0 <= -1e+15) or not (t_0 <= 2.0):
		tmp = x / y
	else:
		tmp = x / (x - -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if ((t_0 <= -1e+15) || !(t_0 <= 2.0))
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	tmp = 0.0;
	if ((t_0 <= -1e+15) || ~((t_0 <= 2.0)))
		tmp = x / y;
	else
		tmp = x / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+15], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e15 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. associate-/l* N/A

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

      10. frac-2neg-rev N/A

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

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

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

      12. sqrt-prod N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

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

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 10.2

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

   4. Applied rewrites 10.2%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot x}}{y}\right) \]

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*r/ N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 86.5

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

   7. Applied rewrites 86.5%

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

   if -1e15 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. rgt-mult-inverse N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. *-rgt-identity N/A

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

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

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

      10. rgt-mult-inverse N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 88.7

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

   5. Applied rewrites 88.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -1 \cdot 10^{+15} \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 2\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+15} \lor \neg \left(t\_0 \leq 0.5\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))))
   (if (or (<= t_0 -1e+15) (not (<= t_0 0.5))) (/ x y) (fma (- x) x x))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if ((t_0 <= -1e+15) || !(t_0 <= 0.5)) {
		tmp = x / y;
	} else {
		tmp = fma(-x, x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if ((t_0 <= -1e+15) || !(t_0 <= 0.5))
		tmp = Float64(x / y);
	else
		tmp = fma(Float64(-x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+15], N[Not[LessEqual[t$95$0, 0.5]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[((-x) * x + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e15 or 0.5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.1%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. associate-/l* N/A

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

      10. frac-2neg-rev N/A

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

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

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

      12. sqrt-prod N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

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

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 15.1

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

   4. Applied rewrites 15.1%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot x}}{y}\right) \]

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*r/ N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 72.1

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

   7. Applied rewrites 72.1%

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

   if -1e15 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. difference-of-squares-rev N/A

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

      15. difference-of-sqr--1-rev N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 N/A

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

      20. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 86.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -1 \cdot 10^{+15} \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 0.5\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+144} \lor \neg \left(x \leq 3.4 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.45e+144) (not (<= x 3.4e+71)))
   (/ x y)
   (* (/ (+ y x) (fma y x y)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.45e+144) || !(x <= 3.4e+71)) {
		tmp = x / y;
	} else {
		tmp = ((y + x) / fma(y, x, y)) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.45e+144) || !(x <= 3.4e+71))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(Float64(y + x) / fma(y, x, y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.45e+144], N[Not[LessEqual[x, 3.4e+71]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999999e144 or 3.3999999999999998e71 < x

   1. Initial program 59.8%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. associate-/l* N/A

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

      10. frac-2neg-rev N/A

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

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

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

      12. sqrt-prod N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

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

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 14.7

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

   4. Applied rewrites 14.7%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot x}}{y}\right) \]

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*r/ N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if -1.44999999999999999e144 < x < 3.3999999999999998e71

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. div-add-rev N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 85.4

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

   7. Applied rewrites 85.4%

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

   9. Applied rewrites 97.6%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+144} \lor \neg \left(x \leq 3.4 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return fma(Float64(-x), x, x)
end

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. flip-+ N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. difference-of-squares-rev N/A

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

   15. difference-of-sqr--1-rev N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 N/A

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

   20. lower--.f64 74.9

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

4. Applied rewrites 74.9%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f64 N/A

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

7. Applied rewrites 58.2%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 47.9%

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

11. Final simplification 47.9%

    \[\leadsto \mathsf{fma}\left(-x, x, x\right) \]
12. Add Preprocessing

Alternative 10: 38.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.9

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 99.9

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around 0

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

   1. div-add-rev N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. +-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. *-rgt-identity N/A

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

   14. lower-fma.f64 71.2

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

7. Applied rewrites 71.2%

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

9. Applied rewrites 87.2%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 43.6%

    \[\leadsto 1 \cdot x \]

13. Final simplification 43.6%

    \[\leadsto 1 \cdot x \]
14. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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 / 1.0d0) * (((x / y) + 1.0d0) / (x + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024352 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1))))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))