Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H

Percentage Accurate: 100.0% → 100.0%

Time: 2.0s

Alternatives: 9

Speedup: 1.0×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \mathsf{fma}\left(-z, x + y, y\right) + x \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (+ (fma (- z) (+ x y) y) x))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return fma(-z, (x + y), y) + x;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(fma(Float64(-z), Float64(x + y), y) + x)
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[((-z) * N[(x + y), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. *-lft-identity N/A

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

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

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt1-in N/A

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

   10. associate-*r* N/A

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

   11. +-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. mul-1-neg N/A

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

   18. lower-neg.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-+r+ N/A

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

   5. lift-neg.f64 N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. mul-1-neg N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lift-neg.f64 N/A

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

   17. lift-+.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-64}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-283}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-64)
   (* x (- z))
   (if (<= (+ x y) 2e-283) (+ y x) (fma (- y) z y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-64) {
		tmp = x * -z;
	} else if ((x + y) <= 2e-283) {
		tmp = y + x;
	} else {
		tmp = fma(-y, z, y);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-64)
		tmp = Float64(x * Float64(-z));
	elseif (Float64(x + y) <= 2e-283)
		tmp = Float64(y + x);
	else
		tmp = fma(Float64(-y), z, y);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-64], N[(x * (-z)), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 2e-283], N[(y + x), $MachinePrecision], N[((-y) * z + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -9.99999999999999965e-65

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 48.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 22.8%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(z\right)\right) \]

       2. lift-neg.f64 27.6

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

   11. Applied rewrites 27.6%

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

   if -9.99999999999999965e-65 < (+.f64 x y) < 1.99999999999999989e-283

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 56.8

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

   5. Applied rewrites 56.8%

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

   if 1.99999999999999989e-283 < (+.f64 x y)

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lift-neg.f64 N/A

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

      17. lift-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 48.0

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

   9. Applied rewrites 48.0%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-64}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;x + y \leq 2 \cdot 10^{-283}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;1 - z \leq -2000 \lor \neg \left(1 - z \leq 5000000000\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (- 1.0 z) -2000.0) (not (<= (- 1.0 z) 5000000000.0)))
   (* x (- z))
   (+ y x)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - z) <= -2000.0) || !((1.0 - z) <= 5000000000.0)) {
		tmp = x * -z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((1.0d0 - z) <= (-2000.0d0)) .or. (.not. ((1.0d0 - z) <= 5000000000.0d0))) then
        tmp = x * -z
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - z) <= -2000.0) || !((1.0 - z) <= 5000000000.0)) {
		tmp = x * -z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if ((1.0 - z) <= -2000.0) or not ((1.0 - z) <= 5000000000.0):
		tmp = x * -z
	else:
		tmp = y + x
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(1.0 - z) <= -2000.0) || !(Float64(1.0 - z) <= 5000000000.0))
		tmp = Float64(x * Float64(-z));
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((1.0 - z) <= -2000.0) || ~(((1.0 - z) <= 5000000000.0)))
		tmp = x * -z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[N[(1.0 - z), $MachinePrecision], -2000.0], N[Not[LessEqual[N[(1.0 - z), $MachinePrecision], 5000000000.0]], $MachinePrecision]], N[(x * (-z)), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) z) < -2e3 or 5e9 < (-.f64 #s(literal 1 binary64) z)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 2.9%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(z\right)\right) \]

       2. lift-neg.f64 50.8

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

   11. Applied rewrites 50.8%

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

   if -2e3 < (-.f64 #s(literal 1 binary64) z) < 5e9

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 94.8

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

   5. Applied rewrites 94.8%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -2000 \lor \neg \left(1 - z \leq 5000000000\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;1 - z \leq -2000:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;1 - z \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 z) -2000.0)
   (* x (- z))
   (if (<= (- 1.0 z) 2.0) (+ y x) (* y (- z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - z) <= -2000.0) {
		tmp = x * -z;
	} else if ((1.0 - z) <= 2.0) {
		tmp = y + x;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((1.0d0 - z) <= (-2000.0d0)) then
        tmp = x * -z
    else if ((1.0d0 - z) <= 2.0d0) then
        tmp = y + x
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - z) <= -2000.0) {
		tmp = x * -z;
	} else if ((1.0 - z) <= 2.0) {
		tmp = y + x;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (1.0 - z) <= -2000.0:
		tmp = x * -z
	elif (1.0 - z) <= 2.0:
		tmp = y + x
	else:
		tmp = y * -z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - z) <= -2000.0)
		tmp = Float64(x * Float64(-z));
	elseif (Float64(1.0 - z) <= 2.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((1.0 - z) <= -2000.0)
		tmp = x * -z;
	elseif ((1.0 - z) <= 2.0)
		tmp = y + x;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(1.0 - z), $MachinePrecision], -2000.0], N[(x * (-z)), $MachinePrecision], If[LessEqual[N[(1.0 - z), $MachinePrecision], 2.0], N[(y + x), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) z) < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 1.6%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(z\right)\right) \]

