Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.3% → 95.6%

Time: 7.4s

Alternatives: 11

Speedup: 1.4×

• 32.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + (y * z)) + (t * a)) + ((a * z) * b);
}

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, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x + (y * z)) + (t * a)) + ((a * z) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x + (y * z)) + (t * a)) + ((a * z) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x + (y * z)) + (t * a)) + ((a * z) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(a * z) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x + (y * z)) + (t * a)) + ((a * z) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Initial Program: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + (y * z)) + (t * a)) + ((a * z) * b);
}

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, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x + (y * z)) + (t * a)) + ((a * z) * b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((x + (y * z)) + (t * a)) + ((a * z) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x + (y * z)) + (t * a)) + ((a * z) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(a * z) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x + (y * z)) + (t * a)) + ((a * z) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, y, x + a \cdot \mathsf{fma}\left(b, z, t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma z y (+ x (* a (fma b z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(z, y, (x + (a * fma(b, z, t))));
}

Julia

function code(x, y, z, t, a, b)
	return fma(z, y, Float64(x + Float64(a * fma(b, z, t))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(z * y + N[(x + N[(a * N[(b * z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.9%

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

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \color{blue}{\left(a \cdot z\right) \cdot b} \]

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

      \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) - \left(\mathsf{neg}\left(a \cdot z\right)\right) \cdot b} \]

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

      \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b} \]

   5. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

   6. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x + y \cdot z\right)} + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

   7. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(y \cdot z + x\right)} + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

   8. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(y \cdot z + \left(x + t \cdot a\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

   9. associate-+l+ N/A

      \[\leadsto \color{blue}{y \cdot z + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \color{blue}{y \cdot z} + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right) \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{z \cdot y} + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right) \]

   12. remove-double-neg N/A

       \[\leadsto z \cdot y + \left(\left(x + t \cdot a\right) + \color{blue}{\left(a \cdot z\right)} \cdot b\right) \]

   13. lift-*.f64 N/A

       \[\leadsto z \cdot y + \left(\left(x + t \cdot a\right) + \color{blue}{\left(a \cdot z\right) \cdot b}\right) \]

   14. associate-+r+ N/A

       \[\leadsto z \cdot y + \color{blue}{\left(x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]

   15. lower-fma.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. associate-*l* N/A

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

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, x + a \cdot \mathsf{fma}\left(b, z, t\right)\right)} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -4.5e+92)
   (fma (fma b a y) z x)
   (if (<= x 2.8e-56) (fma (fma b z t) a (* z y)) (fma (fma b z t) a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -4.5e+92) {
		tmp = fma(fma(b, a, y), z, x);
	} else if (x <= 2.8e-56) {
		tmp = fma(fma(b, z, t), a, (z * y));
	} else {
		tmp = fma(fma(b, z, t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -4.5e+92)
		tmp = fma(fma(b, a, y), z, x);
	elseif (x <= 2.8e-56)
		tmp = fma(fma(b, z, t), a, Float64(z * y));
	else
		tmp = fma(fma(b, z, t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -4.5e+92], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[x, 2.8e-56], N[(N[(b * z + t), $MachinePrecision] * a + N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4999999999999999e92

   1. Initial program 86.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(y + a \cdot b\right)}\right), z, x\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot y + -1 \cdot \left(a \cdot b\right)\right)}\right), z, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right) + -1 \cdot y\right)}\right), z, x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right) + \left(\mathsf{neg}\left(-1 \cdot y\right)\right)}, z, x\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot y\right)\right), z, x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b} + \left(\mathsf{neg}\left(-1 \cdot y\right)\right), z, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(a \cdot b + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot y}, z, x\right) \]

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

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

      16. *-commutative N/A

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

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b + \left(\mathsf{neg}\left(y\right)\right) \cdot -1}, z, x\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot a} + \left(\mathsf{neg}\left(y\right)\right) \cdot -1, z, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(b \cdot a + \color{blue}{\left(\mathsf{neg}\left(y \cdot -1\right)\right)}, z, x\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot a + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right), z, x\right) \]

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower-fma.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if -4.4999999999999999e92 < x < 2.79999999999999993e-56

