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

Percentage Accurate: 77.4% → 100.0%

Time: 2.6s

Alternatives: 8

Speedup: 1.4×

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: 77.4% 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, 1.4× 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 77.4%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

2. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

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

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift--.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-lft-identity N/A

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

   5. associate-*r* N/A

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

   6. distribute-lft-out-- N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

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

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

   10. +-commutative N/A

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

   11. distribute-lft1-in N/A

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

   12. associate-*r* N/A

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

   13. associate--l- N/A

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

   14. associate-*r* N/A

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

   15. mul-1-neg N/A

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

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

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

   17. +-commutative N/A

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

   18. associate--l+ N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f64 N/A

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

   21. lift--.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -210000000000:\\ \;\;\;\;\left(x - 1\right) \cdot y\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;1 - \left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -210000000000.0)
   (* (- x 1.0) y)
   (if (<= y 8.5e-7) (- 1.0 (* (- x) y)) (- (* y x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -210000000000.0) {
		tmp = (x - 1.0) * y;
	} else if (y <= 8.5e-7) {
		tmp = 1.0 - (-x * y);
	} else {
		tmp = (y * x) - y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-210000000000.0d0)) then
        tmp = (x - 1.0d0) * y
    else if (y <= 8.5d-7) then
        tmp = 1.0d0 - (-x * y)
    else
        tmp = (y * x) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -210000000000.0) {
		tmp = (x - 1.0) * y;
	} else if (y <= 8.5e-7) {
		tmp = 1.0 - (-x * y);
	} else {
		tmp = (y * x) - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -210000000000.0:
		tmp = (x - 1.0) * y
	elif y <= 8.5e-7:
		tmp = 1.0 - (-x * y)
	else:
		tmp = (y * x) - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -210000000000.0)
		tmp = Float64(Float64(x - 1.0) * y);
	elseif (y <= 8.5e-7)
		tmp = Float64(1.0 - Float64(Float64(-x) * y));
	else
		tmp = Float64(Float64(y * x) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -210000000000.0)
		tmp = (x - 1.0) * y;
	elseif (y <= 8.5e-7)
		tmp = 1.0 - (-x * y);
	else
		tmp = (y * x) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -210000000000.0], N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 8.5e-7], N[(1.0 - N[((-x) * y), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1e11

   1. Initial program 100.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 52.2

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-out-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate-+l- N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if -2.1e11 < y < 8.50000000000000014e-7

   1. Initial program 55.2%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if 8.50000000000000014e-7 < y

   1. Initial program 99.9%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.8

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

   7. Applied rewrites 48.8%

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

   8. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-out-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate-+l- N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 98.1

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

   10. Applied rewrites 98.1%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

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

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

       8. distribute-rgt-out N/A

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

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

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

       10. metadata-eval N/A

           \[\leadsto x \cdot y - 1 \cdot y \]

       11. *-lft-identity N/A

           \[\leadsto x \cdot y - y \]

       12. lower--.f64 N/A

           \[\leadsto x \cdot y - y \]

       13. *-commutative N/A

           \[\leadsto y \cdot x - y \]

       14. lower-*.f64 98.1

           \[\leadsto y \cdot x - y \]

   12. Applied rewrites 98.1%

       \[\leadsto y \cdot x - y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-53}:\\ \;\;\;\;\left(x - 1\right) \cdot y\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{-65}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e-53)
   (* (- x 1.0) y)
   (if (<= y 6.3e-65) (- 1.0 y) (- (* y x) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-53) {
		tmp = (x - 1.0) * y;
	} else if (y <= 6.3e-65) {
		tmp = 1.0 - y;
	} else {
		tmp = (y * x) - y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.5d-53)) then
        tmp = (x - 1.0d0) * y
    else if (y <= 6.3d-65) then
        tmp = 1.0d0 - y
    else
        tmp = (y * x) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-53) {
		tmp = (x - 1.0) * y;
	} else if (y <= 6.3e-65) {
		tmp = 1.0 - y;
	} else {
		tmp = (y * x) - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e-53:
		tmp = (x - 1.0) * y
	elif y <= 6.3e-65:
		tmp = 1.0 - y
	else:
		tmp = (y * x) - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e-53)
		tmp = Float64(Float64(x - 1.0) * y);
	elseif (y <= 6.3e-65)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(Float64(y * x) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e-53)
		tmp = (x - 1.0) * y;
	elseif (y <= 6.3e-65)
		tmp = 1.0 - y;
	else
		tmp = (y * x) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e-53], N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 6.3e-65], N[(1.0 - y), $MachinePrecision], N[(N[(y * x), $MachinePrecision] - y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5e-53

   1. Initial program 94.4%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 58.2

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

   7. Applied rewrites 58.2%

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

   8. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-out-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate-+l- N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 92.2

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

   10. Applied rewrites 92.2%

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

   if -2.5e-53 < y < 6.2999999999999997e-65

   1. Initial program 53.6%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in x around 0

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

      1. lift--.f64 80.6

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

   4. Applied rewrites 80.6%

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

   if 6.2999999999999997e-65 < y

   1. Initial program 93.0%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 56.4

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

   7. Applied rewrites 56.4%

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

   8. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-out-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate-+l- N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 90.2

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

   10. Applied rewrites 90.2%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

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

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

       8. distribute-rgt-out N/A

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

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

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

       10. metadata-eval N/A

           \[\leadsto x \cdot y - 1 \cdot y \]

       11. *-lft-identity N/A

           \[\leadsto x \cdot y - y \]

       12. lower--.f64 N/A

           \[\leadsto x \cdot y - y \]

       13. *-commutative N/A

           \[\leadsto y \cdot x - y \]

       14. lower-*.f64 90.2

           \[\leadsto y \cdot x - y \]

