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

Percentage Accurate: 99.7% → 99.7%

Time: 3.4s

Alternatives: 14

Speedup: 1.1×

• 21.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift--.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-lft-identity N/A

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

   6. associate-+l+ N/A

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

   7. +-commutative N/A

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

   8. +-commutative N/A

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

   9. associate-*r* N/A

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

   10. +-commutative N/A

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

   11. associate-*r* N/A

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

   12. associate-*r* N/A

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

   13. distribute-rgt-out N/A

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

   14. lower-fma.f64 N/A

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

7. Applied rewrites 99.9%

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

Alternative 2: 61.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+191}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot x\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-26}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+121}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.7e+191)
   (* (* -6.0 z) x)
   (if (<= z -2.4e-26)
     (* (* 6.0 y) z)
     (if (<= z 8e-8)
       x
       (if (<= z 1.3e+121) (* (* z y) 6.0) (* (* -6.0 x) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.7e+191) {
		tmp = (-6.0 * z) * x;
	} else if (z <= -2.4e-26) {
		tmp = (6.0 * y) * z;
	} else if (z <= 8e-8) {
		tmp = x;
	} else if (z <= 1.3e+121) {
		tmp = (z * y) * 6.0;
	} else {
		tmp = (-6.0 * x) * z;
	}
	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 (z <= (-1.7d+191)) then
        tmp = ((-6.0d0) * z) * x
    else if (z <= (-2.4d-26)) then
        tmp = (6.0d0 * y) * z
    else if (z <= 8d-8) then
        tmp = x
    else if (z <= 1.3d+121) then
        tmp = (z * y) * 6.0d0
    else
        tmp = ((-6.0d0) * x) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.7e+191) {
		tmp = (-6.0 * z) * x;
	} else if (z <= -2.4e-26) {
		tmp = (6.0 * y) * z;
	} else if (z <= 8e-8) {
		tmp = x;
	} else if (z <= 1.3e+121) {
		tmp = (z * y) * 6.0;
	} else {
		tmp = (-6.0 * x) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.7e+191:
		tmp = (-6.0 * z) * x
	elif z <= -2.4e-26:
		tmp = (6.0 * y) * z
	elif z <= 8e-8:
		tmp = x
	elif z <= 1.3e+121:
		tmp = (z * y) * 6.0
	else:
		tmp = (-6.0 * x) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.7e+191)
		tmp = Float64(Float64(-6.0 * z) * x);
	elseif (z <= -2.4e-26)
		tmp = Float64(Float64(6.0 * y) * z);
	elseif (z <= 8e-8)
		tmp = x;
	elseif (z <= 1.3e+121)
		tmp = Float64(Float64(z * y) * 6.0);
	else
		tmp = Float64(Float64(-6.0 * x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.7e+191)
		tmp = (-6.0 * z) * x;
	elseif (z <= -2.4e-26)
		tmp = (6.0 * y) * z;
	elseif (z <= 8e-8)
		tmp = x;
	elseif (z <= 1.3e+121)
		tmp = (z * y) * 6.0;
	else
		tmp = (-6.0 * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.7e+191], N[(N[(-6.0 * z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, -2.4e-26], N[(N[(6.0 * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 8e-8], x, If[LessEqual[z, 1.3e+121], N[(N[(z * y), $MachinePrecision] * 6.0), $MachinePrecision], N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.70000000000000004e191

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 72.8

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 72.8

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

   8. Applied rewrites 72.8%

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

   if -1.70000000000000004e191 < z < -2.4000000000000001e-26

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 60.5

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

   9. Applied rewrites 60.5%

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

   if -2.4000000000000001e-26 < z < 8.0000000000000002e-8

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 75.8%

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

   if 8.0000000000000002e-8 < z < 1.2999999999999999e121

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   if 1.2999999999999999e121 < z

