Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D

Percentage Accurate: 99.5% → 99.8%

Time: 3.6s

Alternatives: 13

Speedup: 1.3×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - 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 - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - 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 - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(z, 6, -4\right), x - y, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (fma z 6.0 -4.0) (- x y) x))

C

double code(double x, double y, double z) {
	return fma(fma(z, 6.0, -4.0), (x - y), x);
}

Julia

function code(x, y, z)
	return fma(fma(z, 6.0, -4.0), Float64(x - y), x)
end

Wolfram

code[x_, y_, z_] := N[(N[(z * 6.0 + -4.0), $MachinePrecision] * N[(x - y), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

   3. add-flip N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

   4. sub-flip N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   11. lift--.f64 N/A

       \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   12. sub-flip N/A

       \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   15. remove-double-neg N/A

       \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

   16. associate-+l+ N/A

       \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

   17. *-commutative N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

   18. *-commutative N/A

       \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

   19. associate-*l* N/A

       \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, 6, -4\right), x - y, x\right)} \]
4. Add Preprocessing

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;-6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \mathbf{elif}\;z \leq 0.52:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(6 \cdot z, x - y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.58)
   (* -6.0 (* z (- y x)))
   (if (<= z 0.52) (fma -3.0 x (* 4.0 y)) (fma (* 6.0 z) (- x y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.58) {
		tmp = -6.0 * (z * (y - x));
	} else if (z <= 0.52) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = fma((6.0 * z), (x - y), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.58)
		tmp = Float64(-6.0 * Float64(z * Float64(y - x)));
	elseif (z <= 0.52)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = fma(Float64(6.0 * z), Float64(x - y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.58], N[(-6.0 * N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.52], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * z), $MachinePrecision] * N[(x - y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.57999999999999996

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(z \cdot \left(y - x\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(z \cdot \color{blue}{\left(y - x\right)}\right) \]

      3. lower--.f64 50.4

         \[\leadsto -6 \cdot \left(z \cdot \left(y - \color{blue}{x}\right)\right) \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{-6 \cdot \left(z \cdot \left(y - x\right)\right)} \]

   if -0.57999999999999996 < z < 0.52000000000000002

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, 6, -4\right), x - y, x\right)} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3 - -6 \cdot z, x, \mathsf{fma}\left(-6, z, 4\right) \cdot y\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-3 \cdot x + 4 \cdot y} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-3, \color{blue}{x}, 4 \cdot y\right) \]

      2. lower-*.f64 51.1

         \[\leadsto \mathsf{fma}\left(-3, x, 4 \cdot y\right) \]

   8. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

   if 0.52000000000000002 < z

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, 6, -4\right), x - y, x\right)} \]

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{6 \cdot z}, x - y, x\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 49.9

         \[\leadsto \mathsf{fma}\left(6 \cdot \color{blue}{z}, x - y, x\right) \]

   6. Applied rewrites 49.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{6 \cdot z}, x - y, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;-6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \mathbf{elif}\;z \leq 0.52:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(x - y\right), 6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.58)
   (* -6.0 (* z (- y x)))
   (if (<= z 0.52) (fma -3.0 x (* 4.0 y)) (fma (* z (- x y)) 6.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.58) {
		tmp = -6.0 * (z * (y - x));
	} else if (z <= 0.52) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = fma((z * (x - y)), 6.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.58)
		tmp = Float64(-6.0 * Float64(z * Float64(y - x)));
	elseif (z <= 0.52)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = fma(Float64(z * Float64(x - y)), 6.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.58], N[(-6.0 * N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.52], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] * 6.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.57999999999999996

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(z \cdot \left(y - x\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(z \cdot \color{blue}{\left(y - x\right)}\right) \]

      3. lower--.f64 50.4

         \[\leadsto -6 \cdot \left(z \cdot \left(y - \color{blue}{x}\right)\right) \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{-6 \cdot \left(z \cdot \left(y - x\right)\right)} \]

   if -0.57999999999999996 < z < 0.52000000000000002

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, 6, -4\right), x - y, x\right)} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3 - -6 \cdot z, x, \mathsf{fma}\left(-6, z, 4\right) \cdot y\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-3 \cdot x + 4 \cdot y} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-3, \color{blue}{x}, 4 \cdot y\right) \]

      2. lower-*.f64 51.1

         \[\leadsto \mathsf{fma}\left(-3, x, 4 \cdot y\right) \]

   8. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

   if 0.52000000000000002 < z

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(x - y\right)}, 6, x\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \color{blue}{\left(x - y\right)}, 6, x\right) \]

      2. lower--.f64 49.9

         \[\leadsto \mathsf{fma}\left(z \cdot \left(x - \color{blue}{y}\right), 6, x\right) \]

