Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.7% → 95.1%

Time: 8.2s

Alternatives: 12

Speedup: 0.5×

• 33.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(b \cdot z, a, \mathsf{fma}\left(z, y, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)) INFINITY)
   (fma t a (fma (* b z) a (fma z y x)))
   (* (fma b a y) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((x + (y * z)) + (t * a)) + ((a * z) * b)) <= ((double) INFINITY)) {
		tmp = fma(t, a, fma((b * z), a, fma(z, y, x)));
	} else {
		tmp = fma(b, a, y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(x + Float64(y * z)) + Float64(t * a)) + Float64(Float64(a * z) * b)) <= Inf)
		tmp = fma(t, a, fma(Float64(b * z), a, fma(z, y, x)));
	else
		tmp = Float64(fma(b, a, y) * z);
	end
	return tmp
end

Wolfram

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

   1. Initial program 98.1%

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

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in 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 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 98.4%

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

Alternative 2: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{if}\;a \leq -1.22 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b z t) a)))
   (if (<= a -1.22e+49)
     t_1
     (if (<= a -2e-112) (fma t a x) (if (<= a 8.5e-77) (fma z y x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, z, t) * a;
	double tmp;
	if (a <= -1.22e+49) {
		tmp = t_1;
	} else if (a <= -2e-112) {
		tmp = fma(t, a, x);
	} else if (a <= 8.5e-77) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(b, z, t) * a)
	tmp = 0.0
	if (a <= -1.22e+49)
		tmp = t_1;
	elseif (a <= -2e-112)
		tmp = fma(t, a, x);
	elseif (a <= 8.5e-77)
		tmp = fma(z, y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -1.22e+49], t$95$1, If[LessEqual[a, -2e-112], N[(t * a + x), $MachinePrecision], If[LessEqual[a, 8.5e-77], N[(z * y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.21999999999999988e49 or 8.4999999999999998e-77 < a

   1. Initial program 90.4%

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

   3. Taylor expanded in 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 76.9

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

   5. Applied rewrites 76.9%

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

   if -1.21999999999999988e49 < a < -1.9999999999999999e-112

   1. Initial program 100.0%

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

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 100.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 76.4

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

   7. Applied rewrites 76.4%

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

   if -1.9999999999999999e-112 < a < 8.4999999999999998e-77

   1. Initial program 98.7%

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

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.1

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{fma}\left(t, a, \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{a \cdot \left(b \cdot z\right) + y \cdot z}{x} - 1\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

   7. Applied rewrites 90.3%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 88.2

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

   10. Applied rewrites 88.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.22 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\left(z \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.9e-11)
   (fma t a x)
   (if (<= t 6e-34) (fma z y x) (if (<= t 4.5e-7) (* (* z b) a) (fma t a x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.9e-11) {
		tmp = fma(t, a, x);
	} else if (t <= 6e-34) {
		tmp = fma(z, y, x);
	} else if (t <= 4.5e-7) {
		tmp = (z * b) * a;
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.9e-11)
		tmp = fma(t, a, x);
	elseif (t <= 6e-34)
		tmp = fma(z, y, x);
	elseif (t <= 4.5e-7)
		tmp = Float64(Float64(z * b) * a);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.9e-11], N[(t * a + x), $MachinePrecision], If[LessEqual[t, 6e-34], N[(z * y + x), $MachinePrecision], If[LessEqual[t, 4.5e-7], N[(N[(z * b), $MachinePrecision] * a), $MachinePrecision], N[(t * a + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.8999999999999999e-11 or 4.4999999999999998e-7 < t

   1. Initial program 94.5%

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

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 76.0

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

   7. Applied rewrites 76.0%

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

   if -1.8999999999999999e-11 < t < 6e-34

   1. Initial program 93.3%

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

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 94.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{fma}\left(t, a, \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{a \cdot \left(b \cdot z\right) + y \cdot z}{x} - 1\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

