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

Percentage Accurate: 99.5% → 99.8%

Time: 4.9s

Alternatives: 14

Speedup: 1.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (((y - x) * 6.0d0) * ((2.0d0 / 3.0d0) - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \left(-6 \cdot \left(z \cdot \left(y - x\right)\right) + 4 \cdot \left(y - x\right)\right) + \color{blue}{x} \]

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-fma.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lower-fma.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 2: 97.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.58 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(-6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.58) (not (<= z 0.5)))
   (* (* -6.0 (- y x)) z)
   (fma -3.0 x (* 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.58) || !(z <= 0.5)) {
		tmp = (-6.0 * (y - x)) * z;
	} else {
		tmp = fma(-3.0, x, (4.0 * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.58) || !(z <= 0.5))
		tmp = Float64(Float64(-6.0 * Float64(y - x)) * z);
	else
		tmp = fma(-3.0, x, Float64(4.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.58], N[Not[LessEqual[z, 0.5]], $MachinePrecision]], N[(N[(-6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.57999999999999996 or 0.5 < z

   1. Initial program 99.8%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 98.5

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

   9. Applied rewrites 98.5%

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

   if -0.57999999999999996 < z < 0.5

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto 4 \cdot \left(y - x\right) + \color{blue}{x} \]

      2. lower-fma.f64 N/A

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

      3. lift--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.58 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(-6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;\left(-6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.58)
   (* (* -6.0 (- y x)) z)
   (if (<= z 0.5) (fma -3.0 x (* 4.0 y)) (* (* (- y x) z) -6.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.58) {
		tmp = (-6.0 * (y - x)) * z;
	} else if (z <= 0.5) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = ((y - x) * z) * -6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.58)
		tmp = Float64(Float64(-6.0 * Float64(y - x)) * z);
	elseif (z <= 0.5)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = Float64(Float64(Float64(y - x) * z) * -6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.58], N[(N[(-6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 0.5], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.57999999999999996

   1. Initial program 99.8%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-*.f64 98.3

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

   7. Applied rewrites 98.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 98.5

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

   9. Applied rewrites 98.5%

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

   if -0.57999999999999996 < z < 0.5

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto 4 \cdot \left(y - x\right) + \color{blue}{x} \]

      2. lower-fma.f64 N/A

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

      3. lift--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

   if 0.5 < z

   1. Initial program 99.7%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 4: 75.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.6666666666666666 - z\right) \cdot 6\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e+57)
   (* (fma -6.0 z 4.0) y)
   (if (<= y 3.7e+69)
     (* (fma 6.0 z -3.0) x)
     (* (* (- 0.6666666666666666 z) 6.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+57) {
		tmp = fma(-6.0, z, 4.0) * y;
	} else if (y <= 3.7e+69) {
		tmp = fma(6.0, z, -3.0) * x;
	} else {
		tmp = ((0.6666666666666666 - z) * 6.0) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.3e+57)
		tmp = Float64(fma(-6.0, z, 4.0) * y);
	elseif (y <= 3.7e+69)
		tmp = Float64(fma(6.0, z, -3.0) * x);
	else
		tmp = Float64(Float64(Float64(0.6666666666666666 - z) * 6.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.3e+57], N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 3.7e+69], N[(N[(6.0 * z + -3.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(0.6666666666666666 - z), $MachinePrecision] * 6.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2999999999999999e57

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \left(-6 \cdot \left(z \cdot \left(y - x\right)\right) + 4 \cdot \left(y - x\right)\right) + \color{blue}{x} \]

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 86.1

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

   8. Applied rewrites 86.1%

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

   if -2.2999999999999999e57 < y < 3.6999999999999999e69

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval 77.8

         \[\leadsto \mathsf{fma}\left(-6, 0.6666666666666666 - z, 1\right) \cdot x \]

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 42.7%

      \[\leadsto -3 \cdot x \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(6 \cdot z - 3\right) \cdot x \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(6 \cdot z - 3 \cdot 1\right) \cdot x \]

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

          \[\leadsto \left(6 \cdot z + \left(\mathsf{neg}\left(3\right)\right) \cdot 1\right) \cdot x \]