       2. lift-neg.f64 45.7

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

   11. Applied rewrites 45.7%

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

   if -2e3 < (-.f64 #s(literal 1 binary64) z) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 4.3%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

       2. lift-neg.f64 43.9

          \[\leadsto y \cdot \left(-z\right) \]

   11. Applied rewrites 43.9%

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

Alternative 5: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x + y \leq -4 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -4e-300) (fma (- z) x x) (fma (- y) z y)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -4e-300) {
		tmp = fma(-z, x, x);
	} else {
		tmp = fma(-y, z, y);
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -4e-300)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = fma(Float64(-y), z, y);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -4e-300], N[((-z) * x + x), $MachinePrecision], N[((-y) * z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.0000000000000001e-300

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lift-neg.f64 N/A

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

      17. lift-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot z\right)\right) + x \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(z \cdot x\right)\right) + x \]

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

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot x + x \]

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), x, x\right) \]

      8. lift-neg.f64 48.5

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

   9. Applied rewrites 48.5%

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

   if -4.0000000000000001e-300 < (+.f64 x y)

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lift-neg.f64 N/A

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

      17. lift-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 48.0

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

   9. Applied rewrites 48.0%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -4 \cdot 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) \cdot \left(1 - z\right) \leq -5 \cdot 10^{-227}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* (+ x y) (- 1.0 z)) -5e-227) x y))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (((x + y) * (1.0 - z)) <= -5e-227) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x + y) * (1.0d0 - z)) <= (-5d-227)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (((x + y) * (1.0 - z)) <= -5e-227) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if ((x + y) * (1.0 - z)) <= -5e-227:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + y) * Float64(1.0 - z)) <= -5e-227)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x + y) * (1.0 - z)) <= -5e-227)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], -5e-227], x, y]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) z)) < -4.99999999999999961e-227

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lift-neg.f64 N/A

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

      17. lift-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot z\right)\right) + x \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(z \cdot x\right)\right) + x \]

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

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot x + x \]

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), x, x\right) \]

      8. lift-neg.f64 48.0

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

   9. Applied rewrites 48.0%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 24.5%

       \[\leadsto x \]

   if -4.99999999999999961e-227 < (*.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) z))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt1-in N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lift-neg.f64 N/A

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

      17. lift-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 47.9

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

   9. Applied rewrites 47.9%

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

   10. Taylor expanded in z around 0

       \[\leadsto y \]
   11. Step-by-step derivation

   12. Applied rewrites 24.2%

       \[\leadsto y \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) \cdot \left(1 - z\right) \leq -5 \cdot 10^{-227}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \left(x + y\right) \cdot \left(1 - z\right) \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* (+ x y) (- 1.0 z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return (x + y) * (1.0 - z);
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) * (1.0d0 - z)
end function

Java

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return (x + y) * (1.0 - z)

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(Float64(x + y) * Float64(1.0 - z))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = (x + y) * (1.0 - z);
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

Alternative 8: 50.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ y + x \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (+ y x))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return y + x;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y + x
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return y + x;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return y + x

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(y + x)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = y + x;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(y + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 48.5

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

5. Applied rewrites 48.5%

   \[\leadsto \color{blue}{y + x} \]
6. Add Preprocessing

Alternative 9: 26.4% accurate, 12.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ y \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 y)

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return y;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return y;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return y

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return y
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = y;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := y

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. *-lft-identity N/A

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

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

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt1-in N/A

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

   10. associate-*r* N/A

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

   11. +-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. mul-1-neg N/A

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

   18. lower-neg.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-+r+ N/A

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

   5. lift-neg.f64 N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. mul-1-neg N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lift-neg.f64 N/A

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

   17. lift-+.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

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

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 49.9

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

9. Applied rewrites 49.9%

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

10. Taylor expanded in z around 0

    \[\leadsto y \]
11. Step-by-step derivation

12. Applied rewrites 25.6%

    \[\leadsto y \]

13. Final simplification 25.6%

    \[\leadsto y \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025080 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))