Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.3% → 96.9%

Time: 7.5s

Alternatives: 13

Speedup: 1.4×

• 32.3% 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: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+189} \lor \neg \left(z \leq 1.35 \cdot 10^{+140}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.9e+189) (not (<= z 1.35e+140)))
   (fma (fma b a y) z x)
   (fma a (fma b z t) (fma z y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.9e+189) || !(z <= 1.35e+140)) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(a, fma(b, z, t), fma(z, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.9e+189) || !(z <= 1.35e+140))
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(a, fma(b, z, t), fma(z, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.9e+189], N[Not[LessEqual[z, 1.35e+140]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(a * N[(b * z + t), $MachinePrecision] + N[(z * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8999999999999999e189 or 1.35000000000000009e140 < z

   1. Initial program 73.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 96.4

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

   5. Applied rewrites 96.4%

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

   if -1.8999999999999999e189 < z < 1.35000000000000009e140

   1. Initial program 95.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 98.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 98.5

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

   4. Applied rewrites 98.5%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+189} \lor \neg \left(z \leq 1.35 \cdot 10^{+140}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(b, z, t\right), \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 5 \cdot 10^{+279}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+279)))
     (* (fma b a y) z)
     (fma y z x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x + (y * z)) + (t * a)) + ((a * z) * b);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+279)) {
		tmp = fma(b, a, y) * z;
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(a * z) * b))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+279))
		tmp = Float64(fma(b, a, y) * z);
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+279]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], N[(y * z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < -inf.0 or 5.0000000000000002e279 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b))

   1. Initial program 72.9%

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

   3. Taylor expanded in 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 75.5

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

   5. Applied rewrites 75.5%

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

   if -inf.0 < (+.f64 (+.f64 (+.f64 x (*.f64 y z)) (*.f64 t a)) (*.f64 (*.f64 a z) b)) < 5.0000000000000002e279

   1. Initial program 99.9%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 95.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 86.4

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

   7. Applied rewrites 86.4%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 61.7

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

   10. Applied rewrites 61.7%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \leq -\infty \lor \neg \left(\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \leq 5 \cdot 10^{+279}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+224}:\\ \;\;\;\;\left(z \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{+96}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.2e+224)
   (* (* z b) a)
   (if (<= a -7.5e+96) (* a t) (if (<= a 7e+152) (fma y z x) (* (* b a) z)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.2e+224) {
		tmp = (z * b) * a;
	} else if (a <= -7.5e+96) {
		tmp = a * t;
	} else if (a <= 7e+152) {
		tmp = fma(y, z, x);
	} else {
		tmp = (b * a) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.2e+224)
		tmp = Float64(Float64(z * b) * a);
	elseif (a <= -7.5e+96)
		tmp = Float64(a * t);
	elseif (a <= 7e+152)
		tmp = fma(y, z, x);
	else
		tmp = Float64(Float64(b * a) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.2e+224], N[(N[(z * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, -7.5e+96], N[(a * t), $MachinePrecision], If[LessEqual[a, 7e+152], N[(y * z + x), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.2e224

   1. Initial program 51.2%

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

   3. Taylor expanded in 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 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 85.9%

      \[\leadsto \left(z \cdot b\right) \cdot a \]