   1. Initial program 91.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a \cdot t + a \cdot \left(b \cdot z\right)\right) + y \cdot z} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right)} + y \cdot z \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + y \cdot z \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, y \cdot z\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, y \cdot z\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, y \cdot z\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, \color{blue}{z \cdot y}\right) \]

      8. lower-*.f64 92.2

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, \color{blue}{z \cdot y}\right) \]

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)} \]

   if 2.79999999999999993e-56 < x

   1. Initial program 98.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right) + x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, x\right) \]

      6. lower-fma.f64 90.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, x\right) \]

   5. Applied rewrites 90.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+33} \lor \neg \left(a \leq 6 \cdot 10^{+39}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -3.2e+33) (not (<= a 6e+39)))
   (fma (fma b z t) a x)
   (fma (fma b a y) z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -3.2e+33) || !(a <= 6e+39)) {
		tmp = fma(fma(b, z, t), a, x);
	} else {
		tmp = fma(fma(b, a, y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -3.2e+33) || !(a <= 6e+39))
		tmp = fma(fma(b, z, t), a, x);
	else
		tmp = fma(fma(b, a, y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -3.2e+33], N[Not[LessEqual[a, 6e+39]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.20000000000000017e33 or 5.9999999999999999e39 < a

   1. Initial program 85.0%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right) + x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, x\right) \]

      6. lower-fma.f64 91.8

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, x\right) \]

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)} \]

   if -3.20000000000000017e33 < a < 5.9999999999999999e39

   1. Initial program 98.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(y + a \cdot b\right)}\right), z, x\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot y + -1 \cdot \left(a \cdot b\right)\right)}\right), z, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right) + -1 \cdot y\right)}\right), z, x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right) + \left(\mathsf{neg}\left(-1 \cdot y\right)\right)}, z, x\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot y\right)\right), z, x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b} + \left(\mathsf{neg}\left(-1 \cdot y\right)\right), z, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(a \cdot b + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot y}, z, x\right) \]

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

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

      16. *-commutative N/A

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

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b + \left(\mathsf{neg}\left(y\right)\right) \cdot -1}, z, x\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot a} + \left(\mathsf{neg}\left(y\right)\right) \cdot -1, z, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(b \cdot a + \color{blue}{\left(\mathsf{neg}\left(y \cdot -1\right)\right)}, z, x\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot a + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right), z, x\right) \]

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower-fma.f64 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+33} \lor \neg \left(a \leq 6 \cdot 10^{+39}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+142} \lor \neg \left(t \leq 4.2 \cdot 10^{+46}\right):\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -2e+142) (not (<= t 4.2e+46)))
   (fma a t (fma z y x))
   (fma (fma b a y) z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -2e+142) || !(t <= 4.2e+46)) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = fma(fma(b, a, y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -2e+142) || !(t <= 4.2e+46))
		tmp = fma(a, t, fma(z, y, x));
	else
		tmp = fma(fma(b, a, y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -2e+142], N[Not[LessEqual[t, 4.2e+46]], $MachinePrecision]], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.0000000000000001e142 or 4.2e46 < t

   1. Initial program 88.7%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right) + x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, x\right) \]

      6. lower-fma.f64 74.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, x\right) \]

   5. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(b \cdot z, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 30.8%

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

   9. Taylor expanded in b around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(a, t, z \cdot \color{blue}{\left(1 \cdot y\right)} + x\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, t, z \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot y\right) + x\right) \]

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

          \[\leadsto \mathsf{fma}\left(a, t, z \cdot \color{blue}{\left(\mathsf{neg}\left(-1 \cdot y\right)\right)} + x\right) \]

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

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(-1 \cdot y\right)\right)\right)} + x\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(a, t, \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot y\right) \cdot z}\right)\right) + x\right) \]

       12. mul-1-neg N/A

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

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

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

       14. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(a, t, \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(y \cdot z\right)}\right)\right) + x\right) \]

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

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \left(y \cdot z\right)} + x\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{1} \cdot \left(y \cdot z\right) + x\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(y \cdot z\right) + x\right) \]

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

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(y \cdot z\right)\right)\right)} + x\right) \]

       19. mul-1-neg N/A

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

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

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

       21. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(a, t, \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot y\right)} \cdot z\right)\right) + x\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(a, t, \left(\mathsf{neg}\left(\color{blue}{z \cdot \left(-1 \cdot y\right)}\right)\right) + x\right) \]

