Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.0% → 99.9%

Time: 11.2s

Alternatives: 13

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: 88.0% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+16} \lor \neg \left(x \leq 1.5 \cdot 10^{+16}\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1e+16) (not (<= x 1.5e+16)))
   (- 1.0 (/ (- x) y))
   (/ (fma (/ x y) x x) (- x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1e+16) || !(x <= 1.5e+16)) {
		tmp = 1.0 - (-x / y);
	} else {
		tmp = fma((x / y), x, x) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1e+16) || !(x <= 1.5e+16))
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	else
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1e+16], N[Not[LessEqual[x, 1.5e+16]], $MachinePrecision]], N[(1.0 - N[((-x) / y), $MachinePrecision]), $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 x < -1e16 or 1.5e16 < x

   1. Initial program 78.3%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 100.0%

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

   if -1e16 < x < 1.5e16

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      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

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+16} \lor \neg \left(x \leq 1.5 \cdot 10^{+16}\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1}\\ t_1 := \frac{x - 1}{y}\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-1, x \cdot x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (- (/ x y) -1.0)) (- x -1.0))) (t_1 (/ (- x 1.0) y)))
   (if (<= t_0 -50.0)
     t_1
     (if (<= t_0 5e-9)
       (fma -1.0 (* x x) x)
       (if (<= t_0 2.0) (- 1.0 (/ 1.0 x)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) - -1.0)) / (x - -1.0);
	double t_1 = (x - 1.0) / y;
	double tmp;
	if (t_0 <= -50.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-9) {
		tmp = fma(-1.0, (x * x), x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (1.0 / x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) - -1.0)) / Float64(x - -1.0))
	t_1 = Float64(Float64(x - 1.0) / y)
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = t_1;
	elseif (t_0 <= 5e-9)
		tmp = fma(-1.0, Float64(x * x), x);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(1.0 / x));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -50.0], t$95$1, If[LessEqual[t$95$0, 5e-9], N[(-1.0 * N[(x * x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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

   1. Initial program 73.2%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.0%

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

   if -50 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-9

   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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.7%

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

   if 5.0000000000000001e-9 < (/.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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 91.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -50:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-1, x \cdot x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 2:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -50 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (- (/ x y) -1.0)) (- x -1.0))))
   (if (or (<= t_0 -50.0) (not (<= t_0 2.0)))
     (/ (- x 1.0) 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 <= -50.0) || !(t_0 <= 2.0)) {
		tmp = (x - 1.0) / 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 <= (-50.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = (x - 1.0d0) / 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 <= -50.0) || !(t_0 <= 2.0)) {
		tmp = (x - 1.0) / 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 <= -50.0) or not (t_0 <= 2.0):
		tmp = (x - 1.0) / 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 <= -50.0) || !(t_0 <= 2.0))
		tmp = Float64(Float64(x - 1.0) / 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 <= -50.0) || ~((t_0 <= 2.0)))
		tmp = (x - 1.0) / 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, -50.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[(x - 1.0), $MachinePrecision] / 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))) < -50 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.2%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.0%

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

   if -50 < (/.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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 87.8

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

   5. Applied rewrites 87.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq -50 \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} - -1\right)}{x - -1} \leq 2\right):\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (- (/ x y) -1.0)) (- x -1.0)) -1e+16) (* (- x) x) (* 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) - -1.0)) / (x - -1.0)) <= -1e+16) {
		tmp = -x * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * ((x / y) - (-1.0d0))) / (x - (-1.0d0))) <= (-1d+16)) then
        tmp = -x * x
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * ((x / y) - -1.0)) / (x - -1.0)) <= -1e+16) {
		tmp = -x * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * ((x / y) - -1.0)) / (x - -1.0)) <= -1e+16:
		tmp = -x * x
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(Float64(x / y) - -1.0)) / Float64(x - -1.0)) <= -1e+16)
		tmp = Float64(Float64(-x) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * ((x / y) - -1.0)) / (x - -1.0)) <= -1e+16)
		tmp = -x * x;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], -1e+16], N[((-x) * x), $MachinePrecision], N[(1.0 * 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))) < -1e16

   1. Initial program 72.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 25.3

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

   5. Applied rewrites 25.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 28.1%

      \[\leadsto \left(1 - x\right) \cdot x \]

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 28.3%

       \[\leadsto \left(-x\right) \cdot x \]

   if -1e16 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 92.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.8

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 44.5%

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

4. Final simplification 41.9%

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

Alternative 5: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+15} \lor \neg \left(x \leq 1.7 \cdot 10^{+16}\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.7e+15) (not (<= x 1.7e+16)))
   (- 1.0 (/ (- x) y))
   (* (/ (+ y x) (fma y x y)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.7e+15) || !(x <= 1.7e+16)) {
		tmp = 1.0 - (-x / y);
	} else {
		tmp = ((y + x) / fma(y, x, y)) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.7e+15) || !(x <= 1.7e+16))
		tmp = Float64(1.0 - Float64(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, -2.7e+15], N[Not[LessEqual[x, 1.7e+16]], $MachinePrecision]], N[(1.0 - N[((-x) / y), $MachinePrecision]), $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 < -2.7e15 or 1.7e16 < x

