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

Percentage Accurate: 99.5% → 99.8%

Time: 12.0s

Alternatives: 15

Speedup: 1.7×

• 20.1% 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, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - x, \left(0.6666666666666666 - z\right) \cdot 6, x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (- y x) (* (- 0.6666666666666666 z) 6.0) x))

C

double code(double x, double y, double z) {
	return fma((y - x), ((0.6666666666666666 - z) * 6.0), x);
}

Julia

function code(x, y, z)
	return fma(Float64(y - x), Float64(Float64(0.6666666666666666 - z) * 6.0), x)
end

Wolfram

code[x_, y_, z_] := N[(N[(y - x), $MachinePrecision] * N[(N[(0.6666666666666666 - z), $MachinePrecision] * 6.0), $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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. remove-double-neg N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.8

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

   12. lift-/.f64 N/A

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

   13. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -2000 \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (or (<= t_0 -2000.0) (not (<= t_0 1.0)))
     (* (* z x) 6.0)
     (fma (- y x) 4.0 x))))

C

double code(double x, double y, double z) {
	double t_0 = (2.0 / 3.0) - z;
	double tmp;
	if ((t_0 <= -2000.0) || !(t_0 <= 1.0)) {
		tmp = (z * x) * 6.0;
	} else {
		tmp = fma((y - x), 4.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(2.0 / 3.0) - z)
	tmp = 0.0
	if ((t_0 <= -2000.0) || !(t_0 <= 1.0))
		tmp = Float64(Float64(z * x) * 6.0);
	else
		tmp = fma(Float64(y - x), 4.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2000.0], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[(z * x), $MachinePrecision] * 6.0), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) z) < -2e3 or 1 < (-.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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.9%

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

   if -2e3 < (-.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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -2000 \lor \neg \left(\frac{2}{3} - z \leq 1\right):\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{3} - z\\ \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 3.0) z)))
   (if (<= t_0 -2000.0)
     (* (* z x) 6.0)
     (if (<= t_0 1.0) (fma (- y x) 4.0 x) (* (* 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 <= -2000.0) {
		tmp = (z * x) * 6.0;
	} else if (t_0 <= 1.0) {
		tmp = fma((y - x), 4.0, x);
	} 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 <= -2000.0)
		tmp = Float64(Float64(z * x) * 6.0);
	elseif (t_0 <= 1.0)
		tmp = fma(Float64(y - x), 4.0, x);
	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, -2000.0], N[(N[(z * x), $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], 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) < -2e3

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 49.3%

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

   if -2e3 < (-.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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 1 < (-.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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 53.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{3} - z \leq -2000:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{elif}\;\frac{2}{3} - z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (fma -6.0 z 4.0) y)))
   (if (<= y -1.6e+98)
     t_0
     (if (<= y -3.9e-162)
       (fma -3.0 x (* 4.0 y))
       (if (<= y 2.65e+57) (* (fma 6.0 z -3.0) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-6.0, z, 4.0) * y;
	double tmp;
	if (y <= -1.6e+98) {
		tmp = t_0;
	} else if (y <= -3.9e-162) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else if (y <= 2.65e+57) {
		tmp = fma(6.0, z, -3.0) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(-6.0, z, 4.0) * y)
	tmp = 0.0
	if (y <= -1.6e+98)
		tmp = t_0;
	elseif (y <= -3.9e-162)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	elseif (y <= 2.65e+57)
		tmp = Float64(fma(6.0, z, -3.0) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.6e+98], t$95$0, If[LessEqual[y, -3.9e-162], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.65e+57], N[(N[(6.0 * z + -3.0), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6000000000000001e98 or 2.64999999999999993e57 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(\color{blue}{4} + \left(-1 \cdot z\right) \cdot 6\right) \cdot y \]

      9. fp-cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(4 - \left(\mathsf{neg}\left(-1 \cdot z\right)\right) \cdot 6\right)} \cdot y \]

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

          \[\leadsto \left(4 - \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot z\right)} \cdot 6\right) \cdot y \]

      11. metadata-eval N/A

          \[\leadsto \left(4 - \left(\color{blue}{1} \cdot z\right) \cdot 6\right) \cdot y \]