       3. metadata-eval N/A

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

       4. metadata-eval N/A

          \[\leadsto \left(6 \cdot z + -3\right) \cdot x \]

       5. lower-fma.f64 77.7

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

   11. Applied rewrites 77.7%

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

   if 3.6999999999999999e69 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666 - z, 6, \frac{\mathsf{fma}\left(-6, 0.6666666666666666 - z, 1\right) \cdot x}{y}\right) \cdot y} \]

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval 84.7

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

   8. Applied rewrites 84.7%

      \[\leadsto \left(\left(0.6666666666666666 - z\right) \cdot 6\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+57} \lor \neg \left(y \leq 3.7 \cdot 10^{+69}\right):\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.3e+57) (not (<= y 3.7e+69)))
   (* (fma -6.0 z 4.0) y)
   (* (fma 6.0 z -3.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.3e+57) || !(y <= 3.7e+69)) {
		tmp = fma(-6.0, z, 4.0) * y;
	} else {
		tmp = fma(6.0, z, -3.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.3e+57) || !(y <= 3.7e+69))
		tmp = Float64(fma(-6.0, z, 4.0) * y);
	else
		tmp = Float64(fma(6.0, z, -3.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.3e+57], N[Not[LessEqual[y, 3.7e+69]], $MachinePrecision]], N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(6.0 * z + -3.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2999999999999999e57 or 3.6999999999999999e69 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \left(-6 \cdot \left(z \cdot \left(y - x\right)\right) + 4 \cdot \left(y - x\right)\right) + \color{blue}{x} \]

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 85.4

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

   8. Applied rewrites 85.4%

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

   if -2.2999999999999999e57 < y < 3.6999999999999999e69

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval 77.8

         \[\leadsto \mathsf{fma}\left(-6, 0.6666666666666666 - z, 1\right) \cdot x \]

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 42.7%

      \[\leadsto -3 \cdot x \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(6 \cdot z - 3\right) \cdot x \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(6 \cdot z - 3 \cdot 1\right) \cdot x \]

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

          \[\leadsto \left(6 \cdot z + \left(\mathsf{neg}\left(3\right)\right) \cdot 1\right) \cdot x \]

       3. metadata-eval N/A

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

       4. metadata-eval N/A

          \[\leadsto \left(6 \cdot z + -3\right) \cdot x \]

       5. lower-fma.f64 77.7

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

   11. Applied rewrites 77.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+57} \lor \neg \left(y \leq 3.7 \cdot 10^{+69}\right):\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13.0)
   (* (* 6.0 x) z)
   (if (<= z 5.5e-7) (fma -3.0 x (* 4.0 y)) (* (fma -6.0 z 4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13.0) {
		tmp = (6.0 * x) * z;
	} else if (z <= 5.5e-7) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = fma(-6.0, z, 4.0) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13.0)
		tmp = Float64(Float64(6.0 * x) * z);
	elseif (z <= 5.5e-7)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = Float64(fma(-6.0, z, 4.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13.0], N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 5.5e-7], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval 60.9

         \[\leadsto \mathsf{fma}\left(-6, 0.6666666666666666 - z, 1\right) \cdot x \]

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      6. lower-*.f64 60.9

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in z around inf

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

       1. lift-*.f64 59.6

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

   11. Applied rewrites 59.6%

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

   if -13 < z < 5.5000000000000003e-7

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto 4 \cdot \left(y - x\right) + \color{blue}{x} \]

      2. lower-fma.f64 N/A

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

      3. lift--.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if 5.5000000000000003e-7 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \left(-6 \cdot \left(z \cdot \left(y - x\right)\right) + 4 \cdot \left(y - x\right)\right) + \color{blue}{x} \]

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 55.0

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

   8. Applied rewrites 55.0%

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

Alternative 7: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13.0)
   (* (* 6.0 x) z)
   (if (<= z 0.66) (fma -3.0 x (* 4.0 y)) (* (* y z) -6.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13.0) {
		tmp = (6.0 * x) * z;
	} else if (z <= 0.66) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = (y * z) * -6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13.0)
		tmp = Float64(Float64(6.0 * x) * z);
	elseif (z <= 0.66)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = Float64(Float64(y * z) * -6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13.0], N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 0.66], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * -6.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval 60.9

         \[\leadsto \mathsf{fma}\left(-6, 0.6666666666666666 - z, 1\right) \cdot x \]

