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

Percentage Accurate: 77.5% → 100.0%

Time: 4.7s

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: 77.5% 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 76.4%

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

3. Taylor expanded in x around 0

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

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 - x\right) \cdot \left(1 - y\right)\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* (- 1.0 x) (- 1.0 y)))))
   (if (or (<= t_0 0.0) (not (<= t_0 2.0))) (fma 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 <= 0.0) || !(t_0 <= 2.0)) {
		tmp = fma(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 <= 0.0) || !(t_0 <= 2.0))
		tmp = fma(y, x, Float64(-y));
	else
		tmp = Float64(1.0 - 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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(y * x + (-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))) < 0.0 or 2 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 69.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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. remove-double-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt1-in N/A

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

      12. mul-1-neg N/A

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

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

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. associate--r- N/A

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

      19. metadata-eval N/A

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

      20. +-lft-identity N/A

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

      21. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 86.0%

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

   if 0.0 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(1 - x\right) \cdot \left(1 - y\right) \leq 0 \lor \neg \left(x + \left(1 - x\right) \cdot \left(1 - y\right) \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 - x\right) \cdot \left(1 - y\right)\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\left(-1 + x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* (- 1.0 x) (- 1.0 y)))))
   (if (or (<= t_0 0.0) (not (<= t_0 2.0))) (* (+ -1.0 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 <= 0.0) || !(t_0 <= 2.0)) {
		tmp = (-1.0 + 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 <= 0.0d0) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = ((-1.0d0) + 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 <= 0.0) || !(t_0 <= 2.0)) {
		tmp = (-1.0 + 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 <= 0.0) or not (t_0 <= 2.0):
		tmp = (-1.0 + 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 <= 0.0) || !(t_0 <= 2.0))
		tmp = Float64(Float64(-1.0 + 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 <= 0.0) || ~((t_0 <= 2.0)))
		tmp = (-1.0 + 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, 0.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[(-1.0 + 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))) < 0.0 or 2 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 69.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if 0.0 < (+.f64 x (*.f64 (-.f64 #s(literal 1 binary64) x) (-.f64 #s(literal 1 binary64) y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(1 - x\right) \cdot \left(1 - y\right) \leq 0 \lor \neg \left(x + \left(1 - x\right) \cdot \left(1 - y\right) \leq 2\right):\\ \;\;\;\;\left(-1 + x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq -10000 \lor \neg \left(1 - y \leq 2\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (- 1.0 y) -10000.0) (not (<= (- 1.0 y) 2.0))) (- y) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - y) <= -10000.0) || !((1.0 - y) <= 2.0)) {
		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 (((1.0d0 - y) <= (-10000.0d0)) .or. (.not. ((1.0d0 - y) <= 2.0d0))) 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 (((1.0 - y) <= -10000.0) || !((1.0 - y) <= 2.0)) {
		tmp = -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - y) <= -10000.0) or not ((1.0 - y) <= 2.0):
		tmp = -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(1.0 - y) <= -10000.0) || !(Float64(1.0 - y) <= 2.0))
		tmp = Float64(-y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - y) <= -10000.0) || ~(((1.0 - y) <= 2.0)))
		tmp = -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(1.0 - y), $MachinePrecision], -10000.0], N[Not[LessEqual[N[(1.0 - y), $MachinePrecision], 2.0]], $MachinePrecision]], (-y), 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < -1e4 or 2 < (-.f64 #s(literal 1 binary64) 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.2%

      \[\leadsto -y \]

   if -1e4 < (-.f64 #s(literal 1 binary64) y) < 2

   1. Initial program 52.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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. remove-double-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt1-in N/A

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

      12. mul-1-neg N/A

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

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

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. associate--r- N/A

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

      19. metadata-eval N/A

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

      20. +-lft-identity N/A

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

      21. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.4%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \leq -10000 \lor \neg \left(1 - y \leq 2\right):\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -580000000000 \lor \neg \left(x \leq 5500\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -580000000000.0) (not (<= x 5500.0))) (* y x) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -580000000000.0) || !(x <= 5500.0)) {
		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 <= (-580000000000.0d0)) .or. (.not. (x <= 5500.0d0))) 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 <= -580000000000.0) || !(x <= 5500.0)) {
		tmp = y * x;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -580000000000.0) or not (x <= 5500.0):
		tmp = y * x
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -580000000000.0) || !(x <= 5500.0))
		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 <= -580000000000.0) || ~((x <= 5500.0)))
		tmp = y * x;
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -580000000000.0], N[Not[LessEqual[x, 5500.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e11 or 5500 < x

   1. Initial program 52.8%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
   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. remove-double-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt1-in N/A

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

      12. mul-1-neg N/A

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

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

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. associate--r- N/A

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

      19. metadata-eval N/A

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

      20. +-lft-identity N/A

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

      21. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if -5.8e11 < x < 5500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -580000000000 \lor \neg \left(x \leq 5500\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.5% 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 76.4%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 61.1

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

5. Applied rewrites 61.1%

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

Alternative 7: 38.0% 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 76.4%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]
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. remove-double-neg N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. distribute-rgt1-in N/A

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

   12. mul-1-neg N/A

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

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

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

   14. *-commutative N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. *-lft-identity N/A

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

   18. associate--r- N/A

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

   19. metadata-eval N/A

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

   20. +-lft-identity N/A

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

   21. lower-*.f64 40.5

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

5. Applied rewrites 40.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 34.9%

   \[\leadsto \color{blue}{1} \]
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 2024352 
(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))))