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

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{z \cdot \left(\mathsf{neg}\left(-1 \cdot y\right)\right)} + x\right) \]

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

           \[\leadsto \mathsf{fma}\left(a, t, z \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot y\right)} + x\right) \]

       25. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(a, t, z \cdot \left(\color{blue}{1} \cdot y\right) + x\right) \]

       26. *-lft-identity N/A

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

   11. Applied rewrites 96.7%

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

   if -2.0000000000000001e142 < t < 4.2e46

   1. Initial program 95.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(y + a \cdot b\right)}\right), z, x\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot y + -1 \cdot \left(a \cdot b\right)\right)}\right), z, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right) + -1 \cdot y\right)}\right), z, x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right) + \left(\mathsf{neg}\left(-1 \cdot y\right)\right)}, z, x\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot y\right)\right), z, x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b} + \left(\mathsf{neg}\left(-1 \cdot y\right)\right), z, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(a \cdot b + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot y}, z, x\right) \]

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

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

      16. *-commutative N/A

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

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b + \left(\mathsf{neg}\left(y\right)\right) \cdot -1}, z, x\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot a} + \left(\mathsf{neg}\left(y\right)\right) \cdot -1, z, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(b \cdot a + \color{blue}{\left(\mathsf{neg}\left(y \cdot -1\right)\right)}, z, x\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot a + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right), z, x\right) \]

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower-fma.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+142} \lor \neg \left(t \leq 4.2 \cdot 10^{+46}\right):\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.8e+61)
   (fma z y x)
   (if (<= y -3.8e-74)
     (* (fma b a y) z)
     (if (<= y 1.75e-64) (fma a t x) (fma z y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.8e+61) {
		tmp = fma(z, y, x);
	} else if (y <= -3.8e-74) {
		tmp = fma(b, a, y) * z;
	} else if (y <= 1.75e-64) {
		tmp = fma(a, t, x);
	} else {
		tmp = fma(z, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.8e+61)
		tmp = fma(z, y, x);
	elseif (y <= -3.8e-74)
		tmp = Float64(fma(b, a, y) * z);
	elseif (y <= 1.75e-64)
		tmp = fma(a, t, x);
	else
		tmp = fma(z, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.8e+61], N[(z * y + x), $MachinePrecision], If[LessEqual[y, -3.8e-74], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y, 1.75e-64], N[(a * t + x), $MachinePrecision], N[(z * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.79999999999999995e61 or 1.7500000000000002e-64 < y

   1. Initial program 93.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right) + x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, x\right) \]

      6. lower-fma.f64 62.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, x\right) \]

   5. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(b \cdot z, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 51.0%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{y \cdot z + x} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot y} + x \]

       3. *-lft-identity N/A

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

       4. metadata-eval N/A

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

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

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

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

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

       7. *-commutative N/A

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

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

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

       12. metadata-eval N/A

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

       13. metadata-eval N/A

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

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

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

       15. mul-1-neg N/A

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

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

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

       17. mul-1-neg N/A

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

       18. *-commutative N/A

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

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

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

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

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

       21. metadata-eval N/A

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

       22. *-lft-identity N/A

           \[\leadsto z \cdot \color{blue}{y} + x \]

       23. lower-fma.f64 76.2

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

   11. Applied rewrites 76.2%

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

   if -3.79999999999999995e61 < y < -3.7999999999999996e-74

   1. Initial program 95.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y + a \cdot b\right) \cdot z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y + a \cdot b\right) \cdot z} \]

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(y + a \cdot b\right)}\right)\right) \cdot z \]

      5. distribute-lft-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot y + -1 \cdot \left(a \cdot b\right)\right)}\right)\right) \cdot z \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right) + -1 \cdot y\right)}\right)\right) \cdot z \]

      7. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right) + \left(\mathsf{neg}\left(-1 \cdot y\right)\right)\right)} \cdot z \]

      8. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot y\right)\right)\right) \cdot z \]

      9. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{a \cdot b} + \left(\mathsf{neg}\left(-1 \cdot y\right)\right)\right) \cdot z \]

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

          \[\leadsto \left(a \cdot b + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot y}\right) \cdot z \]