   6. Applied rewrites 49.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(x - y\right)}, 6, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.52:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -6.0 (* z (- y x)))))
   (if (<= z -0.58) t_0 (if (<= z 0.52) (fma -3.0 x (* 4.0 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -6.0 * (z * (y - x));
	double tmp;
	if (z <= -0.58) {
		tmp = t_0;
	} else if (z <= 0.52) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-6.0 * Float64(z * Float64(y - x)))
	tmp = 0.0
	if (z <= -0.58)
		tmp = t_0;
	elseif (z <= 0.52)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-6.0 * N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.58], t$95$0, If[LessEqual[z, 0.52], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.57999999999999996 or 0.52000000000000002 < z

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-6 \cdot \left(z \cdot \left(y - x\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(z \cdot \left(y - x\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(z \cdot \color{blue}{\left(y - x\right)}\right) \]

      3. lower--.f64 50.4

         \[\leadsto -6 \cdot \left(z \cdot \left(y - \color{blue}{x}\right)\right) \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{-6 \cdot \left(z \cdot \left(y - x\right)\right)} \]

   if -0.57999999999999996 < z < 0.52000000000000002

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, 6, -4\right), x - y, x\right)} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3 - -6 \cdot z, x, \mathsf{fma}\left(-6, z, 4\right) \cdot y\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-3 \cdot x + 4 \cdot y} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-3, \color{blue}{x}, 4 \cdot y\right) \]

      2. lower-*.f64 51.1

         \[\leadsto \mathsf{fma}\left(-3, x, 4 \cdot y\right) \]

   8. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ t_1 := \frac{2}{3} - z\\ t_2 := \mathsf{fma}\left(x \cdot z, 6, x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+85}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.666666665:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 4.0) y))
        (t_1 (- (/ 2.0 3.0) z))
        (t_2 (fma (* x z) 6.0 x)))
   (if (<= t_1 -1e+85)
     t_2
     (if (<= t_1 0.666666665)
       t_0
       (if (<= t_1 0.68)
         (fma -3.0 x (* 4.0 y))
         (if (<= t_1 2e+40) t_0 (if (<= t_1 5e+188) t_2 (* -6.0 (* y z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 4.0) * y;
	double t_1 = (2.0 / 3.0) - z;
	double t_2 = fma((x * z), 6.0, x);
	double tmp;
	if (t_1 <= -1e+85) {
		tmp = t_2;
	} else if (t_1 <= 0.666666665) {
		tmp = t_0;
	} else if (t_1 <= 0.68) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else if (t_1 <= 2e+40) {
		tmp = t_0;
	} else if (t_1 <= 5e+188) {
		tmp = t_2;
	} else {
		tmp = -6.0 * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 4.0) * y)
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	t_2 = fma(Float64(x * z), 6.0, x)
	tmp = 0.0
	if (t_1 <= -1e+85)
		tmp = t_2;
	elseif (t_1 <= 0.666666665)
		tmp = t_0;
	elseif (t_1 <= 0.68)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	elseif (t_1 <= 2e+40)
		tmp = t_0;
	elseif (t_1 <= 5e+188)
		tmp = t_2;
	else
		tmp = Float64(-6.0 * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * z), $MachinePrecision] * 6.0 + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+85], t$95$2, If[LessEqual[t$95$1, 0.666666665], t$95$0, If[LessEqual[t$95$1, 0.68], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+40], t$95$0, If[LessEqual[t$95$1, 5e+188], t$95$2, N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e85 or 2.00000000000000006e40 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 5.0000000000000001e188

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(x - y\right)}, 6, x\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z \cdot \color{blue}{\left(x - y\right)}, 6, x\right) \]

      2. lower--.f64 49.9

         \[\leadsto \mathsf{fma}\left(z \cdot \left(x - \color{blue}{y}\right), 6, x\right) \]

   6. Applied rewrites 49.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(x - y\right)}, 6, x\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{z}, 6, x\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 27.2

         \[\leadsto \mathsf{fma}\left(x \cdot z, 6, x\right) \]

   9. Applied rewrites 27.2%

      \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{z}, 6, x\right) \]

   if -1e85 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.666666665000000047 or 0.680000000000000049 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 2.00000000000000006e40

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto -6 \cdot \left(\left(z - \frac{2}{3}\right) \cdot \color{blue}{y}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(-6 \cdot \left(z - \frac{2}{3}\right)\right) \cdot \color{blue}{y} \]

      6. lift--.f64 N/A

         \[\leadsto \left(-6 \cdot \left(z - \frac{2}{3}\right)\right) \cdot y \]

      7. sub-negate-rev N/A

         \[\leadsto \left(-6 \cdot \left(\mathsf{neg}\left(\left(\frac{2}{3} - z\right)\right)\right)\right) \cdot y \]

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

         \[\leadsto \left(\mathsf{neg}\left(-6 \cdot \left(\frac{2}{3} - z\right)\right)\right) \cdot y \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(-6\right)\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

      10. metadata-eval N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

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

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      12. metadata-eval N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(4 - 6 \cdot z\right) \cdot y \]

      14. lift-*.f64 N/A

          \[\leadsto \left(4 - 6 \cdot z\right) \cdot y \]

      15. metadata-eval N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      16. metadata-eval N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      17. lift-*.f64 N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