   7. Applied rewrites 91.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 64.3

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

   10. Applied rewrites 64.3%

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

   if 6e-34 < t < 4.4999999999999998e-7

   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. 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 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 75.5%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\left(z \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+63} \lor \neg \left(b \leq 9.4 \cdot 10^{+30}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -2.6e+63) (not (<= b 9.4e+30)))
   (fma (fma b z t) a x)
   (fma t a (fma y z x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -2.6e+63) || !(b <= 9.4e+30)) {
		tmp = fma(fma(b, z, t), a, x);
	} else {
		tmp = fma(t, a, fma(y, z, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -2.6e+63) || !(b <= 9.4e+30))
		tmp = fma(fma(b, z, t), a, x);
	else
		tmp = fma(t, a, fma(y, z, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -2.6e+63], N[Not[LessEqual[b, 9.4e+30]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], N[(t * a + N[(y * z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.6000000000000001e63 or 9.39999999999999979e30 < b

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if -2.6000000000000001e63 < b < 9.39999999999999979e30

   1. Initial program 94.9%

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

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.3

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 92.0

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

   7. Applied rewrites 92.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+63} \lor \neg \left(b \leq 9.4 \cdot 10^{+30}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-103} \lor \neg \left(t \leq 1.5 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -6.5e-103) (not (<= t 1.5e-10)))
   (fma t a (fma y z x))
   (fma (fma b a y) z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -6.5e-103) || !(t <= 1.5e-10)) {
		tmp = fma(t, a, fma(y, z, x));
	} else {
		tmp = fma(fma(b, a, y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -6.5e-103) || !(t <= 1.5e-10))
		tmp = fma(t, a, fma(y, z, x));
	else
		tmp = fma(fma(b, a, y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -6.5e-103], N[Not[LessEqual[t, 1.5e-10]], $MachinePrecision]], N[(t * a + N[(y * z + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.49999999999999966e-103 or 1.5e-10 < t

   1. Initial program 93.9%

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

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 87.4

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

   7. Applied rewrites 87.4%

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

   if -6.49999999999999966e-103 < t < 1.5e-10

   1. Initial program 94.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-lft-out N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

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

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

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

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

      16. *-commutative N/A

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

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

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

      21. mul-1-neg N/A

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

      22. remove-double-neg N/A

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

      23. lower-fma.f64 94.0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-103} \lor \neg \left(t \leq 1.5 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{+129} \lor \neg \left(a \leq 2.8 \cdot 10^{+37}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -9.2e+129) (not (<= a 2.8e+37)))
   (* (fma b z t) a)
   (fma 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 <= -9.2e+129) || !(a <= 2.8e+37)) {
		tmp = fma(b, z, t) * a;
	} else {
		tmp = fma(t, a, fma(y, z, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -9.2e+129) || !(a <= 2.8e+37))
		tmp = Float64(fma(b, z, t) * a);
	else
		tmp = fma(t, a, fma(y, z, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -9.2e+129], N[Not[LessEqual[a, 2.8e+37]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision], N[(t * a + N[(y * z + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.19999999999999961e129 or 2.7999999999999998e37 < a

   1. Initial program 86.4%

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

   3. Taylor expanded in 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 85.0

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

   5. Applied rewrites 85.0%

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

   if -9.19999999999999961e129 < a < 2.7999999999999998e37

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 97.6

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{+129} \lor \neg \left(a \leq 2.8 \cdot 10^{+37}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \mathbf{elif}\;a \leq 12.5:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.55e+48)
   (fma (fma b z t) a (* z y))
   (if (<= a 12.5) (fma t a (fma 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 (a <= -1.55e+48) {
		tmp = fma(fma(b, z, t), a, (z * y));
	} else if (a <= 12.5) {
		tmp = fma(t, a, fma(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 (a <= -1.55e+48)
		tmp = fma(fma(b, z, t), a, Float64(z * y));
	elseif (a <= 12.5)
		tmp = fma(t, a, fma(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[LessEqual[a, -1.55e+48], N[(N[(b * z + t), $MachinePrecision] * a + N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 12.5], N[(t * a + N[(y * z + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.55000000000000003e48

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -1.55000000000000003e48 < a < 12.5

   1. Initial program 99.2%

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

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 92.4

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

   7. Applied rewrites 92.4%

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

   if 12.5 < a

   1. Initial program 84.1%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \mathbf{elif}\;a \leq 12.5:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+55} \lor \neg \left(z \leq 9.8 \cdot 10^{+23}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3.5e+55) (not (<= z 9.8e+23))) (* (fma b a y) z) (fma t a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3.5e+55) || !(z <= 9.8e+23)) {
		tmp = fma(b, a, y) * z;
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -3.5e+55) || !(z <= 9.8e+23))
		tmp = Float64(fma(b, a, y) * z);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -3.5e+55], N[Not[LessEqual[z, 9.8e+23]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], N[(t * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.5000000000000001e55 or 9.8000000000000006e23 < z