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

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

      12. *-commutative N/A

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

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

          \[\leadsto \color{blue}{\left(a \cdot b + \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right)} \cdot z \]

      14. *-commutative N/A

          \[\leadsto \left(\color{blue}{b \cdot a} + \left(\mathsf{neg}\left(y\right)\right) \cdot -1\right) \cdot z \]

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

          \[\leadsto \left(b \cdot a + \color{blue}{\left(\mathsf{neg}\left(y \cdot -1\right)\right)}\right) \cdot z \]

      16. *-commutative N/A

          \[\leadsto \left(b \cdot a + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right)\right) \cdot z \]

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

          \[\leadsto \left(b \cdot a + \color{blue}{y}\right) \cdot z \]

      19. lower-fma.f64 73.4

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

   5. Applied rewrites 73.4%

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

   if -3.7999999999999996e-74 < y < 1.7500000000000002e-64

   1. Initial program 92.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right) + x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, x\right) \]

      6. lower-fma.f64 95.3

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, x\right) \]

   5. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(b \cdot z, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + a \cdot t} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot t + x} \]

       2. lower-fma.f64 73.5

          \[\leadsto \color{blue}{\mathsf{fma}\left(a, t, x\right)} \]

   11. Applied rewrites 73.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, z, x\right)\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot b, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -4.6e+141)
   (fma (* b a) z x)
   (if (<= b 1.15e+201) (fma a t (fma z y x)) (fma (* z b) a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -4.6e+141) {
		tmp = fma((b * a), z, x);
	} else if (b <= 1.15e+201) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = fma((z * b), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -4.6e+141)
		tmp = fma(Float64(b * a), z, x);
	elseif (b <= 1.15e+201)
		tmp = fma(a, t, fma(z, y, x));
	else
		tmp = fma(Float64(z * b), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -4.6e+141], N[(N[(b * a), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[b, 1.15e+201], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(z * b), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.6000000000000003e141

   1. Initial program 96.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(y + a \cdot b\right)}\right), z, x\right) \]

      9. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot y + -1 \cdot \left(a \cdot b\right)\right)}\right), z, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \left(a \cdot b\right) + -1 \cdot y\right)}\right), z, x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right) + \left(\mathsf{neg}\left(-1 \cdot y\right)\right)}, z, x\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot y\right)\right), z, x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b} + \left(\mathsf{neg}\left(-1 \cdot y\right)\right), z, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(a \cdot b + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot y}, z, x\right) \]

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

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

      16. *-commutative N/A

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

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot b + \left(\mathsf{neg}\left(y\right)\right) \cdot -1}, z, x\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot a} + \left(\mathsf{neg}\left(y\right)\right) \cdot -1, z, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(b \cdot a + \color{blue}{\left(\mathsf{neg}\left(y \cdot -1\right)\right)}, z, x\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b \cdot a + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right), z, x\right) \]

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower-fma.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(a \cdot b, z, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 85.1%

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

   if -4.6000000000000003e141 < b < 1.1500000000000001e201

   1. Initial program 93.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right) + x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, x\right) \]

      6. lower-fma.f64 73.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, x\right) \]

   5. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(b \cdot z, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 49.8%

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

   9. Taylor expanded in b around 0

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

       1. associate-+r+ N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(a, t, z \cdot \color{blue}{\left(1 \cdot y\right)} + x\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, t, z \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot y\right) + x\right) \]

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

          \[\leadsto \mathsf{fma}\left(a, t, z \cdot \color{blue}{\left(\mathsf{neg}\left(-1 \cdot y\right)\right)} + x\right) \]

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

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(-1 \cdot y\right)\right)\right)} + x\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(a, t, \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot y\right) \cdot z}\right)\right) + x\right) \]

       12. mul-1-neg N/A

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

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

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

       14. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(a, t, \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(y \cdot z\right)}\right)\right) + x\right) \]

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

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \left(y \cdot z\right)} + x\right) \]

       16. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{1} \cdot \left(y \cdot z\right) + x\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(y \cdot z\right) + x\right) \]

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

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \left(y \cdot z\right)\right)\right)} + x\right) \]

