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

Percentage Accurate: 99.5% → 99.8%

Time: 4.1s

Alternatives: 15

Speedup: 1.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.9× 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 99.6%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
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 99.8

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 75.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;t\_0 \leq 0.665:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.66666666666669:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(-6 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (fma -6.0 z 4.0) y)))
   (if (<= t_0 0.665)
     t_1
     (if (<= t_0 0.66666666666669)
       (fma -3.0 x (* 4.0 y))
       (if (<= t_0 5e+140) t_1 (* (- x) (* -6.0 z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (t_0 <= 0.665) {
		tmp = t_1;
	} else if (t_0 <= 0.66666666666669) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else if (t_0 <= 5e+140) {
		tmp = t_1;
	} else {
		tmp = -x * (-6.0 * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (t_0 <= 0.665)
		tmp = t_1;
	elseif (t_0 <= 0.66666666666669)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	elseif (t_0 <= 5e+140)
		tmp = t_1;
	else
		tmp = Float64(Float64(-x) * Float64(-6.0 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, 0.665], t$95$1, If[LessEqual[t$95$0, 0.66666666666669], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+140], t$95$1, N[((-x) * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.66500000000000004 or 0.66666666666669006 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 5.00000000000000008e140

   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 99.7

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

   5. Applied rewrites 99.7%

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

   6. 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 58.5

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

   8. Applied rewrites 58.5%

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

   if 0.66500000000000004 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.66666666666669006

   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 99.6

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

   5. Applied rewrites 99.6%

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

   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 99.6

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

   8. Applied rewrites 99.6%

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

   if 5.00000000000000008e140 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) 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 99.7

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

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]
   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 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 72.2

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

   10. Applied rewrites 72.2%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq 0.665:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{elif}\;\frac{2}{3} - z \leq 0.66666666666669:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(-6 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;t\_0 \leq 0.665:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.66666666666669:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (fma -6.0 z 4.0) y)))
   (if (<= t_0 0.665)
     t_1
     (if (<= t_0 0.66666666666669)
       (fma -3.0 x (* 4.0 y))
       (if (<= t_0 5e+140) t_1 (* (* 6.0 x) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (t_0 <= 0.665) {
		tmp = t_1;
	} else if (t_0 <= 0.66666666666669) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else if (t_0 <= 5e+140) {
		tmp = t_1;
	} else {
		tmp = (6.0 * x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (t_0 <= 0.665)
		tmp = t_1;
	elseif (t_0 <= 0.66666666666669)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	elseif (t_0 <= 5e+140)
		tmp = t_1;
	else
		tmp = Float64(Float64(6.0 * x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, 0.665], t$95$1, If[LessEqual[t$95$0, 0.66666666666669], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+140], t$95$1, N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.66500000000000004 or 0.66666666666669006 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 5.00000000000000008e140

   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 99.7

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

   5. Applied rewrites 99.7%

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

   6. 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 58.5

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

   8. Applied rewrites 58.5%

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

   if 0.66500000000000004 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 0.66666666666669006

   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 99.6

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

   5. Applied rewrites 99.6%

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

   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 99.6

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

   8. Applied rewrites 99.6%

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

   if 5.00000000000000008e140 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) 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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x 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 72.0

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

   8. Applied rewrites 72.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      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 72.0

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

   10. Applied rewrites 72.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq 0.665:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{elif}\;\frac{2}{3} - z \leq 0.66666666666669:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := \left(y \cdot -6\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* (* y -6.0) z)))
   (if (<= t_0 -2000000.0)
     t_1
     (if (<= t_0 1.0)
       (fma -3.0 x (* 4.0 y))
       (if (<= t_0 5e+140) t_1 (* (* 6.0 x) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = (y * -6.0) * z;
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else if (t_0 <= 5e+140) {
		tmp = t_1;
	} else {
		tmp = (6.0 * x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(Float64(y * -6.0) * z)
	tmp = 0.0
	if (t_0 <= -2000000.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	elseif (t_0 <= 5e+140)
		tmp = t_1;
	else
		tmp = Float64(Float64(6.0 * x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * -6.0), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000.0], t$95$1, If[LessEqual[t$95$0, 1.0], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+140], t$95$1, N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -2e6 or 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 5.00000000000000008e140