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

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

      19. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(-6\right)\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(-6\right)\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{y} \]

   8. Applied rewrites 51.7%

      \[\leadsto \mathsf{fma}\left(-6, z, 4\right) \cdot \color{blue}{y} \]

   if 0.666666665000000047 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.680000000000000049

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, 6, -4\right), x - y, x\right)} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3 - -6 \cdot z, x, \mathsf{fma}\left(-6, z, 4\right) \cdot y\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-3 \cdot x + 4 \cdot y} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-3, \color{blue}{x}, 4 \cdot y\right) \]

      2. lower-*.f64 51.1

         \[\leadsto \mathsf{fma}\left(-3, x, 4 \cdot y\right) \]

   8. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

   if 5.0000000000000001e188 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]

   7. Taylor expanded in z around inf

      \[\leadsto -6 \cdot \left(y \cdot \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 27.1

         \[\leadsto -6 \cdot \left(y \cdot z\right) \]

   9. Applied rewrites 27.1%

      \[\leadsto -6 \cdot \left(y \cdot \color{blue}{z}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ t_1 := \frac{2}{3} - z\\ t_2 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+85}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.666666665:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 4.0) y))
        (t_1 (- (/ 2.0 3.0) z))
        (t_2 (* 6.0 (* x z))))
   (if (<= t_1 -1e+85)
     t_2
     (if (<= t_1 0.666666665)
       t_0
       (if (<= t_1 0.68)
         (fma -3.0 x (* 4.0 y))
         (if (<= t_1 2e+40) t_0 (if (<= t_1 5e+188) t_2 (* -6.0 (* y z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 4.0) * y;
	double t_1 = (2.0 / 3.0) - z;
	double t_2 = 6.0 * (x * z);
	double tmp;
	if (t_1 <= -1e+85) {
		tmp = t_2;
	} else if (t_1 <= 0.666666665) {
		tmp = t_0;
	} else if (t_1 <= 0.68) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else if (t_1 <= 2e+40) {
		tmp = t_0;
	} else if (t_1 <= 5e+188) {
		tmp = t_2;
	} else {
		tmp = -6.0 * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 4.0) * y)
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	t_2 = Float64(6.0 * Float64(x * z))
	tmp = 0.0
	if (t_1 <= -1e+85)
		tmp = t_2;
	elseif (t_1 <= 0.666666665)
		tmp = t_0;
	elseif (t_1 <= 0.68)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	elseif (t_1 <= 2e+40)
		tmp = t_0;
	elseif (t_1 <= 5e+188)
		tmp = t_2;
	else
		tmp = Float64(-6.0 * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$2 = N[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+85], t$95$2, If[LessEqual[t$95$1, 0.666666665], t$95$0, If[LessEqual[t$95$1, 0.68], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+40], t$95$0, If[LessEqual[t$95$1, 5e+188], t$95$2, N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e85 or 2.00000000000000006e40 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 5.0000000000000001e188

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-6 \cdot \left(\frac{2}{3} - z\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \color{blue}{\left(\frac{2}{3} - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - \color{blue}{z}\right)\right) \]

      6. metadata-eval 51.6

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto 6 \cdot \color{blue}{\left(x \cdot z\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left(x \cdot \color{blue}{z}\right) \]

      2. lower-*.f64 27.6

         \[\leadsto 6 \cdot \left(x \cdot z\right) \]

   7. Applied rewrites 27.6%

      \[\leadsto 6 \cdot \color{blue}{\left(x \cdot z\right)} \]

   if -1e85 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.666666665000000047 or 0.680000000000000049 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 2.00000000000000006e40

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto -6 \cdot \left(\left(z - \frac{2}{3}\right) \cdot \color{blue}{y}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(-6 \cdot \left(z - \frac{2}{3}\right)\right) \cdot \color{blue}{y} \]

      6. lift--.f64 N/A

         \[\leadsto \left(-6 \cdot \left(z - \frac{2}{3}\right)\right) \cdot y \]

      7. sub-negate-rev N/A

         \[\leadsto \left(-6 \cdot \left(\mathsf{neg}\left(\left(\frac{2}{3} - z\right)\right)\right)\right) \cdot y \]

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

         \[\leadsto \left(\mathsf{neg}\left(-6 \cdot \left(\frac{2}{3} - z\right)\right)\right) \cdot y \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(-6\right)\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

      10. metadata-eval N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

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

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      12. metadata-eval N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(4 - 6 \cdot z\right) \cdot y \]

      14. lift-*.f64 N/A

          \[\leadsto \left(4 - 6 \cdot z\right) \cdot y \]

      15. metadata-eval N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      16. metadata-eval N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      17. lift-*.f64 N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

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

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

      19. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(-6\right)\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(-6\right)\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{y} \]

   8. Applied rewrites 51.7%

      \[\leadsto \mathsf{fma}\left(-6, z, 4\right) \cdot \color{blue}{y} \]

   if 0.666666665000000047 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.680000000000000049