   12. Applied rewrites 90.2%

       \[\leadsto y \cdot x - y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - 1\right) \cdot y\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{-65}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x 1.0) y)))
   (if (<= y -2.5e-53) t_0 (if (<= y 6.3e-65) (- 1.0 y) t_0))))

C

double code(double x, double y) {
	double t_0 = (x - 1.0) * y;
	double tmp;
	if (y <= -2.5e-53) {
		tmp = t_0;
	} else if (y <= 6.3e-65) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - 1.0d0) * y
    if (y <= (-2.5d-53)) then
        tmp = t_0
    else if (y <= 6.3d-65) then
        tmp = 1.0d0 - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - 1.0) * y;
	double tmp;
	if (y <= -2.5e-53) {
		tmp = t_0;
	} else if (y <= 6.3e-65) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - 1.0) * y
	tmp = 0
	if y <= -2.5e-53:
		tmp = t_0
	elif y <= 6.3e-65:
		tmp = 1.0 - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - 1.0) * y)
	tmp = 0.0
	if (y <= -2.5e-53)
		tmp = t_0;
	elseif (y <= 6.3e-65)
		tmp = Float64(1.0 - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - 1.0) * y;
	tmp = 0.0;
	if (y <= -2.5e-53)
		tmp = t_0;
	elseif (y <= 6.3e-65)
		tmp = 1.0 - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.5e-53], t$95$0, If[LessEqual[y, 6.3e-65], N[(1.0 - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5e-53 or 6.2999999999999997e-65 < y

   1. Initial program 93.7%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 57.3

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

   7. Applied rewrites 57.3%

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

   8. Taylor expanded in y around inf

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-out-- N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate-+l- N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 91.2

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

   10. Applied rewrites 91.2%

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

   if -2.5e-53 < y < 6.2999999999999997e-65

   1. Initial program 53.6%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in x around 0

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

      1. lift--.f64 80.6

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

   4. Applied rewrites 80.6%

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

Alternative 5: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -21000000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+72}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -21000000000.0) (* y x) (if (<= x 1.1e+72) (- 1.0 y) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -21000000000.0) {
		tmp = y * x;
	} else if (x <= 1.1e+72) {
		tmp = 1.0 - y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-21000000000.0d0)) then
        tmp = y * x
    else if (x <= 1.1d+72) then
        tmp = 1.0d0 - y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -21000000000.0) {
		tmp = y * x;
	} else if (x <= 1.1e+72) {
		tmp = 1.0 - y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -21000000000.0:
		tmp = y * x
	elif x <= 1.1e+72:
		tmp = 1.0 - y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -21000000000.0)
		tmp = Float64(y * x);
	elseif (x <= 1.1e+72)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -21000000000.0)
		tmp = y * x;
	elseif (x <= 1.1e+72)
		tmp = 1.0 - y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -21000000000.0], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.1e+72], N[(1.0 - y), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e10 or 1.1e72 < x

   1. Initial program 53.1%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.8

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

   4. Applied rewrites 76.8%

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

   if -2.1e10 < x < 1.1e72

   1. Initial program 96.1%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in x around 0

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

      1. lift--.f64 92.7

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

   4. Applied rewrites 92.7%

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

Alternative 6: 63.3% accurate, 3.3× 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 77.4%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

2. Taylor expanded in x around 0

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

   1. lift--.f64 63.3

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

4. Applied rewrites 63.3%

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

Alternative 7: 60.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - y \leq 0.9999995:\\ \;\;\;\;-y\\ \mathbf{elif}\;1 - y \leq 10^{+57}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- 1.0 y) 0.9999995) (- y) (if (<= (- 1.0 y) 1e+57) 1.0 (- y))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - y) <= 0.9999995) {
		tmp = -y;
	} else if ((1.0 - y) <= 1e+57) {
		tmp = 1.0;
	} else {
		tmp = -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 ((1.0d0 - y) <= 0.9999995d0) then
        tmp = -y
    else if ((1.0d0 - y) <= 1d+57) then
        tmp = 1.0d0
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - y) <= 0.9999995) {
		tmp = -y;
	} else if ((1.0 - y) <= 1e+57) {
		tmp = 1.0;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - y) <= 0.9999995:
		tmp = -y
	elif (1.0 - y) <= 1e+57:
		tmp = 1.0
	else:
		tmp = -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - y) <= 0.9999995)
		tmp = Float64(-y);
	elseif (Float64(1.0 - y) <= 1e+57)
		tmp = 1.0;
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - y) <= 0.9999995)
		tmp = -y;
	elseif ((1.0 - y) <= 1e+57)
		tmp = 1.0;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - y), $MachinePrecision], 0.9999995], (-y), If[LessEqual[N[(1.0 - y), $MachinePrecision], 1e+57], 1.0, (-y)]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) y) < 0.999999500000000041 or 1.00000000000000005e57 < (-.f64 #s(literal 1 binary64) y)

   1. Initial program 99.9%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in x around 0

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

      1. lift--.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(y\right) \]

      2. lift-neg.f64 50.3

         \[\leadsto -y \]

   7. Applied rewrites 50.3%

      \[\leadsto -y \]

   if 0.999999500000000041 < (-.f64 #s(literal 1 binary64) y) < 1.00000000000000005e57

   1. Initial program 58.5%

      \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 68.8%

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

Alternative 8: 38.6% accurate, 11.9× 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 77.4%

   \[x + \left(1 - x\right) \cdot \left(1 - y\right) \]

2. Taylor expanded in y around 0

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

4. Applied rewrites 38.6%

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

Reproduce

herbie shell --seed 2025112 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.Vectors:renderPlotVectors from Chart-1.5.3"
  :precision binary64
  (+ x (* (- 1.0 x) (- 1.0 y))))