   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 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]
   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 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 56.3%

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

       1. metadata-eval 56.3

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

       2. associate-*l* 56.3

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

       3. *-commutative 56.3

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

       4. associate-*l* 56.3

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

       5. +-commutative 56.3

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 56.3

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

   12. Applied rewrites 56.3%

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

   if -2e6 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   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 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 98.7

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

   5. Applied rewrites 98.7%

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

   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 98.7

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

   8. Applied rewrites 98.7%

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

   if 5.00000000000000008e140 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) 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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x 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 72.0

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

   8. Applied rewrites 72.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      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 72.0

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

   10. Applied rewrites 72.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -2000000:\\ \;\;\;\;\left(y \cdot -6\right) \cdot z\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\left(y \cdot -6\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -2000000.0)
     (* (* y z) -6.0)
     (if (<= t_0 1.0)
       (fma 4.0 (- y x) x)
       (if (<= t_0 5e+140) (* y (* -6.0 z)) (* (* 6.0 x) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = (y * z) * -6.0;
	} else if (t_0 <= 1.0) {
		tmp = fma(4.0, (y - x), x);
	} else if (t_0 <= 5e+140) {
		tmp = y * (-6.0 * z);
	} else {
		tmp = (6.0 * x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if (t_0 <= -2000000.0)
		tmp = Float64(Float64(y * z) * -6.0);
	elseif (t_0 <= 1.0)
		tmp = fma(4.0, Float64(y - x), x);
	elseif (t_0 <= 5e+140)
		tmp = Float64(y * Float64(-6.0 * z));
	else
		tmp = Float64(Float64(6.0 * x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000.0], N[(N[(y * z), $MachinePrecision] * -6.0), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 5e+140], N[(y * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -2e6

   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.7%

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

   if -2e6 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   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 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 98.7

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

   5. Applied rewrites 98.7%

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

   if 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 5.00000000000000008e140

   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 99.6

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]
   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 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.6%

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

   if 5.00000000000000008e140 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) 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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x 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 72.0

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

   8. Applied rewrites 72.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      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 72.0

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

   10. Applied rewrites 72.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -2000000:\\ \;\;\;\;\left(y \cdot z\right) \cdot -6\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 5 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ t_1 := y \cdot \left(-6 \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)) (t_1 (* y (* -6.0 z))))
   (if (<= t_0 -2000000.0)
     t_1
     (if (<= t_0 1.0)
       (fma 4.0 (- y x) x)
       (if (<= t_0 5e+140) t_1 (* (* 6.0 x) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double t_1 = y * (-6.0 * z);
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = fma(4.0, (y - x), x);
	} else if (t_0 <= 5e+140) {
		tmp = t_1;
	} else {
		tmp = (6.0 * x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	t_1 = Float64(y * Float64(-6.0 * z))
	tmp = 0.0
	if (t_0 <= -2000000.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = fma(4.0, Float64(y - x), x);
	elseif (t_0 <= 5e+140)
		tmp = t_1;
	else
		tmp = Float64(Float64(6.0 * x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000.0], t$95$1, If[LessEqual[t$95$0, 1.0], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 5e+140], t$95$1, N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -2e6 or 1 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 5.00000000000000008e140

   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 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]
   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 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 56.3%

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

   if -2e6 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < 1

   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 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 98.7

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

   5. Applied rewrites 98.7%

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

   if 5.00000000000000008e140 < (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) 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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x 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 72.0

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

   8. Applied rewrites 72.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      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 72.0

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

   10. Applied rewrites 72.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -2000000:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \mathbf{elif}\;\frac{2}{3} - z \leq 5 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(-6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+142}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -65 \lor \neg \left(z \leq 0.65\right):\\ \;\;\;\;\left(y \cdot -6\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.55e+142)
   (* (* 6.0 x) z)
   (if (or (<= z -65.0) (not (<= z 0.65)))
     (* (* y -6.0) z)
     (fma 4.0 (- y x) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.55e+142) {
		tmp = (6.0 * x) * z;
	} else if ((z <= -65.0) || !(z <= 0.65)) {
		tmp = (y * -6.0) * z;
	} else {
		tmp = fma(4.0, (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.55e+142)
		tmp = Float64(Float64(6.0 * x) * z);
	elseif ((z <= -65.0) || !(z <= 0.65))
		tmp = Float64(Float64(y * -6.0) * z);
	else
		tmp = fma(4.0, Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.55e+142], N[(N[(6.0 * x), $MachinePrecision] * z), $MachinePrecision], If[Or[LessEqual[z, -65.0], N[Not[LessEqual[z, 0.65]], $MachinePrecision]], N[(N[(y * -6.0), $MachinePrecision] * z), $MachinePrecision], N[(4.0 * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55e142