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, 6, -4\right), x - y, x\right)} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3 - -6 \cdot z, x, \mathsf{fma}\left(-6, z, 4\right) \cdot y\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-3 \cdot x + 4 \cdot y} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-3, \color{blue}{x}, 4 \cdot y\right) \]

      2. lower-*.f64 51.1

         \[\leadsto \mathsf{fma}\left(-3, x, 4 \cdot y\right) \]

   8. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x, 4 \cdot y\right)} \]

   if 5.0000000000000001e188 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]

   7. Taylor expanded in z around inf

      \[\leadsto -6 \cdot \left(y \cdot \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 27.1

         \[\leadsto -6 \cdot \left(y \cdot z\right) \]

   9. Applied rewrites 27.1%

      \[\leadsto -6 \cdot \left(y \cdot \color{blue}{z}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ t_1 := \frac{2}{3} - z\\ t_2 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+85}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.666666665:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(-4, x - y, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 4.0) y))
        (t_1 (- (/ 2.0 3.0) z))
        (t_2 (* 6.0 (* x z))))
   (if (<= t_1 -1e+85)
     t_2
     (if (<= t_1 0.666666665)
       t_0
       (if (<= t_1 0.68)
         (fma -4.0 (- x y) x)
         (if (<= t_1 2e+40) t_0 (if (<= t_1 5e+188) t_2 (* -6.0 (* y z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 4.0) * y;
	double t_1 = (2.0 / 3.0) - z;
	double t_2 = 6.0 * (x * z);
	double tmp;
	if (t_1 <= -1e+85) {
		tmp = t_2;
	} else if (t_1 <= 0.666666665) {
		tmp = t_0;
	} else if (t_1 <= 0.68) {
		tmp = fma(-4.0, (x - y), x);
	} else if (t_1 <= 2e+40) {
		tmp = t_0;
	} else if (t_1 <= 5e+188) {
		tmp = t_2;
	} else {
		tmp = -6.0 * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 4.0) * y)
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	t_2 = Float64(6.0 * Float64(x * z))
	tmp = 0.0
	if (t_1 <= -1e+85)
		tmp = t_2;
	elseif (t_1 <= 0.666666665)
		tmp = t_0;
	elseif (t_1 <= 0.68)
		tmp = fma(-4.0, Float64(x - y), x);
	elseif (t_1 <= 2e+40)
		tmp = t_0;
	elseif (t_1 <= 5e+188)
		tmp = t_2;
	else
		tmp = Float64(-6.0 * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$2 = N[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+85], t$95$2, If[LessEqual[t$95$1, 0.666666665], t$95$0, If[LessEqual[t$95$1, 0.68], N[(-4.0 * N[(x - y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+40], t$95$0, If[LessEqual[t$95$1, 5e+188], t$95$2, N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e85 or 2.00000000000000006e40 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 5.0000000000000001e188

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-6 \cdot \left(\frac{2}{3} - z\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \color{blue}{\left(\frac{2}{3} - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - \color{blue}{z}\right)\right) \]

      6. metadata-eval 51.6

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto 6 \cdot \color{blue}{\left(x \cdot z\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left(x \cdot \color{blue}{z}\right) \]

      2. lower-*.f64 27.6

         \[\leadsto 6 \cdot \left(x \cdot z\right) \]

   7. Applied rewrites 27.6%

      \[\leadsto 6 \cdot \color{blue}{\left(x \cdot z\right)} \]

   if -1e85 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.666666665000000047 or 0.680000000000000049 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 2.00000000000000006e40

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto -6 \cdot \left(\left(z - \frac{2}{3}\right) \cdot \color{blue}{y}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(-6 \cdot \left(z - \frac{2}{3}\right)\right) \cdot \color{blue}{y} \]

      6. lift--.f64 N/A

         \[\leadsto \left(-6 \cdot \left(z - \frac{2}{3}\right)\right) \cdot y \]

      7. sub-negate-rev N/A

         \[\leadsto \left(-6 \cdot \left(\mathsf{neg}\left(\left(\frac{2}{3} - z\right)\right)\right)\right) \cdot y \]

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

         \[\leadsto \left(\mathsf{neg}\left(-6 \cdot \left(\frac{2}{3} - z\right)\right)\right) \cdot y \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(-6\right)\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

      10. metadata-eval N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

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

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      12. metadata-eval N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(4 - 6 \cdot z\right) \cdot y \]

      14. lift-*.f64 N/A

          \[\leadsto \left(4 - 6 \cdot z\right) \cdot y \]

      15. metadata-eval N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      16. metadata-eval N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

      17. lift-*.f64 N/A

          \[\leadsto \left(6 \cdot \frac{2}{3} - 6 \cdot z\right) \cdot y \]

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

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

      19. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(-6\right)\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot y \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(-6\right)\right) \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{y} \]

   8. Applied rewrites 51.7%

      \[\leadsto \mathsf{fma}\left(-6, z, 4\right) \cdot \color{blue}{y} \]

   if 0.666666665000000047 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.680000000000000049