       19. mul-1-neg N/A

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

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

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

       21. mul-1-neg N/A

           \[\leadsto \mathsf{fma}\left(a, t, \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot y\right)} \cdot z\right)\right) + x\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(a, t, \left(\mathsf{neg}\left(\color{blue}{z \cdot \left(-1 \cdot y\right)}\right)\right) + x\right) \]

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

           \[\leadsto \mathsf{fma}\left(a, t, \color{blue}{z \cdot \left(\mathsf{neg}\left(-1 \cdot y\right)\right)} + x\right) \]

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

           \[\leadsto \mathsf{fma}\left(a, t, z \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot y\right)} + x\right) \]

       25. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(a, t, z \cdot \left(\color{blue}{1} \cdot y\right) + x\right) \]

       26. *-lft-identity N/A

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

   11. Applied rewrites 86.4%

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

   if 1.1500000000000001e201 < b

   1. Initial program 84.2%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right) + x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, x\right) \]

      6. lower-fma.f64 95.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, x\right) \]

   5. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(b \cdot z, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 88.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, z, x\right)\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot b, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+76} \lor \neg \left(a \leq 2.05 \cdot 10^{+39}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -8.2e+76) (not (<= a 2.05e+39))) (* (fma b z t) a) (fma z y x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -8.2e+76) || !(a <= 2.05e+39)) {
		tmp = fma(b, z, t) * a;
	} else {
		tmp = fma(z, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -8.2e+76) || !(a <= 2.05e+39))
		tmp = Float64(fma(b, z, t) * a);
	else
		tmp = fma(z, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -8.2e+76], N[Not[LessEqual[a, 2.05e+39]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision], N[(z * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.1999999999999997e76 or 2.05000000000000002e39 < a

   1. Initial program 85.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot z + t\right)} \cdot a \]

      4. lower-fma.f64 79.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, z, t\right)} \cdot a \]

   5. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, z, t\right) \cdot a} \]

   if -8.1999999999999997e76 < a < 2.05000000000000002e39

   1. Initial program 97.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right) + x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, x\right) \]

      6. lower-fma.f64 66.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, x\right) \]

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(b \cdot z, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 57.5%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{y \cdot z + x} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot y} + x \]

       3. *-lft-identity N/A

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

       4. metadata-eval N/A

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

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

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

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

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

       7. *-commutative N/A

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

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

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

       12. metadata-eval N/A

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

       13. metadata-eval N/A

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

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

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

       15. mul-1-neg N/A

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

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

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

       17. mul-1-neg N/A

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

       18. *-commutative N/A

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

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

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

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

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

       21. metadata-eval N/A

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

       22. *-lft-identity N/A

           \[\leadsto z \cdot \color{blue}{y} + x \]

       23. lower-fma.f64 78.3

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

   11. Applied rewrites 78.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+76} \lor \neg \left(a \leq 2.05 \cdot 10^{+39}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+97} \lor \neg \left(y \leq 1.75 \cdot 10^{-64}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.3e+97) (not (<= y 1.75e-64))) (fma z y x) (fma a t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.3e+97) || !(y <= 1.75e-64)) {
		tmp = fma(z, y, x);
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.3e+97) || !(y <= 1.75e-64))
		tmp = fma(z, y, x);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.3e+97], N[Not[LessEqual[y, 1.75e-64]], $MachinePrecision]], N[(z * y + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e97 or 1.7500000000000002e-64 < y

   1. Initial program 92.6%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right) + x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, x\right) \]

      6. lower-fma.f64 60.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, x\right) \]

   5. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(b \cdot z, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 48.9%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{y \cdot z + x} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot y} + x \]

       3. *-lft-identity N/A

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

       4. metadata-eval N/A

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

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

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

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

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

       7. *-commutative N/A

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

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

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

       12. metadata-eval N/A

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

       13. metadata-eval N/A

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

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

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

       15. mul-1-neg N/A

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

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

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

       17. mul-1-neg N/A

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

       18. *-commutative N/A

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

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

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

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

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

       21. metadata-eval N/A

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

       22. *-lft-identity N/A

           \[\leadsto z \cdot \color{blue}{y} + x \]

       23. lower-fma.f64 76.7

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

   11. Applied rewrites 76.7%

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

   if -1.3e97 < y < 1.7500000000000002e-64

   1. Initial program 93.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right) + x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, x\right) \]

