Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3

Percentage Accurate: 77.9% → 100.0%

Time: 4.0s

Alternatives: 7

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

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

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. associate--r- N/A

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

   8. metadata-eval N/A

      \[\leadsto \left(\color{blue}{0} + y\right) \cdot x + \left(1 - y\right) \]

   9. +-lft-identity N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 - x\right) \cdot \left(1 - y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+77} \lor \neg \left(t\_0 \leq 500000000000\right):\\ \;\;\;\;y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* (- 1.0 x) (- 1.0 y)))))
   (if (or (<= t_0 -2e+77) (not (<= t_0 500000000000.0)))
     (- (* y x) y)
     (- 1.0 y))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) * (1.0 - y));
	double tmp;
	if ((t_0 <= -2e+77) || !(t_0 <= 500000000000.0)) {
		tmp = (y * x) - y;
	} else {
		tmp = 1.0 - 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) :: t_0
    real(8) :: tmp
    t_0 = x + ((1.0d0 - x) * (1.0d0 - y))
    if ((t_0 <= (-2d+77)) .or. (.not. (t_0 <= 500000000000.0d0))) then
        tmp = (y * x) - y
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + ((1.0 - x) * (1.0 - y));
	double tmp;
	if ((t_0 <= -2e+77) || !(t_0 <= 500000000000.0)) {
		tmp = (y * x) - y;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + ((1.0 - x) * (1.0 - y))
	tmp = 0
	if (t_0 <= -2e+77) or not (t_0 <= 500000000000.0):
		tmp = (y * x) - y
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) * Float64(1.0 - y)))
	tmp = 0.0
	if ((t_0 <= -2e+77) || !(t_0 <= 500000000000.0))
		tmp = Float64(Float64(y * x) - y);
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + ((1.0 - x) * (1.0 - y));
	tmp = 0.0;
	if ((t_0 <= -2e+77) || ~((t_0 <= 500000000000.0)))
		tmp = (y * x) - y;
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+77], N[Not[LessEqual[t$95$0, 500000000000.0]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < -1.99999999999999997e77 or 5e11 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. sub0-neg N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(1 - x\right)} \]

      3. remove-double-neg N/A

         \[\leadsto 0 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \cdot \left(1 - x\right) \]

      4. mul-1-neg N/A

         \[\leadsto 0 - \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right) \cdot \left(1 - x\right) \]

      5. mul-1-neg N/A

         \[\leadsto 0 - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \cdot \left(1 - x\right) \]

      6. remove-double-neg N/A

         \[\leadsto 0 - \color{blue}{y} \cdot \left(1 - x\right) \]

      7. *-lft-identity N/A

         \[\leadsto 0 - y \cdot \left(1 - \color{blue}{1 \cdot x}\right) \]

      8. metadata-eval N/A

         \[\leadsto 0 - y \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot x\right) \]

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

         \[\leadsto 0 - y \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]

      10. distribute-rgt-in N/A

          \[\leadsto 0 - \color{blue}{\left(1 \cdot y + \left(-1 \cdot x\right) \cdot y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto 0 - \left(\color{blue}{y} + \left(-1 \cdot x\right) \cdot y\right) \]

      12. associate--r+ N/A

          \[\leadsto \color{blue}{\left(0 - y\right) - \left(-1 \cdot x\right) \cdot y} \]

      13. mul-1-neg N/A

          \[\leadsto \left(0 - y\right) - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot y \]

      14. +-lft-identity N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(0 + y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\left(1 - 1\right)} + y\right) \]

      16. associate--r- N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(1 - \left(1 - y\right)\right)} \]

      17. *-lft-identity N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(1 - \color{blue}{1 \cdot \left(1 - y\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(1 - y\right)\right) \]

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

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(1 - y\right)\right)} \]

      20. associate--r+ N/A

          \[\leadsto \color{blue}{0 - \left(y + \left(\mathsf{neg}\left(x\right)\right) \cdot \left(1 + -1 \cdot \left(1 - y\right)\right)\right)} \]