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, 6, -4\right), x - y, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{-4}, x - y, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-4}, x - y, x\right) \]

   if 5.0000000000000001e188 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]

   7. Taylor expanded in z around inf

      \[\leadsto -6 \cdot \left(y \cdot \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 27.1

         \[\leadsto -6 \cdot \left(y \cdot z\right) \]

   9. Applied rewrites 27.1%

      \[\leadsto -6 \cdot \left(y \cdot \color{blue}{z}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ t_1 := \frac{2}{3} - z\\ t_2 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+85}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-4, x - y, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -6.0 (* y z))) (t_1 (- (/ 2.0 3.0) z)) (t_2 (* 6.0 (* x z))))
   (if (<= t_1 -1e+85)
     t_2
     (if (<= t_1 -2.0)
       t_0
       (if (<= t_1 1.0) (fma -4.0 (- x y) x) (if (<= t_1 5e+188) t_2 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = -6.0 * (y * z);
	double t_1 = (2.0 / 3.0) - z;
	double t_2 = 6.0 * (x * z);
	double tmp;
	if (t_1 <= -1e+85) {
		tmp = t_2;
	} else if (t_1 <= -2.0) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = fma(-4.0, (x - y), x);
	} else if (t_1 <= 5e+188) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-6.0 * Float64(y * z))
	t_1 = Float64(Float64(2.0 / 3.0) - z)
	t_2 = Float64(6.0 * Float64(x * z))
	tmp = 0.0
	if (t_1 <= -1e+85)
		tmp = t_2;
	elseif (t_1 <= -2.0)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = fma(-4.0, Float64(x - y), x);
	elseif (t_1 <= 5e+188)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$2 = N[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+85], t$95$2, If[LessEqual[t$95$1, -2.0], t$95$0, If[LessEqual[t$95$1, 1.0], N[(-4.0 * N[(x - y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+188], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -1e85 or 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 5.0000000000000001e188

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-6 \cdot \left(\frac{2}{3} - z\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \color{blue}{\left(\frac{2}{3} - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - \color{blue}{z}\right)\right) \]

      6. metadata-eval 51.6

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto 6 \cdot \color{blue}{\left(x \cdot z\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left(x \cdot \color{blue}{z}\right) \]

      2. lower-*.f64 27.6

         \[\leadsto 6 \cdot \left(x \cdot z\right) \]

   7. Applied rewrites 27.6%

      \[\leadsto 6 \cdot \color{blue}{\left(x \cdot z\right)} \]

   if -1e85 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -2 or 5.0000000000000001e188 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z)

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]

   7. Taylor expanded in z around inf

      \[\leadsto -6 \cdot \left(y \cdot \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 27.1

         \[\leadsto -6 \cdot \left(y \cdot z\right) \]

   9. Applied rewrites 27.1%

      \[\leadsto -6 \cdot \left(y \cdot \color{blue}{z}\right) \]

   if -2 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, 6, -4\right), x - y, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{-4}, x - y, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 51.1%