   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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x 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 72.0

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

   8. Applied rewrites 72.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      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 72.0

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

   10. Applied rewrites 72.0%

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

   if -1.55e142 < z < -65 or 0.650000000000000022 < 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 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]
   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 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 56.3%

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

       1. metadata-eval 56.3

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

       2. associate-*l* 56.3

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

       3. *-commutative 56.3

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

       4. associate-*l* 56.3

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

       5. +-commutative 56.3

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 56.3

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

   12. Applied rewrites 56.3%

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

   if -65 < z < 0.650000000000000022

   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 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 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+142}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -65 \lor \neg \left(z \leq 0.65\right):\\ \;\;\;\;\left(y \cdot -6\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.52 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.52) (not (<= z 0.5)))
   (* (* (- y x) z) -6.0)
   (fma -3.0 x (* 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.52) || !(z <= 0.5)) {
		tmp = ((y - x) * z) * -6.0;
	} else {
		tmp = fma(-3.0, x, (4.0 * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.52) || !(z <= 0.5))
		tmp = Float64(Float64(Float64(y - x) * z) * -6.0);
	else
		tmp = fma(-3.0, x, Float64(4.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.52], N[Not[LessEqual[z, 0.5]], $MachinePrecision]], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.52000000000000002 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 99.0

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

   5. Applied rewrites 99.0%

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

   if -0.52000000000000002 < z < 0.5

   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 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 98.7

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

   5. Applied rewrites 98.7%

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

   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 98.7

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

   8. Applied rewrites 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.52 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-101} \lor \neg \left(x \leq 10^{-9}\right):\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(-6, z, 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.9e-101) (not (<= x 1e-9)))
   (* (- x) (fma -6.0 z 3.0))
   (* (fma -6.0 z 4.0) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.9e-101) || !(x <= 1e-9)) {
		tmp = -x * fma(-6.0, z, 3.0);
	} else {
		tmp = fma(-6.0, z, 4.0) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.9e-101) || !(x <= 1e-9))
		tmp = Float64(Float64(-x) * fma(-6.0, z, 3.0));
	else
		tmp = Float64(fma(-6.0, z, 4.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.9e-101], N[Not[LessEqual[x, 1e-9]], $MachinePrecision]], N[((-x) * N[(-6.0 * z + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.90000000000000005e-101 or 1.00000000000000006e-9 < 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 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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 78.6

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

   8. Applied rewrites 78.6%

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

   if -1.90000000000000005e-101 < x < 1.00000000000000006e-9

   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 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 81.2

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

   8. Applied rewrites 81.2%

      \[\leadsto \mathsf{fma}\left(-6, z, 4\right) \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-101} \lor \neg \left(x \leq 10^{-9}\right):\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(-6, z, 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.52:\\ \;\;\;\;\left(y - x\right) \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.52)
   (* (- y x) (* -6.0 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 <= -0.52) {
		tmp = (y - x) * (-6.0 * 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 <= -0.52)
		tmp = Float64(Float64(y - x) * Float64(-6.0 * 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, -0.52], N[(N[(y - x), $MachinePrecision] * N[(-6.0 * z), $MachinePrecision]), $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 < -0.52000000000000002

   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 99.7

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

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot -6} \]
   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 99.8

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

   7. Applied rewrites 99.8%

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

   if -0.52000000000000002 < z < 0.5

   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 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 98.7