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

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

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

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

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

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

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if -1.6000000000000001e98 < y < -3.8999999999999999e-162

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.9

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

      12. lift-/.f64 N/A

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

      13. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   if -3.8999999999999999e-162 < y < 2.64999999999999993e57

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

      12. lift-/.f64 N/A

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

      13. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 84.6

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

   7. Applied rewrites 84.6%

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

   9. Applied rewrites 84.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+146}:\\ \;\;\;\;\left(z \cdot y\right) \cdot -6\\ \mathbf{elif}\;z \leq -40:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.3e+146)
   (* (* z y) -6.0)
   (if (<= z -40.0)
     (* (* z x) 6.0)
     (if (<= z 2.25e-14) (fma (- y x) 4.0 x) (* (fma -6.0 z 4.0) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.3e+146) {
		tmp = (z * y) * -6.0;
	} else if (z <= -40.0) {
		tmp = (z * x) * 6.0;
	} else if (z <= 2.25e-14) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = fma(-6.0, z, 4.0) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.3e+146)
		tmp = Float64(Float64(z * y) * -6.0);
	elseif (z <= -40.0)
		tmp = Float64(Float64(z * x) * 6.0);
	elseif (z <= 2.25e-14)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = Float64(fma(-6.0, z, 4.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.3e+146], N[(N[(z * y), $MachinePrecision] * -6.0), $MachinePrecision], If[LessEqual[z, -40.0], N[(N[(z * x), $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[z, 2.25e-14], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.30000000000000016e146

   1. Initial program 99.9%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.2%

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

   if -3.30000000000000016e146 < z < -40

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.1%

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

   if -40 < z < 2.2499999999999999e-14

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 2.2499999999999999e-14 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(\color{blue}{4} + \left(-1 \cdot z\right) \cdot 6\right) \cdot y \]

      9. fp-cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(4 - \left(\mathsf{neg}\left(-1 \cdot z\right)\right) \cdot 6\right)} \cdot y \]

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

          \[\leadsto \left(4 - \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot z\right)} \cdot 6\right) \cdot y \]

      11. metadata-eval N/A

          \[\leadsto \left(4 - \left(\color{blue}{1} \cdot z\right) \cdot 6\right) \cdot y \]

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

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

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

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

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

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

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 56.4

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

   5. Applied rewrites 56.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+146}:\\ \;\;\;\;\left(z \cdot y\right) \cdot -6\\ \mathbf{elif}\;z \leq -40:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+146}:\\ \;\;\;\;\left(z \cdot y\right) \cdot -6\\ \mathbf{elif}\;z \leq -40:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.65:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.3e+146)
   (* (* z y) -6.0)
   (if (<= z -40.0)
     (* (* z x) 6.0)
     (if (<= z 0.65) (fma (- y x) 4.0 x) (* (* -6.0 z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.3e+146) {
		tmp = (z * y) * -6.0;
	} else if (z <= -40.0) {
		tmp = (z * x) * 6.0;
	} else if (z <= 0.65) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = (-6.0 * z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.3e+146)
		tmp = Float64(Float64(z * y) * -6.0);
	elseif (z <= -40.0)
		tmp = Float64(Float64(z * x) * 6.0);
	elseif (z <= 0.65)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = Float64(Float64(-6.0 * z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.3e+146], N[(N[(z * y), $MachinePrecision] * -6.0), $MachinePrecision], If[LessEqual[z, -40.0], N[(N[(z * x), $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[z, 0.65], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], N[(N[(-6.0 * z), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.30000000000000016e146

   1. Initial program 99.9%

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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.2%

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

   if -3.30000000000000016e146 < z < -40

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.1%

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

   if -40 < 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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 0.650000000000000022 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

      12. lift-/.f64 N/A

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

      13. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(\color{blue}{4} + 6 \cdot \left(-1 \cdot z\right)\right) \cdot y \]