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 67.6

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 67.7

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

   10. Applied rewrites 67.7%

       \[\leadsto \left(-6 \cdot x\right) \cdot z \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+191}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot x\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-26}:\\ \;\;\;\;\left(6 \cdot y\right) \cdot z\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+121}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot y\right) \cdot 6\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+191}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot x\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z y) 6.0)))
   (if (<= z -1.7e+191)
     (* (* -6.0 z) x)
     (if (<= z -2.4e-26)
       t_0
       (if (<= z 8e-8) x (if (<= z 1.3e+121) t_0 (* (* -6.0 x) z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (z * y) * 6.0;
	double tmp;
	if (z <= -1.7e+191) {
		tmp = (-6.0 * z) * x;
	} else if (z <= -2.4e-26) {
		tmp = t_0;
	} else if (z <= 8e-8) {
		tmp = x;
	} else if (z <= 1.3e+121) {
		tmp = t_0;
	} else {
		tmp = (-6.0 * x) * z;
	}
	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 = (z * y) * 6.0d0
    if (z <= (-1.7d+191)) then
        tmp = ((-6.0d0) * z) * x
    else if (z <= (-2.4d-26)) then
        tmp = t_0
    else if (z <= 8d-8) then
        tmp = x
    else if (z <= 1.3d+121) then
        tmp = t_0
    else
        tmp = ((-6.0d0) * x) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * y) * 6.0;
	double tmp;
	if (z <= -1.7e+191) {
		tmp = (-6.0 * z) * x;
	} else if (z <= -2.4e-26) {
		tmp = t_0;
	} else if (z <= 8e-8) {
		tmp = x;
	} else if (z <= 1.3e+121) {
		tmp = t_0;
	} else {
		tmp = (-6.0 * x) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * y) * 6.0
	tmp = 0
	if z <= -1.7e+191:
		tmp = (-6.0 * z) * x
	elif z <= -2.4e-26:
		tmp = t_0
	elif z <= 8e-8:
		tmp = x
	elif z <= 1.3e+121:
		tmp = t_0
	else:
		tmp = (-6.0 * x) * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * y) * 6.0)
	tmp = 0.0
	if (z <= -1.7e+191)
		tmp = Float64(Float64(-6.0 * z) * x);
	elseif (z <= -2.4e-26)
		tmp = t_0;
	elseif (z <= 8e-8)
		tmp = x;
	elseif (z <= 1.3e+121)
		tmp = t_0;
	else
		tmp = Float64(Float64(-6.0 * x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * y) * 6.0;
	tmp = 0.0;
	if (z <= -1.7e+191)
		tmp = (-6.0 * z) * x;
	elseif (z <= -2.4e-26)
		tmp = t_0;
	elseif (z <= 8e-8)
		tmp = x;
	elseif (z <= 1.3e+121)
		tmp = t_0;
	else
		tmp = (-6.0 * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * y), $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[z, -1.7e+191], N[(N[(-6.0 * z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, -2.4e-26], t$95$0, If[LessEqual[z, 8e-8], x, If[LessEqual[z, 1.3e+121], t$95$0, N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.70000000000000004e191

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 72.8

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 72.8

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

   8. Applied rewrites 72.8%

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

   if -1.70000000000000004e191 < z < -2.4000000000000001e-26 or 8.0000000000000002e-8 < z < 1.2999999999999999e121

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 58.8

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

   5. Applied rewrites 58.8%

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

   if -2.4000000000000001e-26 < z < 8.0000000000000002e-8

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 75.8%

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

   if 1.2999999999999999e121 < z

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 67.6

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 67.7

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

   10. Applied rewrites 67.7%

       \[\leadsto \left(-6 \cdot x\right) \cdot z \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+191}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot x\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-26}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+121}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-11} \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot 6\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.1e-11) (not (<= z 0.165)))
   (* (* (- y x) 6.0) z)
   (fma (* y z) 6.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e-11) || !(z <= 0.165)) {
		tmp = ((y - x) * 6.0) * z;
	} else {
		tmp = fma((y * z), 6.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.1e-11) || !(z <= 0.165))
		tmp = Float64(Float64(Float64(y - x) * 6.0) * z);
	else
		tmp = fma(Float64(y * z), 6.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.1e-11], N[Not[LessEqual[z, 0.165]], $MachinePrecision]], N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * z), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * 6.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1000000000000001e-11 or 0.165000000000000008 < z

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -1.1000000000000001e-11 < z < 0.165000000000000008