      21. metadata-eval N/A

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

      22. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   if -1.99999999999999997e77 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < 5e11

   1. Initial program 62.7%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 81.8

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

   5. Applied rewrites 81.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(1 - x\right) \cdot \left(1 - y\right) \leq -2 \cdot 10^{+77} \lor \neg \left(x + \left(1 - x\right) \cdot \left(1 - y\right) \leq 500000000000\right):\\ \;\;\;\;y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 - x\right) \cdot \left(1 - y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+77}:\\ \;\;\;\;y \cdot x - y\\ \mathbf{elif}\;t\_0 \leq 500000000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* (- 1.0 x) (- 1.0 y)))))
   (if (<= t_0 -2e+77)
     (- (* y x) y)
     (if (<= t_0 500000000000.0) (- 1.0 y) (fma y x (- y))))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) * (1.0 - y));
	double tmp;
	if (t_0 <= -2e+77) {
		tmp = (y * x) - y;
	} else if (t_0 <= 500000000000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = fma(y, x, -y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) * Float64(1.0 - y)))
	tmp = 0.0
	if (t_0 <= -2e+77)
		tmp = Float64(Float64(y * x) - y);
	elseif (t_0 <= 500000000000.0)
		tmp = Float64(1.0 - y);
	else
		tmp = fma(y, x, Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+77], N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[t$95$0, 500000000000.0], N[(1.0 - y), $MachinePrecision], N[(y * x + (-y)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < -1.99999999999999997e77

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. sub0-neg N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(1 - x\right)} \]

      3. remove-double-neg N/A

         \[\leadsto 0 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \cdot \left(1 - x\right) \]

      4. mul-1-neg N/A

         \[\leadsto 0 - \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right) \cdot \left(1 - x\right) \]

      5. mul-1-neg N/A

         \[\leadsto 0 - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \cdot \left(1 - x\right) \]

      6. remove-double-neg N/A

         \[\leadsto 0 - \color{blue}{y} \cdot \left(1 - x\right) \]

      7. *-lft-identity N/A

         \[\leadsto 0 - y \cdot \left(1 - \color{blue}{1 \cdot x}\right) \]

      8. metadata-eval N/A

         \[\leadsto 0 - y \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot x\right) \]

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

         \[\leadsto 0 - y \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]

      10. distribute-rgt-in N/A

          \[\leadsto 0 - \color{blue}{\left(1 \cdot y + \left(-1 \cdot x\right) \cdot y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto 0 - \left(\color{blue}{y} + \left(-1 \cdot x\right) \cdot y\right) \]

      12. associate--r+ N/A

          \[\leadsto \color{blue}{\left(0 - y\right) - \left(-1 \cdot x\right) \cdot y} \]

      13. mul-1-neg N/A

          \[\leadsto \left(0 - y\right) - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot y \]

      14. +-lft-identity N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(0 + y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\left(1 - 1\right)} + y\right) \]

      16. associate--r- N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(1 - \left(1 - y\right)\right)} \]

      17. *-lft-identity N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(1 - \color{blue}{1 \cdot \left(1 - y\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(1 - y\right)\right) \]

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

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(1 - y\right)\right)} \]

      20. associate--r+ N/A

          \[\leadsto \color{blue}{0 - \left(y + \left(\mathsf{neg}\left(x\right)\right) \cdot \left(1 + -1 \cdot \left(1 - y\right)\right)\right)} \]

      21. metadata-eval N/A

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

      22. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   if -1.99999999999999997e77 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < 5e11

   1. Initial program 62.7%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if 5e11 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{0} + y\right) \cdot x \]

      7. +-lft-identity N/A

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

      8. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

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

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

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

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

      4. mul-1-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lft-mult-inverse N/A