      9. mul-1-neg N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 53.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+146}:\\ \;\;\;\;\left(z \cdot y\right) \cdot -6\\ \mathbf{elif}\;z \leq -40:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.65:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot y\right) \cdot -6\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -40:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.65:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z y) -6.0)))
   (if (<= z -3.3e+146)
     t_0
     (if (<= z -40.0)
       (* (* z x) 6.0)
       (if (<= z 0.65) (fma (- y x) 4.0 x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (z * y) * -6.0;
	double tmp;
	if (z <= -3.3e+146) {
		tmp = t_0;
	} else if (z <= -40.0) {
		tmp = (z * x) * 6.0;
	} else if (z <= 0.65) {
		tmp = fma((y - x), 4.0, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * y) * -6.0)
	tmp = 0.0
	if (z <= -3.3e+146)
		tmp = t_0;
	elseif (z <= -40.0)
		tmp = Float64(Float64(z * x) * 6.0);
	elseif (z <= 0.65)
		tmp = fma(Float64(y - x), 4.0, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * y), $MachinePrecision] * -6.0), $MachinePrecision]}, If[LessEqual[z, -3.3e+146], t$95$0, If[LessEqual[z, -40.0], N[(N[(z * x), $MachinePrecision] * 6.0), $MachinePrecision], If[LessEqual[z, 0.65], N[(N[(y - x), $MachinePrecision] * 4.0 + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.30000000000000016e146 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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.1%

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

   if -3.30000000000000016e146 < z < -40

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.1%

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

   if -40 < 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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+146}:\\ \;\;\;\;\left(z \cdot y\right) \cdot -6\\ \mathbf{elif}\;z \leq -40:\\ \;\;\;\;\left(z \cdot x\right) \cdot 6\\ \mathbf{elif}\;z \leq 0.65:\\ \;\;\;\;\mathsf{fma}\left(y - x, 4, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot -6\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.55 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.55) (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 <= -0.55) || !(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 <= -0.55) || !(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, -0.55], 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 < -0.55000000000000004 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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if -0.55000000000000004 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

      12. lift-/.f64 N/A

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

      13. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 97.1

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

   7. Applied rewrites 97.1%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.55 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.55 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(-6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.55) (not (<= z 0.5)))
   (* (* -6.0 (- y x)) z)
   (fma -3.0 x (* 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.55) || !(z <= 0.5)) {
		tmp = (-6.0 * (y - x)) * z;
	} else {
		tmp = fma(-3.0, x, (4.0 * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.55) || !(z <= 0.5))
		tmp = Float64(Float64(-6.0 * Float64(y - x)) * z);
	else
		tmp = fma(-3.0, x, Float64(4.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.55], N[Not[LessEqual[z, 0.5]], $MachinePrecision]], N[(N[(-6.0 * N[(y - x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.55000000000000004 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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 98.0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

   if -0.55000000000000004 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

      12. lift-/.f64 N/A

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

      13. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 97.1

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

   7. Applied rewrites 97.1%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.55 \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\left(-6 \cdot \left(y - x\right)\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.55:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(-6 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.55)
   (* (* (- y x) z) -6.0)
   (if (<= z 0.5) (fma -3.0 x (* 4.0 y)) (* (- y x) (* -6.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.55) {
		tmp = ((y - x) * z) * -6.0;
	} else if (z <= 0.5) {
		tmp = fma(-3.0, x, (4.0 * y));
	} else {
		tmp = (y - x) * (-6.0 * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.55)
		tmp = Float64(Float64(Float64(y - x) * z) * -6.0);
	elseif (z <= 0.5)
		tmp = fma(-3.0, x, Float64(4.0 * y));
	else
		tmp = Float64(Float64(y - x) * Float64(-6.0 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.55], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * -6.0), $MachinePrecision], If[LessEqual[z, 0.5], N[(-3.0 * x + N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.55000000000000004

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -0.55000000000000004 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.8

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

      12. lift-/.f64 N/A

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

      13. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-4 + 1\right) \cdot x} + 4 \cdot y \]

      9. metadata-eval N/A

         \[\leadsto \color{blue}{-3} \cdot x + 4 \cdot y \]