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

Alternative 9: 50.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ t_1 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -5.6 \cdot 10^{+189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.06 \cdot 10^{-169}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-225}:\\ \;\;\;\;4 \cdot y\\ \mathbf{elif}\;z \leq 0.52:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -6.0 (* y z))) (t_1 (* 6.0 (* x z))))
   (if (<= z -5.6e+189)
     t_0
     (if (<= z -2.35e-33)
       t_1
       (if (<= z -1.06e-169)
         (* x -3.0)
         (if (<= z -1.2e-225)
           (* 4.0 y)
           (if (<= z 0.52) (* x -3.0) (if (<= z 3.4e+79) t_0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = -6.0 * (y * z);
	double t_1 = 6.0 * (x * z);
	double tmp;
	if (z <= -5.6e+189) {
		tmp = t_0;
	} else if (z <= -2.35e-33) {
		tmp = t_1;
	} else if (z <= -1.06e-169) {
		tmp = x * -3.0;
	} else if (z <= -1.2e-225) {
		tmp = 4.0 * y;
	} else if (z <= 0.52) {
		tmp = x * -3.0;
	} else if (z <= 3.4e+79) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-6.0d0) * (y * z)
    t_1 = 6.0d0 * (x * z)
    if (z <= (-5.6d+189)) then
        tmp = t_0
    else if (z <= (-2.35d-33)) then
        tmp = t_1
    else if (z <= (-1.06d-169)) then
        tmp = x * (-3.0d0)
    else if (z <= (-1.2d-225)) then
        tmp = 4.0d0 * y
    else if (z <= 0.52d0) then
        tmp = x * (-3.0d0)
    else if (z <= 3.4d+79) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -6.0 * (y * z);
	double t_1 = 6.0 * (x * z);
	double tmp;
	if (z <= -5.6e+189) {
		tmp = t_0;
	} else if (z <= -2.35e-33) {
		tmp = t_1;
	} else if (z <= -1.06e-169) {
		tmp = x * -3.0;
	} else if (z <= -1.2e-225) {
		tmp = 4.0 * y;
	} else if (z <= 0.52) {
		tmp = x * -3.0;
	} else if (z <= 3.4e+79) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -6.0 * (y * z)
	t_1 = 6.0 * (x * z)
	tmp = 0
	if z <= -5.6e+189:
		tmp = t_0
	elif z <= -2.35e-33:
		tmp = t_1
	elif z <= -1.06e-169:
		tmp = x * -3.0
	elif z <= -1.2e-225:
		tmp = 4.0 * y
	elif z <= 0.52:
		tmp = x * -3.0
	elif z <= 3.4e+79:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-6.0 * Float64(y * z))
	t_1 = Float64(6.0 * Float64(x * z))
	tmp = 0.0
	if (z <= -5.6e+189)
		tmp = t_0;
	elseif (z <= -2.35e-33)
		tmp = t_1;
	elseif (z <= -1.06e-169)
		tmp = Float64(x * -3.0);
	elseif (z <= -1.2e-225)
		tmp = Float64(4.0 * y);
	elseif (z <= 0.52)
		tmp = Float64(x * -3.0);
	elseif (z <= 3.4e+79)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -6.0 * (y * z);
	t_1 = 6.0 * (x * z);
	tmp = 0.0;
	if (z <= -5.6e+189)
		tmp = t_0;
	elseif (z <= -2.35e-33)
		tmp = t_1;
	elseif (z <= -1.06e-169)
		tmp = x * -3.0;
	elseif (z <= -1.2e-225)
		tmp = 4.0 * y;
	elseif (z <= 0.52)
		tmp = x * -3.0;
	elseif (z <= 3.4e+79)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.6e+189], t$95$0, If[LessEqual[z, -2.35e-33], t$95$1, If[LessEqual[z, -1.06e-169], N[(x * -3.0), $MachinePrecision], If[LessEqual[z, -1.2e-225], N[(4.0 * y), $MachinePrecision], If[LessEqual[z, 0.52], N[(x * -3.0), $MachinePrecision], If[LessEqual[z, 3.4e+79], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.60000000000000013e189 or 0.52000000000000002 < z < 3.40000000000000032e79

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]

   7. Taylor expanded in z around inf

      \[\leadsto -6 \cdot \left(y \cdot \color{blue}{z}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 27.1

         \[\leadsto -6 \cdot \left(y \cdot z\right) \]

   9. Applied rewrites 27.1%

      \[\leadsto -6 \cdot \left(y \cdot \color{blue}{z}\right) \]

   if -5.60000000000000013e189 < z < -2.3500000000000001e-33 or 3.40000000000000032e79 < z

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-6 \cdot \left(\frac{2}{3} - z\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \color{blue}{\left(\frac{2}{3} - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - \color{blue}{z}\right)\right) \]

      6. metadata-eval 51.6

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto 6 \cdot \color{blue}{\left(x \cdot z\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left(x \cdot \color{blue}{z}\right) \]

      2. lower-*.f64 27.6

         \[\leadsto 6 \cdot \left(x \cdot z\right) \]

   7. Applied rewrites 27.6%

      \[\leadsto 6 \cdot \color{blue}{\left(x \cdot z\right)} \]

   if -2.3500000000000001e-33 < z < -1.06e-169 or -1.19999999999999998e-225 < z < 0.52000000000000002

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-6 \cdot \left(\frac{2}{3} - z\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \color{blue}{\left(\frac{2}{3} - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - \color{blue}{z}\right)\right) \]

      6. metadata-eval 51.6

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot -3 \]
   6. Step-by-step derivation

   7. Applied rewrites 26.3%

      \[\leadsto x \cdot -3 \]

   if -1.06e-169 < z < -1.19999999999999998e-225

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]

   7. Taylor expanded in z around 0

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

      1. lower-*.f64 26.7

         \[\leadsto 4 \cdot y \]