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y} \cdot z, 6, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-11} \lor \neg \left(z \leq 0.165\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot 6\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-11}:\\ \;\;\;\;\left(\left(y - x\right) \cdot 6\right) \cdot z\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(y - x\right)\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.1e-11)
   (* (* (- y x) 6.0) z)
   (if (<= z 3.6e-6) (fma (* y z) 6.0 x) (* (* z (- y x)) 6.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.1e-11) {
		tmp = ((y - x) * 6.0) * z;
	} else if (z <= 3.6e-6) {
		tmp = fma((y * z), 6.0, x);
	} else {
		tmp = (z * (y - x)) * 6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.1e-11)
		tmp = Float64(Float64(Float64(y - x) * 6.0) * z);
	elseif (z <= 3.6e-6)
		tmp = fma(Float64(y * z), 6.0, x);
	else
		tmp = Float64(Float64(z * Float64(y - x)) * 6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.1e-11], N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 3.6e-6], N[(N[(y * z), $MachinePrecision] * 6.0 + x), $MachinePrecision], N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1000000000000001e-11

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -1.1000000000000001e-11 < z < 3.59999999999999984e-6

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

   if 3.59999999999999984e-6 < z

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lft-identity N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 97.2

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

   10. Applied rewrites 97.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-11}:\\ \;\;\;\;\left(\left(y - x\right) \cdot 6\right) \cdot z\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(y - x\right)\right) \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -140000000 \lor \neg \left(y \leq 2.15 \cdot 10^{-41}\right):\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -140000000.0) (not (<= y 2.15e-41)))
   (fma y (* z 6.0) x)
   (fma (* -6.0 x) z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -140000000.0) || !(y <= 2.15e-41)) {
		tmp = fma(y, (z * 6.0), x);
	} else {
		tmp = fma((-6.0 * x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -140000000.0) || !(y <= 2.15e-41))
		tmp = fma(y, Float64(z * 6.0), x);
	else
		tmp = fma(Float64(-6.0 * x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -140000000.0], N[Not[LessEqual[y, 2.15e-41]], $MachinePrecision]], N[(y * N[(z * 6.0), $MachinePrecision] + x), $MachinePrecision], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e8 or 2.1499999999999999e-41 < y

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 91.9%

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

   if -1.4e8 < y < 2.1499999999999999e-41

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 89.8

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot \color{blue}{x} \]

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 89.9

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

   7. Applied rewrites 89.9%

      \[\leadsto \mathsf{fma}\left(-6 \cdot x, \color{blue}{z}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -140000000 \lor \neg \left(y \leq 2.15 \cdot 10^{-41}\right):\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot 6, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-117} \lor \neg \left(x \leq 9.5 \cdot 10^{-55}\right):\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4e-117) (not (<= x 9.5e-55)))
   (* (fma -6.0 z 1.0) x)
   (* (* z y) 6.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4e-117) || !(x <= 9.5e-55)) {
		tmp = fma(-6.0, z, 1.0) * x;
	} else {
		tmp = (z * y) * 6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4e-117) || !(x <= 9.5e-55))
		tmp = Float64(fma(-6.0, z, 1.0) * x);
	else
		tmp = Float64(Float64(z * y) * 6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4e-117], N[Not[LessEqual[x, 9.5e-55]], $MachinePrecision]], N[(N[(-6.0 * z + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * 6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000012e-117 or 9.5000000000000006e-55 < x

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 83.1

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]

   if -4.00000000000000012e-117 < x < 9.5000000000000006e-55

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\left(z \cdot y\right) \cdot 6} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-117} \lor \neg \left(x \leq 9.5 \cdot 10^{-55}\right):\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -140000000:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot 6, x\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -140000000.0)
   (fma y (* z 6.0) x)
   (if (<= y 2.15e-41) (fma (* -6.0 x) z x) (fma (* y z) 6.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -140000000.0) {
		tmp = fma(y, (z * 6.0), x);
	} else if (y <= 2.15e-41) {
		tmp = fma((-6.0 * x), z, x);
	} else {
		tmp = fma((y * z), 6.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -140000000.0)
		tmp = fma(y, Float64(z * 6.0), x);
	elseif (y <= 2.15e-41)
		tmp = fma(Float64(-6.0 * x), z, x);
	else
		tmp = fma(Float64(y * z), 6.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -140000000.0], N[(y * N[(z * 6.0), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.15e-41], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * 6.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4e8

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.1%

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

   if -1.4e8 < y < 2.1499999999999999e-41

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 89.8

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot \color{blue}{x} \]