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 97.1

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

   7. Applied rewrites 97.1%

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

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 98.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.55:\\ \;\;\;\;\left(\left(y - x\right) \cdot z\right) \cdot -6\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-3, x, 4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(-6 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+98} \lor \neg \left(y \leq 2.65 \cdot 10^{+57}\right):\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.4e+98) (not (<= y 2.65e+57)))
   (* (fma -6.0 z 4.0) y)
   (* (fma 6.0 z -3.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.4e+98) || !(y <= 2.65e+57)) {
		tmp = fma(-6.0, z, 4.0) * y;
	} else {
		tmp = fma(6.0, z, -3.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.4e+98) || !(y <= 2.65e+57))
		tmp = Float64(fma(-6.0, z, 4.0) * y);
	else
		tmp = Float64(fma(6.0, z, -3.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.4e+98], N[Not[LessEqual[y, 2.65e+57]], $MachinePrecision]], N[(N[(-6.0 * z + 4.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(6.0 * z + -3.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e98 or 2.64999999999999993e57 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(\color{blue}{4} + \left(-1 \cdot z\right) \cdot 6\right) \cdot y \]

      9. fp-cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(4 - \left(\mathsf{neg}\left(-1 \cdot z\right)\right) \cdot 6\right)} \cdot y \]

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

          \[\leadsto \left(4 - \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot z\right)} \cdot 6\right) \cdot y \]

      11. metadata-eval N/A

          \[\leadsto \left(4 - \left(\color{blue}{1} \cdot z\right) \cdot 6\right) \cdot y \]

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

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

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

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

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

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

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. lower-fma.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if -1.4e98 < y < 2.64999999999999993e57

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. remove-double-neg N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

      12. lift-/.f64 N/A

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

      13. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 75.8

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

   7. Applied rewrites 75.8%

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

   9. Applied rewrites 75.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+98} \lor \neg \left(y \leq 2.65 \cdot 10^{+57}\right):\\ \;\;\;\;\mathsf{fma}\left(-6, z, 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(6, z, -3\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+63} \lor \neg \left(x \leq 9.2 \cdot 10^{+26}\right):\\ \;\;\;\;-3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.8e+63) (not (<= x 9.2e+26))) (* -3.0 x) (* 4.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.8e+63) || !(x <= 9.2e+26)) {
		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 <= (-4.8d+63)) .or. (.not. (x <= 9.2d+26))) 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 <= -4.8e+63) || !(x <= 9.2e+26)) {
		tmp = -3.0 * x;
	} else {
		tmp = 4.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.8e+63) or not (x <= 9.2e+26):
		tmp = -3.0 * x
	else:
		tmp = 4.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.8e+63) || !(x <= 9.2e+26))
		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 <= -4.8e+63) || ~((x <= 9.2e+26)))
		tmp = -3.0 * x;
	else
		tmp = 4.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.8e+63], N[Not[LessEqual[x, 9.2e+26]], $MachinePrecision]], N[(-3.0 * x), $MachinePrecision], N[(4.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8e63 or 9.2000000000000002e26 < 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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.3%

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

   if -4.8e63 < x < 9.2000000000000002e26

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.0%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+63} \lor \neg \left(x \leq 9.2 \cdot 10^{+26}\right):\\ \;\;\;\;-3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(0.6666666666666666 - z\right) \cdot \left(y - x\right), 6, x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (* (- 0.6666666666666666 z) (- y x)) 6.0 x))

C

double code(double x, double y, double z) {
	return fma(((0.6666666666666666 - z) * (y - x)), 6.0, x);
}

Julia

function code(x, y, z)
	return fma(Float64(Float64(0.6666666666666666 - z) * Float64(y - x)), 6.0, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(0.6666666666666666 - z), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] * 6.0 + 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. remove-double-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.6

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

   14. lift-/.f64 N/A

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

   15. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

Alternative 14: 51.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 51.9

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

5. Applied rewrites 51.9%

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

6. Final simplification 51.9%

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

Alternative 15: 26.1% 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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 51.9

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

5. Applied rewrites 51.9%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 27.9%

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

9. Final simplification 27.9%

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

Reproduce

herbie shell --seed 2024359 
(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))))