   if -1.2e224 < a < -7.4999999999999996e96

   1. Initial program 90.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 51.6

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

   5. Applied rewrites 51.6%

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

   if -7.4999999999999996e96 < a < 6.99999999999999963e152

   1. Initial program 96.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 86.3

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 65.2

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

   10. Applied rewrites 65.2%

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

   if 6.99999999999999963e152 < a

   1. Initial program 79.8%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 59.5

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

   5. Applied rewrites 59.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 54.9%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+224}:\\ \;\;\;\;\left(z \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{+96}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot a\right) \cdot z\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{+224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{+96}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* b a) z)))
   (if (<= a -1.2e+224)
     t_1
     (if (<= a -7.5e+96) (* a t) (if (<= a 7e+152) (fma y z x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * a) * z;
	double tmp;
	if (a <= -1.2e+224) {
		tmp = t_1;
	} else if (a <= -7.5e+96) {
		tmp = a * t;
	} else if (a <= 7e+152) {
		tmp = fma(y, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * a) * z)
	tmp = 0.0
	if (a <= -1.2e+224)
		tmp = t_1;
	elseif (a <= -7.5e+96)
		tmp = Float64(a * t);
	elseif (a <= 7e+152)
		tmp = fma(y, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * a), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[a, -1.2e+224], t$95$1, If[LessEqual[a, -7.5e+96], N[(a * t), $MachinePrecision], If[LessEqual[a, 7e+152], N[(y * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.2e224 or 6.99999999999999963e152 < a

   1. Initial program 70.9%

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

   3. Taylor expanded in 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 66.4

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.8%

      \[\leadsto \left(b \cdot a\right) \cdot z \]

   if -1.2e224 < a < -7.4999999999999996e96

   1. Initial program 90.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 51.6

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

   5. Applied rewrites 51.6%

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

   if -7.4999999999999996e96 < a < 6.99999999999999963e152

   1. Initial program 96.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 86.3

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 65.2

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

   10. Applied rewrites 65.2%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+224}:\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{+96}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{+47} \lor \neg \left(y \leq 1.95 \cdot 10^{-16}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.35e+47) (not (<= y 1.95e-16)))
   (fma z y (fma t a x))
   (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 ((y <= -2.35e+47) || !(y <= 1.95e-16)) {
		tmp = fma(z, y, fma(t, a, x));
	} else {
		tmp = fma(fma(b, z, t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.35e+47) || !(y <= 1.95e-16))
		tmp = fma(z, y, fma(t, a, x));
	else
		tmp = fma(fma(b, z, t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.35e+47], N[Not[LessEqual[y, 1.95e-16]], $MachinePrecision]], N[(z * y + N[(t * a + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.34999999999999982e47 or 1.94999999999999989e-16 < 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 96.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if -2.34999999999999982e47 < y < 1.94999999999999989e-16

   1. Initial program 92.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{+47} \lor \neg \left(y \leq 1.95 \cdot 10^{-16}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(t, a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -14000000 \lor \neg \left(z \leq 9.2 \cdot 10^{+82}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -14000000.0) (not (<= z 9.2e+82)))
   (fma (fma b a y) z x)
   (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 ((z <= -14000000.0) || !(z <= 9.2e+82)) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(fma(b, z, t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -14000000.0) || !(z <= 9.2e+82))
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(fma(b, z, t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -14000000.0], N[Not[LessEqual[z, 9.2e+82]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4e7 or 9.19999999999999953e82 < z

   1. Initial program 78.0%

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

   3. Taylor expanded in 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 89.3

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

   5. Applied rewrites 89.3%

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

   if -1.4e7 < z < 9.19999999999999953e82

   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 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 85.7

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

   5. Applied rewrites 85.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -14000000 \lor \neg \left(z \leq 9.2 \cdot 10^{+82}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+105} \lor \neg \left(t \leq 1.4 \cdot 10^{+21}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -5.8e+105) (not (<= t 1.4e+21)))
   (fma z y (* t a))
   (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 <= -5.8e+105) || !(t <= 1.4e+21)) {
		tmp = fma(z, y, (t * a));
	} 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 <= -5.8e+105) || !(t <= 1.4e+21))
		tmp = fma(z, y, Float64(t * a));
	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, -5.8e+105], N[Not[LessEqual[t, 1.4e+21]], $MachinePrecision]], N[(z * y + N[(t * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.8000000000000002e105 or 1.4e21 < t

   1. Initial program 83.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 96.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 88.2

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 76.0%

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

   if -5.8000000000000002e105 < t < 1.4e21

   1. Initial program 93.7%

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

   3. Taylor expanded in 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 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+105} \lor \neg \left(t \leq 1.4 \cdot 10^{+21}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-19} \lor \neg \left(a \leq 1.22 \cdot 10^{+51}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.45e-19) (not (<= a 1.22e+51)))
   (* (fma b z t) a)
   (fma y z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.45e-19) || !(a <= 1.22e+51)) {
		tmp = fma(b, z, t) * a;
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.45e-19) || !(a <= 1.22e+51))
		tmp = Float64(fma(b, z, t) * a);
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.45e-19], N[Not[LessEqual[a, 1.22e+51]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision], N[(y * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45e-19 or 1.22000000000000005e51 < a