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 89.9

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

   7. Applied rewrites 89.9%

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

   if 2.1499999999999999e-41 < y

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 87.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -140000000:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot 6, x\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, 6, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-55}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4e-117)
   (* (fma -6.0 z 1.0) x)
   (if (<= x 9.5e-55) (* (* z y) 6.0) (fma (* -6.0 x) z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4e-117) {
		tmp = fma(-6.0, z, 1.0) * x;
	} else if (x <= 9.5e-55) {
		tmp = (z * y) * 6.0;
	} else {
		tmp = fma((-6.0 * x), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4e-117)
		tmp = Float64(fma(-6.0, z, 1.0) * x);
	elseif (x <= 9.5e-55)
		tmp = Float64(Float64(z * y) * 6.0);
	else
		tmp = fma(Float64(-6.0 * x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4e-117], N[(N[(-6.0 * z + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 9.5e-55], N[(N[(z * y), $MachinePrecision] * 6.0), $MachinePrecision], N[(N[(-6.0 * x), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.00000000000000012e-117

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 84.4

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]

   if -4.00000000000000012e-117 < x < 9.5000000000000006e-55

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if 9.5000000000000006e-55 < x

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 81.7

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot \color{blue}{x} \]

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 81.8

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

   7. Applied rewrites 81.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 1\right) \cdot x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-55}:\\ \;\;\;\;\left(z \cdot y\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-11} \lor \neg \left(z \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.1e-11) (not (<= z 3.6e-6))) (* (* -6.0 x) z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e-11) || !(z <= 3.6e-6)) {
		tmp = (-6.0 * x) * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.1d-11)) .or. (.not. (z <= 3.6d-6))) then
        tmp = ((-6.0d0) * x) * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e-11) || !(z <= 3.6e-6)) {
		tmp = (-6.0 * x) * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.1e-11) or not (z <= 3.6e-6):
		tmp = (-6.0 * x) * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.1e-11) || !(z <= 3.6e-6))
		tmp = Float64(Float64(-6.0 * x) * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.1e-11) || ~((z <= 3.6e-6)))
		tmp = (-6.0 * x) * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.1e-11], N[Not[LessEqual[z, 3.6e-6]], $MachinePrecision]], N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.1000000000000001e-11 or 3.59999999999999984e-6 < z

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 54.9

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 52.4

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

   8. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 52.5

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

   10. Applied rewrites 52.5%

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

   if -1.1000000000000001e-11 < z < 3.59999999999999984e-6

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 75.2%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-11} \lor \neg \left(z \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-11}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot x\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.1e-11) (* (* -6.0 z) x) (if (<= z 3.6e-6) x (* (* -6.0 x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.1e-11) {
		tmp = (-6.0 * z) * x;
	} else if (z <= 3.6e-6) {
		tmp = x;
	} else {
		tmp = (-6.0 * x) * z;
	}
	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 (z <= (-1.1d-11)) then
        tmp = ((-6.0d0) * z) * x
    else if (z <= 3.6d-6) then
        tmp = x
    else
        tmp = ((-6.0d0) * x) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.1e-11) {
		tmp = (-6.0 * z) * x;
	} else if (z <= 3.6e-6) {
		tmp = x;
	} else {
		tmp = (-6.0 * x) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.1e-11:
		tmp = (-6.0 * z) * x
	elif z <= 3.6e-6:
		tmp = x
	else:
		tmp = (-6.0 * x) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.1e-11)
		tmp = Float64(Float64(-6.0 * z) * x);
	elseif (z <= 3.6e-6)
		tmp = x;
	else
		tmp = Float64(Float64(-6.0 * x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.1e-11)
		tmp = (-6.0 * z) * x;
	elseif (z <= 3.6e-6)
		tmp = x;
	else
		tmp = (-6.0 * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.1e-11], N[(N[(-6.0 * z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 3.6e-6], x, N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1000000000000001e-11

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 53.6

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 52.4

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

   8. Applied rewrites 52.4%

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

   if -1.1000000000000001e-11 < z < 3.59999999999999984e-6

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 75.2%

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

   if 3.59999999999999984e-6 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 56.4

         \[\leadsto \mathsf{fma}\left(-6, z, 1\right) \cdot x \]

   5. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, z, 1\right) \cdot x} \]

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 53.9

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

   8. Applied rewrites 53.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 54.0

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

   10. Applied rewrites 54.0%

       \[\leadsto \left(-6 \cdot x\right) \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-11}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot x\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 13: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift--.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 14: 37.0% accurate, 17.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 x)

C

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

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
end function

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

5. Applied rewrites 35.4%

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

6. Final simplification 35.4%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (* 6 z) (- x y))))

  (+ x (* (* (- y x) 6.0) z)))