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

      18. associate-*l* N/A

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

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

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

   8. Applied rewrites 100.0%

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

Alternative 4: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+44} \lor \neg \left(x \leq 4.6 \cdot 10^{+19}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.15e+44) (not (<= x 4.6e+19))) (* y x) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.15e+44) || !(x <= 4.6e+19)) {
		tmp = y * x;
	} else {
		tmp = 1.0 - 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 <= (-2.15d+44)) .or. (.not. (x <= 4.6d+19))) then
        tmp = y * x
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.15e+44) || !(x <= 4.6e+19)) {
		tmp = y * x;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.15e+44) or not (x <= 4.6e+19):
		tmp = y * x
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.15e+44) || !(x <= 4.6e+19))
		tmp = Float64(y * x);
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.15e+44) || ~((x <= 4.6e+19)))
		tmp = y * x;
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.15e+44], N[Not[LessEqual[x, 4.6e+19]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.14999999999999991e44 or 4.6e19 < x

   1. Initial program 52.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{0} + y\right) \cdot x \]

      7. +-lft-identity N/A

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

      8. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -2.14999999999999991e44 < x < 4.6e19

   1. Initial program 96.1%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 94.2

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

   5. Applied rewrites 94.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+44} \lor \neg \left(x \leq 4.6 \cdot 10^{+19}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = -y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. sub0-neg N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(1 - x\right)} \]

      3. remove-double-neg N/A

         \[\leadsto 0 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \cdot \left(1 - x\right) \]

      4. mul-1-neg N/A

         \[\leadsto 0 - \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right) \cdot \left(1 - x\right) \]

      5. mul-1-neg N/A

         \[\leadsto 0 - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \cdot \left(1 - x\right) \]

      6. remove-double-neg N/A

         \[\leadsto 0 - \color{blue}{y} \cdot \left(1 - x\right) \]

      7. *-lft-identity N/A

         \[\leadsto 0 - y \cdot \left(1 - \color{blue}{1 \cdot x}\right) \]

      8. metadata-eval N/A

         \[\leadsto 0 - y \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot x\right) \]

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

         \[\leadsto 0 - y \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]

      10. distribute-rgt-in N/A

          \[\leadsto 0 - \color{blue}{\left(1 \cdot y + \left(-1 \cdot x\right) \cdot y\right)} \]

      11. *-lft-identity N/A

          \[\leadsto 0 - \left(\color{blue}{y} + \left(-1 \cdot x\right) \cdot y\right) \]

      12. associate--r+ N/A

          \[\leadsto \color{blue}{\left(0 - y\right) - \left(-1 \cdot x\right) \cdot y} \]

      13. mul-1-neg N/A

          \[\leadsto \left(0 - y\right) - \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot y \]

      14. +-lft-identity N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(0 + y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\left(1 - 1\right)} + y\right) \]

      16. associate--r- N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(1 - \left(1 - y\right)\right)} \]

      17. *-lft-identity N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(1 - \color{blue}{1 \cdot \left(1 - y\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(1 - y\right)\right) \]

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

          \[\leadsto \left(0 - y\right) - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(1 - y\right)\right)} \]

      20. associate--r+ N/A

          \[\leadsto \color{blue}{0 - \left(y + \left(\mathsf{neg}\left(x\right)\right) \cdot \left(1 + -1 \cdot \left(1 - y\right)\right)\right)} \]

      21. metadata-eval N/A

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

      22. +-commutative N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.7%

      \[\leadsto -y \]

   if -1 < y < 1

   1. Initial program 59.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate--r- N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{0} + y\right) \cdot x \]

      7. +-lft-identity N/A

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

      8. lower-*.f64 21.8

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

   5. Applied rewrites 21.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 77.9%

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

4. Final simplification 63.6%

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

Alternative 6: 62.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 64.9

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

5. Applied rewrites 64.9%

   \[\leadsto \color{blue}{1 - y} \]
6. Add Preprocessing

Alternative 7: 37.6% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. associate--r- N/A

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

   6. metadata-eval N/A

      \[\leadsto \left(\color{blue}{0} + y\right) \cdot x \]

   7. +-lft-identity N/A

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

   8. lower-*.f64 35.3

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

5. Applied rewrites 35.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 44.4%

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

Developer Target 1: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025015 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64

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

  (+ x (* (- 1.0 x) (- 1.0 y))))