      6. lower-fma.f64 91.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, x\right) \]

   5. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(b \cdot z, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 62.2%

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

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + a \cdot t} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot t + x} \]

       2. lower-fma.f64 68.0

          \[\leadsto \color{blue}{\mathsf{fma}\left(a, t, x\right)} \]

   11. Applied rewrites 68.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+97} \lor \neg \left(y \leq 1.75 \cdot 10^{-64}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+140} \lor \neg \left(y \leq 9 \cdot 10^{+167}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7.2e+140) (not (<= y 9e+167))) (* y z) (fma a t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7.2e+140) || !(y <= 9e+167)) {
		tmp = y * z;
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -7.2e+140) || !(y <= 9e+167))
		tmp = Float64(y * z);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -7.2e+140], N[Not[LessEqual[y, 9e+167]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.1999999999999999e140 or 8.9999999999999998e167 < y

   1. Initial program 86.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \color{blue}{\left(a \cdot z\right) \cdot b} \]

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

         \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) - \left(\mathsf{neg}\left(a \cdot z\right)\right) \cdot b} \]

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

         \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b} \]

      5. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x + y \cdot z\right)} + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

      7. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y \cdot z + x\right)} + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(x + t \cdot a\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

      9. associate-+l+ N/A

         \[\leadsto \color{blue}{y \cdot z + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot z} + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right) \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot y} + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right) \]

      12. remove-double-neg N/A

          \[\leadsto z \cdot y + \left(\left(x + t \cdot a\right) + \color{blue}{\left(a \cdot z\right)} \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto z \cdot y + \left(\left(x + t \cdot a\right) + \color{blue}{\left(a \cdot z\right) \cdot b}\right) \]

      14. associate-+r+ N/A

          \[\leadsto z \cdot y + \color{blue}{\left(x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]

      15. lower-fma.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 65.8

         \[\leadsto \color{blue}{y \cdot z} \]

   7. Applied rewrites 65.8%

      \[\leadsto \color{blue}{y \cdot z} \]

   if -7.1999999999999999e140 < y < 8.9999999999999998e167

   1. Initial program 94.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

         \[\leadsto x + \color{blue}{a \cdot \left(t + b \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right) + x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + b \cdot z\right) \cdot a} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + b \cdot z, a, x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot z + t}, a, x\right) \]

      6. lower-fma.f64 85.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, z, t\right)}, a, x\right) \]

   5. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(b \cdot z, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 61.8%

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

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + a \cdot t} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{a \cdot t + x} \]

       2. lower-fma.f64 65.8

          \[\leadsto \color{blue}{\mathsf{fma}\left(a, t, x\right)} \]

   11. Applied rewrites 65.8%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+140} \lor \neg \left(y \leq 9 \cdot 10^{+167}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+81} \lor \neg \left(y \leq 1.75 \cdot 10^{-64}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3.8e+81) (not (<= y 1.75e-64))) (* y z) (* a t)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.8e+81) || !(y <= 1.75e-64)) {
		tmp = y * z;
	} else {
		tmp = a * t;
	}
	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, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-3.8d+81)) .or. (.not. (y <= 1.75d-64))) then
        tmp = y * z
    else
        tmp = a * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.8e+81) || !(y <= 1.75e-64)) {
		tmp = y * z;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3.8e+81) or not (y <= 1.75e-64):
		tmp = y * z
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3.8e+81) || !(y <= 1.75e-64))
		tmp = Float64(y * z);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3.8e+81) || ~((y <= 1.75e-64)))
		tmp = y * z;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3.8e+81], N[Not[LessEqual[y, 1.75e-64]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(a * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8e81 or 1.7500000000000002e-64 < y

   1. Initial program 92.8%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \color{blue}{\left(a \cdot z\right) \cdot b} \]

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

         \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) - \left(\mathsf{neg}\left(a \cdot z\right)\right) \cdot b} \]

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

         \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b} \]

      5. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x + y \cdot z\right)} + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

      7. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(y \cdot z + x\right)} + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(y \cdot z + \left(x + t \cdot a\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

      9. associate-+l+ N/A

         \[\leadsto \color{blue}{y \cdot z + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot z} + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right) \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{z \cdot y} + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right) \]