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      6. lower-*.f64 60.9

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in z around inf

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

       1. lift-*.f64 59.6

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

   11. Applied rewrites 59.6%

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

   if -13 < z < 0.660000000000000031

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto 4 \cdot \left(y - x\right) + \color{blue}{x} \]

      2. lower-fma.f64 N/A

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

      3. lift--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

   if 0.660000000000000031 < z

   1. Initial program 99.7%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.3%

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

Alternative 8: 74.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -13.0) (not (<= z 0.5))) (* (* 6.0 x) z) (fma 4.0 (- y x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -13.0) || !(z <= 0.5)) {
		tmp = (6.0 * x) * z;
	} else {
		tmp = fma(4.0, (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -13.0) || !(z <= 0.5))
		tmp = Float64(Float64(6.0 * x) * z);
	else
		tmp = fma(4.0, Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -13.0], N[Not[LessEqual[z, 0.5]], $MachinePrecision]], N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -13 or 0.5 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval 56.1

         \[\leadsto \mathsf{fma}\left(-6, 0.6666666666666666 - z, 1\right) \cdot x \]

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      6. lower-*.f64 56.1

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

   8. Applied rewrites 56.1%

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

   9. Taylor expanded in z around inf

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

       1. lift-*.f64 54.8

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

   11. Applied rewrites 54.8%

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

   if -13 < z < 0.5

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto 4 \cdot \left(y - x\right) + \color{blue}{x} \]

      2. lower-fma.f64 N/A

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

      3. lift--.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -13 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13.0)
   (* (* 6.0 x) z)
   (if (<= z 0.66) (fma 4.0 (- y x) x) (* (* y z) -6.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13.0) {
		tmp = (6.0 * x) * z;
	} else if (z <= 0.66) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = (y * z) * -6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13.0)
		tmp = Float64(Float64(6.0 * x) * z);
	elseif (z <= 0.66)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = Float64(Float64(y * z) * -6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13.0], N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 0.66], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y * z), $MachinePrecision] * -6.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval 60.9

         \[\leadsto \mathsf{fma}\left(-6, 0.6666666666666666 - z, 1\right) \cdot x \]

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      6. lower-*.f64 60.9

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in z around inf

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

       1. lift-*.f64 59.6

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

   11. Applied rewrites 59.6%

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

   if -13 < z < 0.660000000000000031

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto 4 \cdot \left(y - x\right) + \color{blue}{x} \]

      2. lower-fma.f64 N/A

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

      3. lift--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.660000000000000031 < z

   1. Initial program 99.7%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.3%

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

Alternative 10: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 0.66:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13.0)
   (* (* 6.0 x) z)
   (if (<= z 0.66) (fma 4.0 (- y x) x) (* y (* -6.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13.0) {
		tmp = (6.0 * x) * z;
	} else if (z <= 0.66) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = y * (-6.0 * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13.0)
		tmp = Float64(Float64(6.0 * x) * z);
	elseif (z <= 0.66)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = Float64(y * Float64(-6.0 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13.0], N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 0.66], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval 60.9

         \[\leadsto \mathsf{fma}\left(-6, 0.6666666666666666 - z, 1\right) \cdot x \]

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      6. lower-*.f64 60.9

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in z around inf

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

       1. lift-*.f64 59.6

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

   11. Applied rewrites 59.6%

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

   if -13 < z < 0.660000000000000031

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto 4 \cdot \left(y - x\right) + \color{blue}{x} \]

      2. lower-fma.f64 N/A

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

      3. lift--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.660000000000000031 < z