   1. Initial program 78.5%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 100.0%

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

   if -2.7e15 < x < 1.7e16

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. 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. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.8

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

      15. lift-+.f64 N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. div-add-rev N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+15} \lor \neg \left(x \leq 1.7 \cdot 10^{+16}\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

   7. lift-+.f64 N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval 99.9

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

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

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower--.f64 N/A

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

   22. metadata-eval 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\frac{x}{y} - -1}{x - -1} \cdot x} \]
5. Add Preprocessing

Alternative 7: 98.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (- 1.0 (/ (- x) y))
   (if (<= x 0.85) (fma (- (/ x y) x) x x) (/ (- y (- x)) y))))

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.85], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(y - (-x)), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 78.3%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.4%

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

   10. Applied rewrites 98.4%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 98.4%

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

   if -1 < x < 0.849999999999999978

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.5%

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

   if 0.849999999999999978 < x

   1. Initial program 80.6%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.3%

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

Alternative 8: 86.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+15} \lor \neg \left(x \leq 40000000\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -9.2e+15) (not (<= x 40000000.0)))
   (- 1.0 (/ (- x) y))
   (/ x (- x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -9.2e+15) || !(x <= 40000000.0)) {
		tmp = 1.0 - (-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) :: tmp
    if ((x <= (-9.2d+15)) .or. (.not. (x <= 40000000.0d0))) then
        tmp = 1.0d0 - (-x / y)
    else
        tmp = x / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -9.2e+15) || !(x <= 40000000.0)) {
		tmp = 1.0 - (-x / y);
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -9.2e+15) or not (x <= 40000000.0):
		tmp = 1.0 - (-x / y)
	else:
		tmp = x / (x - -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -9.2e+15) || !(x <= 40000000.0))
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	else
		tmp = Float64(x / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -9.2e+15) || ~((x <= 40000000.0)))
		tmp = 1.0 - (-x / y);
	else
		tmp = x / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -9.2e+15], N[Not[LessEqual[x, 40000000.0]], $MachinePrecision]], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.2e15 or 4e7 < x

   1. Initial program 78.7%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.9%

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

   if -9.2e15 < x < 4e7

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+15} \lor \neg \left(x \leq 40000000\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+15}:\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{elif}\;x \leq 40000000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \left(-x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.2e+15)
   (- 1.0 (/ (- x) y))
   (if (<= x 40000000.0) (/ x (- x -1.0)) (/ (- y (- x)) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.2e+15) {
		tmp = 1.0 - (-x / y);
	} else if (x <= 40000000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = (y - -x) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.2d+15)) then
        tmp = 1.0d0 - (-x / y)
    else if (x <= 40000000.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = (y - -x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.2e+15) {
		tmp = 1.0 - (-x / y);
	} else if (x <= 40000000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = (y - -x) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.2e+15:
		tmp = 1.0 - (-x / y)
	elif x <= 40000000.0:
		tmp = x / (x - -1.0)
	else:
		tmp = (y - -x) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.2e+15)
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	elseif (x <= 40000000.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(Float64(y - Float64(-x)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.2e+15)
		tmp = 1.0 - (-x / y);
	elseif (x <= 40000000.0)
		tmp = x / (x - -1.0);
	else
		tmp = (y - -x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.2e+15], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 40000000.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y - (-x)), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.2e15

   1. Initial program 76.9%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 100.0%

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

   if -9.2e15 < x < 4e7

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if 4e7 < x

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.9%

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

Alternative 10: 74.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 79.5%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.1%

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

   if -1 < x < 1

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. *-rgt-identity N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 75.6%

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

4. Final simplification 70.1%

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

Alternative 11: 43.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.3%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. *-rgt-identity N/A

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

   8. rgt-mult-inverse N/A

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

   9. metadata-eval N/A

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

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

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 54.0

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

5. Applied rewrites 54.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 43.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 41.1%

    \[\leadsto \mathsf{fma}\left(-1, x \cdot x, x\right) \]
12. Add Preprocessing

Alternative 12: 43.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 54.5

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 41.1%

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

Alternative 13: 39.1% 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 89.3%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

   7. lift-+.f64 N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval 99.9

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

   15. lift-+.f64 N/A

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

   16. metadata-eval N/A

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

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

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower--.f64 N/A

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

   22. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 37.9%

   \[\leadsto \color{blue}{1} \cdot x \]
8. 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 2025016 
(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)))