      12. remove-double-neg N/A

          \[\leadsto z \cdot y + \left(\left(x + t \cdot a\right) + \color{blue}{\left(a \cdot z\right)} \cdot b\right) \]

      13. lift-*.f64 N/A

          \[\leadsto z \cdot y + \left(\left(x + t \cdot a\right) + \color{blue}{\left(a \cdot z\right) \cdot b}\right) \]

      14. associate-+r+ N/A

          \[\leadsto z \cdot y + \color{blue}{\left(x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]

      15. lower-fma.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 45.2

         \[\leadsto \color{blue}{y \cdot z} \]

   7. Applied rewrites 45.2%

      \[\leadsto \color{blue}{y \cdot z} \]

   if -3.8e81 < y < 1.7500000000000002e-64

   1. Initial program 93.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 33.0

         \[\leadsto \color{blue}{a \cdot t} \]

   5. Applied rewrites 33.0%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+81} \lor \neg \left(y \leq 1.75 \cdot 10^{-64}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.1% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (* y z))

C

double code(double x, double y, double z, double t, double a, double b) {
	return y * 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, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = y * z
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return y * z;
}

Python

def code(x, y, z, t, a, b):
	return y * z

Julia

function code(x, y, z, t, a, b)
	return Float64(y * z)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = y * z;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(y * z), $MachinePrecision]

Derivation

1. Initial program 92.9%

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

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \color{blue}{\left(a \cdot z\right) \cdot b} \]

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

      \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) - \left(\mathsf{neg}\left(a \cdot z\right)\right) \cdot b} \]

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

      \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b} \]

   5. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x + y \cdot z\right) + t \cdot a\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

   6. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x + y \cdot z\right)} + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

   7. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(y \cdot z + x\right)} + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

   8. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(y \cdot z + \left(x + t \cdot a\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b \]

   9. associate-+l+ N/A

      \[\leadsto \color{blue}{y \cdot z + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \color{blue}{y \cdot z} + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right) \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{z \cdot y} + \left(\left(x + t \cdot a\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot z\right)\right)\right)\right) \cdot b\right) \]

   12. remove-double-neg N/A

       \[\leadsto z \cdot y + \left(\left(x + t \cdot a\right) + \color{blue}{\left(a \cdot z\right)} \cdot b\right) \]

   13. lift-*.f64 N/A

       \[\leadsto z \cdot y + \left(\left(x + t \cdot a\right) + \color{blue}{\left(a \cdot z\right) \cdot b}\right) \]

   14. associate-+r+ N/A

       \[\leadsto z \cdot y + \color{blue}{\left(x + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)\right)} \]

   15. lower-fma.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. associate-*l* N/A

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in y around inf

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

   1. lower-*.f64 26.2

      \[\leadsto \color{blue}{y \cdot z} \]

7. Applied rewrites 26.2%

   \[\leadsto \color{blue}{y \cdot z} \]
8. Add Preprocessing

Developer Target 1: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* z (+ (* b a) y)) (+ x (* t a)))))
   (if (< z -11820553527347888000.0)
     t_1
     (if (< z 4.7589743188364287e-122)
       (+ (* (+ (* b z) t) a) (+ (* z y) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * ((b * a) + y)) + (x + (t * a))
    if (z < (-11820553527347888000.0d0)) then
        tmp = t_1
    else if (z < 4.7589743188364287d-122) then
        tmp = (((b * z) + t) * a) + ((z * y) + x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z * ((b * a) + y)) + (x + (t * a))
	tmp = 0
	if z < -11820553527347888000.0:
		tmp = t_1
	elif z < 4.7589743188364287e-122:
		tmp = (((b * z) + t) * a) + ((z * y) + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * Float64(Float64(b * a) + y)) + Float64(x + Float64(t * a)))
	tmp = 0.0
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = Float64(Float64(Float64(Float64(b * z) + t) * a) + Float64(Float64(z * y) + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z * ((b * a) + y)) + (x + (t * a));
	tmp = 0.0;
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * N[(N[(b * a), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -11820553527347888000.0], t$95$1, If[Less[z, 4.7589743188364287e-122], N[(N[(N[(N[(b * z), $MachinePrecision] + t), $MachinePrecision] * a), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024350 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -11820553527347888000) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 47589743188364287/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a))))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))