   9. Applied rewrites 26.7%

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

Alternative 10: 50.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -2.35 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.06 \cdot 10^{-169}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-225}:\\ \;\;\;\;4 \cdot y\\ \mathbf{elif}\;z \leq 3.1:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* x z))))
   (if (<= z -2.35e-33)
     t_0
     (if (<= z -1.06e-169)
       (* x -3.0)
       (if (<= z -1.2e-225) (* 4.0 y) (if (<= z 3.1) (* x -3.0) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 6.0 * (x * z);
	double tmp;
	if (z <= -2.35e-33) {
		tmp = t_0;
	} else if (z <= -1.06e-169) {
		tmp = x * -3.0;
	} else if (z <= -1.2e-225) {
		tmp = 4.0 * y;
	} else if (z <= 3.1) {
		tmp = x * -3.0;
	} 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, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 6.0d0 * (x * z)
    if (z <= (-2.35d-33)) then
        tmp = t_0
    else if (z <= (-1.06d-169)) then
        tmp = x * (-3.0d0)
    else if (z <= (-1.2d-225)) then
        tmp = 4.0d0 * y
    else if (z <= 3.1d0) then
        tmp = x * (-3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 6.0 * (x * z);
	double tmp;
	if (z <= -2.35e-33) {
		tmp = t_0;
	} else if (z <= -1.06e-169) {
		tmp = x * -3.0;
	} else if (z <= -1.2e-225) {
		tmp = 4.0 * y;
	} else if (z <= 3.1) {
		tmp = x * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 6.0 * (x * z)
	tmp = 0
	if z <= -2.35e-33:
		tmp = t_0
	elif z <= -1.06e-169:
		tmp = x * -3.0
	elif z <= -1.2e-225:
		tmp = 4.0 * y
	elif z <= 3.1:
		tmp = x * -3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(6.0 * Float64(x * z))
	tmp = 0.0
	if (z <= -2.35e-33)
		tmp = t_0;
	elseif (z <= -1.06e-169)
		tmp = Float64(x * -3.0);
	elseif (z <= -1.2e-225)
		tmp = Float64(4.0 * y);
	elseif (z <= 3.1)
		tmp = Float64(x * -3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 6.0 * (x * z);
	tmp = 0.0;
	if (z <= -2.35e-33)
		tmp = t_0;
	elseif (z <= -1.06e-169)
		tmp = x * -3.0;
	elseif (z <= -1.2e-225)
		tmp = 4.0 * y;
	elseif (z <= 3.1)
		tmp = x * -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.35e-33], t$95$0, If[LessEqual[z, -1.06e-169], N[(x * -3.0), $MachinePrecision], If[LessEqual[z, -1.2e-225], N[(4.0 * y), $MachinePrecision], If[LessEqual[z, 3.1], N[(x * -3.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.3500000000000001e-33 or 3.10000000000000009 < z

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-6 \cdot \left(\frac{2}{3} - z\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \color{blue}{\left(\frac{2}{3} - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - \color{blue}{z}\right)\right) \]

      6. metadata-eval 51.6

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto 6 \cdot \color{blue}{\left(x \cdot z\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left(x \cdot \color{blue}{z}\right) \]

      2. lower-*.f64 27.6

         \[\leadsto 6 \cdot \left(x \cdot z\right) \]

   7. Applied rewrites 27.6%

      \[\leadsto 6 \cdot \color{blue}{\left(x \cdot z\right)} \]

   if -2.3500000000000001e-33 < z < -1.06e-169 or -1.19999999999999998e-225 < z < 3.10000000000000009

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-6 \cdot \left(\frac{2}{3} - z\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \color{blue}{\left(\frac{2}{3} - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - \color{blue}{z}\right)\right) \]

      6. metadata-eval 51.6

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot -3 \]
   6. Step-by-step derivation

   7. Applied rewrites 26.3%

      \[\leadsto x \cdot -3 \]

   if -1.06e-169 < z < -1.19999999999999998e-225

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]

   7. Taylor expanded in z around 0

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

      1. lower-*.f64 26.7

         \[\leadsto 4 \cdot y \]

   9. Applied rewrites 26.7%

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

Alternative 11: 36.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+36}:\\ \;\;\;\;x + 4 \cdot y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+161}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.7e+36) (+ x (* 4.0 y)) (if (<= y 6e+161) (* x -3.0) (* 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.7e+36) {
		tmp = x + (4.0 * y);
	} else if (y <= 6e+161) {
		tmp = x * -3.0;
	} else {
		tmp = 4.0 * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.7d+36)) then
        tmp = x + (4.0d0 * y)
    else if (y <= 6d+161) then
        tmp = x * (-3.0d0)
    else
        tmp = 4.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.7e+36) {
		tmp = x + (4.0 * y);
	} else if (y <= 6e+161) {
		tmp = x * -3.0;
	} else {
		tmp = 4.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.7e+36:
		tmp = x + (4.0 * y)
	elif y <= 6e+161:
		tmp = x * -3.0
	else:
		tmp = 4.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.7e+36)
		tmp = Float64(x + Float64(4.0 * y));
	elseif (y <= 6e+161)
		tmp = Float64(x * -3.0);
	else
		tmp = Float64(4.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.7e+36)
		tmp = x + (4.0 * y);
	elseif (y <= 6e+161)
		tmp = x * -3.0;
	else
		tmp = 4.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.7e+36], N[(x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+161], N[(x * -3.0), $MachinePrecision], N[(4.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.70000000000000029e36

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

         \[\leadsto x + 4 \cdot \color{blue}{\left(y - x\right)} \]

      2. lower--.f64 51.1

         \[\leadsto x + 4 \cdot \left(y - \color{blue}{x}\right) \]

   4. Applied rewrites 51.1%

      \[\leadsto x + \color{blue}{4 \cdot \left(y - x\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto x + 4 \cdot y \]
   6. Step-by-step derivation