   1. Initial program 99.7%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 54.3

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

   10. Applied rewrites 54.3%

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

Alternative 11: 74.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13.0)
   (* (* 6.0 x) z)
   (if (<= z 0.5) (fma 4.0 (- y x) x) (* (* z x) 6.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13.0) {
		tmp = (6.0 * x) * z;
	} else if (z <= 0.5) {
		tmp = fma(4.0, (y - x), x);
	} else {
		tmp = (z * x) * 6.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13.0)
		tmp = Float64(Float64(6.0 * x) * z);
	elseif (z <= 0.5)
		tmp = fma(4.0, Float64(y - x), x);
	else
		tmp = Float64(Float64(z * x) * 6.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13.0], N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 0.5], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * 6.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval 60.9

         \[\leadsto \mathsf{fma}\left(-6, 0.6666666666666666 - z, 1\right) \cdot x \]

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

      6. lower-*.f64 60.9

         \[\leadsto \mathsf{fma}\left(\frac{x}{z}, -3, 6 \cdot x\right) \cdot z \]

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in z around inf

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

       1. lift-*.f64 59.6

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

   11. Applied rewrites 59.6%

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

   if -13 < z < 0.5

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto 4 \cdot \left(y - x\right) + \color{blue}{x} \]

      2. lower-fma.f64 N/A

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

      3. lift--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.5 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval 51.4

         \[\leadsto \mathsf{fma}\left(-6, 0.6666666666666666 - z, 1\right) \cdot x \]

   5. Applied rewrites 51.4%

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

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 50.1

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

   8. Applied rewrites 50.1%

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

Alternative 12: 37.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-6} \lor \neg \left(x \leq 1.45 \cdot 10^{+65}\right):\\ \;\;\;\;-3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.75e-6) (not (<= x 1.45e+65))) (* -3.0 x) (* 4.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.75e-6) || !(x <= 1.45e+65)) {
		tmp = -3.0 * x;
	} else {
		tmp = 4.0 * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2.75d-6)) .or. (.not. (x <= 1.45d+65))) then
        tmp = (-3.0d0) * x
    else
        tmp = 4.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.75e-6) || !(x <= 1.45e+65)) {
		tmp = -3.0 * x;
	} else {
		tmp = 4.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.75e-6) or not (x <= 1.45e+65):
		tmp = -3.0 * x
	else:
		tmp = 4.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.75e-6) || !(x <= 1.45e+65))
		tmp = Float64(-3.0 * x);
	else
		tmp = Float64(4.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.75e-6) || ~((x <= 1.45e+65)))
		tmp = -3.0 * x;
	else
		tmp = 4.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.75e-6], N[Not[LessEqual[x, 1.45e+65]], $MachinePrecision]], N[(-3.0 * x), $MachinePrecision], N[(4.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7499999999999999e-6 or 1.45e65 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval 83.1

         \[\leadsto \mathsf{fma}\left(-6, 0.6666666666666666 - z, 1\right) \cdot x \]

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 44.9%

      \[\leadsto -3 \cdot x \]

   if -2.7499999999999999e-6 < x < 1.45e65

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto 4 \cdot \left(y - x\right) + \color{blue}{x} \]

      2. lower-fma.f64 N/A

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

      3. lift--.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 40.0

         \[\leadsto 4 \cdot y \]

   8. Applied rewrites 40.0%

      \[\leadsto 4 \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-6} \lor \neg \left(x \leq 1.45 \cdot 10^{+65}\right):\\ \;\;\;\;-3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto 4 \cdot \left(y - x\right) + \color{blue}{x} \]

   2. lower-fma.f64 N/A

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

   3. lift--.f64 52.7

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

5. Applied rewrites 52.7%

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

Alternative 14: 26.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ 4 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 4.0 y))

C

double code(double x, double y, double z) {
	return 4.0 * y;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 4.0d0 * y
end function

Java

public static double code(double x, double y, double z) {
	return 4.0 * y;
}

Python

def code(x, y, z):
	return 4.0 * y

Julia

function code(x, y, z)
	return Float64(4.0 * y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 4.0 * y;
end

Wolfram

code[x_, y_, z_] := N[(4.0 * y), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto 4 \cdot \left(y - x\right) + \color{blue}{x} \]

   2. lower-fma.f64 N/A

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

   3. lift--.f64 52.7

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

5. Applied rewrites 52.7%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 25.6

      \[\leadsto 4 \cdot y \]

8. Applied rewrites 25.6%

   \[\leadsto 4 \cdot \color{blue}{y} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025037 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  :precision binary64
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))