   1. Initial program 81.0%

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

   3. Taylor expanded in 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 77.5

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

   5. Applied rewrites 77.5%

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

   if -1.45e-19 < a < 1.22000000000000005e51

   1. Initial program 98.4%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 93.2%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 72.9

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

   10. Applied rewrites 72.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-19} \lor \neg \left(a \leq 1.22 \cdot 10^{+51}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+99} \lor \neg \left(a \leq 3.5 \cdot 10^{+152}\right):\\ \;\;\;\;\left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -4.5e+99) (not (<= a 3.5e+152))) (* (* a z) b) (fma y z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -4.5e+99) || !(a <= 3.5e+152)) {
		tmp = (a * z) * b;
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -4.5e+99) || !(a <= 3.5e+152))
		tmp = Float64(Float64(a * z) * b);
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -4.5e+99], N[Not[LessEqual[a, 3.5e+152]], $MachinePrecision]], N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision], N[(y * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e99 or 3.49999999999999981e152 < a

   1. Initial program 75.2%

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

   3. Taylor expanded in 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 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 52.9%

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

   if -4.5e99 < a < 3.49999999999999981e152

   1. Initial program 96.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 86.3

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 64.9

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

   10. Applied rewrites 64.9%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+99} \lor \neg \left(a \leq 3.5 \cdot 10^{+152}\right):\\ \;\;\;\;\left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.7% 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 89.6%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
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 95.9%

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

Alternative 11: 59.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+167} \lor \neg \left(t \leq 1.3 \cdot 10^{+109}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -6.5e+167) (not (<= t 1.3e+109))) (* a t) (fma y z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -6.5e+167) || !(t <= 1.3e+109)) {
		tmp = a * t;
	} else {
		tmp = fma(y, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -6.5e+167) || !(t <= 1.3e+109))
		tmp = Float64(a * t);
	else
		tmp = fma(y, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -6.5e+167], N[Not[LessEqual[t, 1.3e+109]], $MachinePrecision]], N[(a * t), $MachinePrecision], N[(y * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5e167 or 1.2999999999999999e109 < t

   1. Initial program 82.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if -6.5e167 < t < 1.2999999999999999e109

   1. Initial program 92.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 94.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 70.5

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 56.5

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

   10. Applied rewrites 56.5%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+167} \lor \neg \left(t \leq 1.3 \cdot 10^{+109}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+140} \lor \neg \left(y \leq 0.75\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.8e+140) (not (<= y 0.75))) (* y z) (* a t)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.8e+140) || !(y <= 0.75)) {
		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 <= (-1.8d+140)) .or. (.not. (y <= 0.75d0))) 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 <= -1.8e+140) || !(y <= 0.75)) {
		tmp = y * z;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.8e+140) or not (y <= 0.75):
		tmp = y * z
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.8e+140) || !(y <= 0.75))
		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 <= -1.8e+140) || ~((y <= 0.75)))
		tmp = y * z;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.8e+140], N[Not[LessEqual[y, 0.75]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(a * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e140 or 0.75 < y

   1. Initial program 87.5%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 96.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 89.5

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 53.3

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

   10. Applied rewrites 53.3%

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

   if -1.8e140 < y < 0.75

   1. Initial program 90.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 33.5

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

   5. Applied rewrites 33.5%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+140} \lor \neg \left(y \leq 0.75\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.5% 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 89.6%

   \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
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 95.9%

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

5. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 75.6

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

7. Applied rewrites 75.6%

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

8. Taylor expanded in y around inf

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

   1. lower-*.f64 26.3

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

10. Applied rewrites 26.3%

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

Developer Target 1: 97.0% 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 2024358 
(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)))