   7. Applied rewrites 26.4%

      \[\leadsto x + 4 \cdot y \]

   if -3.70000000000000029e36 < y < 6.00000000000000023e161

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-6 \cdot \left(\frac{2}{3} - z\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \color{blue}{\left(\frac{2}{3} - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - \color{blue}{z}\right)\right) \]

      6. metadata-eval 51.6

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot -3 \]
   6. Step-by-step derivation

   7. Applied rewrites 26.3%

      \[\leadsto x \cdot -3 \]

   if 6.00000000000000023e161 < y

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]

   7. Taylor expanded in z around 0

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

      1. lower-*.f64 26.7

         \[\leadsto 4 \cdot y \]

   9. Applied rewrites 26.7%

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

Alternative 12: 36.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+36}:\\ \;\;\;\;4 \cdot y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+161}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.7e+36) (* 4.0 y) (if (<= y 6e+161) (* x -3.0) (* 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.7e+36) {
		tmp = 4.0 * y;
	} else if (y <= 6e+161) {
		tmp = x * -3.0;
	} else {
		tmp = 4.0 * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.7d+36)) then
        tmp = 4.0d0 * y
    else if (y <= 6d+161) then
        tmp = x * (-3.0d0)
    else
        tmp = 4.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.7e+36) {
		tmp = 4.0 * y;
	} else if (y <= 6e+161) {
		tmp = x * -3.0;
	} else {
		tmp = 4.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.7e+36:
		tmp = 4.0 * y
	elif y <= 6e+161:
		tmp = x * -3.0
	else:
		tmp = 4.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.7e+36)
		tmp = Float64(4.0 * y);
	elseif (y <= 6e+161)
		tmp = Float64(x * -3.0);
	else
		tmp = Float64(4.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.7e+36)
		tmp = 4.0 * y;
	elseif (y <= 6e+161)
		tmp = x * -3.0;
	else
		tmp = 4.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.7e+36], N[(4.0 * y), $MachinePrecision], If[LessEqual[y, 6e+161], N[(x * -3.0), $MachinePrecision], N[(4.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.70000000000000029e36 or 6.00000000000000023e161 < y

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} - z\right) \cdot \left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(\left(y - x\right) \cdot 6\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} - z\right) \cdot \color{blue}{\left(6 \cdot \left(y - x\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} - z\right) \cdot 6\right) \cdot \left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right)} \cdot \left(y - x\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      12. sub-flip N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y\right) + \color{blue}{x} \]

      16. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + \left(\left(6 \cdot \left(\frac{2}{3} - z\right)\right) \cdot y + x\right) \]

      18. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{y \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right)} + x\right) \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(6 \cdot \left(\frac{2}{3} - z\right)\right) + \left(\color{blue}{\left(y \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)} + x\right) \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x - y\right) \cdot \left(z - 0.6666666666666666\right), 6, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \frac{2}{\color{blue}{3}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -6 \cdot \color{blue}{\left(y \cdot \left(z - \frac{2}{3}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \color{blue}{\left(z - \frac{2}{3}\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto -6 \cdot \left(y \cdot \left(z - \color{blue}{\frac{2}{3}}\right)\right) \]

      5. metadata-eval 51.5

         \[\leadsto -6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right) \]

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{-6 \cdot \left(y \cdot \left(z - 0.6666666666666666\right)\right)} \]

   7. Taylor expanded in z around 0

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

      1. lower-*.f64 26.7

         \[\leadsto 4 \cdot y \]

   9. Applied rewrites 26.7%

      \[\leadsto 4 \cdot \color{blue}{y} \]

   if -3.70000000000000029e36 < y < 6.00000000000000023e161

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{-6 \cdot \left(\frac{2}{3} - z\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \color{blue}{\left(\frac{2}{3} - z\right)}\right) \]

      5. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - \color{blue}{z}\right)\right) \]

      6. metadata-eval 51.6

         \[\leadsto x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto x \cdot -3 \]
   6. Step-by-step derivation

   7. Applied rewrites 26.3%

      \[\leadsto x \cdot -3 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 26.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ x \cdot -3 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x -3.0))

C

double code(double x, double y, double z) {
	return x * -3.0;
}

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, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (-3.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return x * -3.0;
}

Python

def code(x, y, z):
	return x * -3.0

Julia

function code(x, y, z)
	return Float64(x * -3.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * -3.0;
end

Wolfram

code[x_, y_, z_] := N[(x * -3.0), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(1 + -6 \cdot \left(\frac{2}{3} - z\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto x \cdot \left(1 + \color{blue}{-6 \cdot \left(\frac{2}{3} - z\right)}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \left(1 + -6 \cdot \color{blue}{\left(\frac{2}{3} - z\right)}\right) \]

   5. lower--.f64 N/A

      \[\leadsto x \cdot \left(1 + -6 \cdot \left(\frac{2}{3} - \color{blue}{z}\right)\right) \]

   6. metadata-eval 51.6

      \[\leadsto x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto x \cdot -3 \]
6. Step-by-step derivation

7. Applied rewrites 26.3%

   \[\leadsto x \cdot -3 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  :precision binary64
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))