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

   5. Applied rewrites 98.7%

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

   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 98.7

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

   8. Applied rewrites 98.7%

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

   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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.52:\\ \;\;\;\;\left(y - x\right) \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-101}:\\ \;\;\;\;\left(z \cdot 6 - 3\right) \cdot x\\ \mathbf{elif}\;x \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(-6, z, 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.9e-101)
   (* (- (* z 6.0) 3.0) x)
   (if (<= x 1e-9) (* (fma -6.0 z 4.0) y) (* (- x) (fma -6.0 z 3.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e-101) {
		tmp = ((z * 6.0) - 3.0) * x;
	} else if (x <= 1e-9) {
		tmp = fma(-6.0, z, 4.0) * y;
	} else {
		tmp = -x * fma(-6.0, z, 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.9e-101)
		tmp = Float64(Float64(Float64(z * 6.0) - 3.0) * x);
	elseif (x <= 1e-9)
		tmp = Float64(fma(-6.0, z, 4.0) * y);
	else
		tmp = Float64(Float64(-x) * fma(-6.0, z, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.9e-101], N[(N[(N[(z * 6.0), $MachinePrecision] - 3.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1e-9], N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision], N[((-x) * N[(-6.0 * z + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.90000000000000005e-101

   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 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*r* N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 73.5

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

   10. Applied rewrites 73.5%

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

   if -1.90000000000000005e-101 < x < 1.00000000000000006e-9

   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 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 81.2

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

   8. Applied rewrites 81.2%

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

   if 1.00000000000000006e-9 < x

   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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 84.9

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

   8. Applied rewrites 84.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-101}:\\ \;\;\;\;\left(z \cdot 6 - 3\right) \cdot x\\ \mathbf{elif}\;x \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(-6, z, 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+29} \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.6e+29) (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 <= -2.6e+29) || !(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 <= -2.6e+29) || !(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, -2.6e+29], 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 < -2.6e29 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 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x 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 55.5

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

   8. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      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 55.6

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

   10. Applied rewrites 55.6%

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

   if -2.6e29 < z < 0.5

   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 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 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+29} \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-5} \lor \neg \left(x \leq 10^{-9}\right):\\ \;\;\;\;-3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.1e-5) (not (<= x 1e-9))) (* -3.0 x) (* 4.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e-5) || !(x <= 1e-9)) {
		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 <= (-1.1d-5)) .or. (.not. (x <= 1d-9))) 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 <= -1.1e-5) || !(x <= 1e-9)) {
		tmp = -3.0 * x;
	} else {
		tmp = 4.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.1e-5) or not (x <= 1e-9):
		tmp = -3.0 * x
	else:
		tmp = 4.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.1e-5) || !(x <= 1e-9))
		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 <= -1.1e-5) || ~((x <= 1e-9)))
		tmp = -3.0 * x;
	else
		tmp = 4.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.1e-5], N[Not[LessEqual[x, 1e-9]], $MachinePrecision]], N[(-3.0 * x), $MachinePrecision], N[(4.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e-5 or 1.00000000000000006e-9 < 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 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 49.0

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

   5. Applied rewrites 49.0%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 41.7

         \[\leadsto -3 \cdot x \]

   8. Applied rewrites 41.7%

      \[\leadsto -3 \cdot \color{blue}{x} \]

   if -1.1e-5 < x < 1.00000000000000006e-9

   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 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 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 42.4

         \[\leadsto 4 \cdot y \]

   8. Applied rewrites 42.4%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-5} \lor \neg \left(x \leq 10^{-9}\right):\\ \;\;\;\;-3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.6% 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 99.6%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
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 50.9

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

5. Applied rewrites 50.9%

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

6. Final simplification 50.9%

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

Alternative 15: 25.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ -3 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* -3.0 x))

C

double code(double x, double y, double z) {
	return -3.0 * 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 = (-3.0d0) * x
end function

Java

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

Python

def code(x, y, z):
	return -3.0 * x

Julia

function code(x, y, z)
	return Float64(-3.0 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -3.0 * x;
end

Wolfram

code[x_, y_, z_] := N[(-3.0 * x), $MachinePrecision]

Derivation

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 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 50.9

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

5. Applied rewrites 50.9%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 26.8

      \[\leadsto -3 \cdot x \]

8. Applied rewrites 26.8%

   \[\leadsto -3 \cdot \color{blue}{x} \]

9. Final simplification 26.8%

   \[\leadsto -3 \cdot x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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))))