   1. Initial program 87.8%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 78.6

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

   5. Applied rewrites 78.6%

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

   if -3.5000000000000001e55 < z < 9.8000000000000006e23

   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. 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

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.3

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 76.4

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

   7. Applied rewrites 76.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+55} \lor \neg \left(z \leq 9.8 \cdot 10^{+23}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-11} \lor \neg \left(t \leq 8 \cdot 10^{+40}\right):\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.9e-11) (not (<= t 8e+40))) (fma t a x) (fma z y x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.9e-11) || !(t <= 8e+40)) {
		tmp = fma(t, a, x);
	} else {
		tmp = fma(z, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.9e-11) || !(t <= 8e+40))
		tmp = fma(t, a, x);
	else
		tmp = fma(z, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.9e-11], N[Not[LessEqual[t, 8e+40]], $MachinePrecision]], N[(t * a + x), $MachinePrecision], N[(z * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8999999999999999e-11 or 8.00000000000000024e40 < t

   1. Initial program 94.2%

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

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.6

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 77.6

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

   7. Applied rewrites 77.6%

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

   if -1.8999999999999999e-11 < t < 8.00000000000000024e40

   1. Initial program 94.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{fma}\left(t, a, \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{a \cdot \left(b \cdot z\right) + y \cdot z}{x} - 1\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

   7. Applied rewrites 89.6%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 61.3

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

   10. Applied rewrites 61.3%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-11} \lor \neg \left(t \leq 8 \cdot 10^{+40}\right):\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{+112} \lor \neg \left(z \leq 9.5 \cdot 10^{+185}\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.35e+112) (not (<= z 9.5e+185))) (* z y) (fma t a x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.35e+112) || !(z <= 9.5e+185)) {
		tmp = z * y;
	} else {
		tmp = fma(t, a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.35e+112) || !(z <= 9.5e+185))
		tmp = Float64(z * y);
	else
		tmp = fma(t, a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.35e+112], N[Not[LessEqual[z, 9.5e+185]], $MachinePrecision]], N[(z * y), $MachinePrecision], N[(t * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.34999999999999999e112 or 9.4999999999999995e185 < z

   1. Initial program 83.9%

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

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.4

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{fma}\left(t, a, \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{a \cdot \left(b \cdot z\right) + y \cdot z}{x} - 1\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

   7. Applied rewrites 88.6%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 58.6

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

   10. Applied rewrites 58.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 48.0%

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

   if -2.34999999999999999e112 < z < 9.4999999999999995e185

   1. Initial program 97.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.4

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 70.2

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

   7. Applied rewrites 70.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{+112} \lor \neg \left(z \leq 9.5 \cdot 10^{+185}\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-11} \lor \neg \left(t \leq 8 \cdot 10^{+40}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.9e-11) (not (<= t 8e+40))) (* a t) (* z y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.9e-11) || !(t <= 8e+40)) {
		tmp = a * t;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, 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 ((t <= (-1.9d-11)) .or. (.not. (t <= 8d+40))) then
        tmp = a * t
    else
        tmp = z * y
    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 ((t <= -1.9e-11) || !(t <= 8e+40)) {
		tmp = a * t;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -1.9e-11) or not (t <= 8e+40):
		tmp = a * t
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.9e-11) || !(t <= 8e+40))
		tmp = Float64(a * t);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -1.9e-11) || ~((t <= 8e+40)))
		tmp = a * t;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.9e-11], N[Not[LessEqual[t, 8e+40]], $MachinePrecision]], N[(a * t), $MachinePrecision], N[(z * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8999999999999999e-11 or 8.00000000000000024e40 < t

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 55.7

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

   5. Applied rewrites 55.7%

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

   if -1.8999999999999999e-11 < t < 8.00000000000000024e40

   1. Initial program 94.3%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{fma}\left(t, a, \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{a \cdot \left(b \cdot z\right) + y \cdot z}{x} - 1\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

   7. Applied rewrites 89.6%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 61.3

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

   10. Applied rewrites 61.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 29.7%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-11} \lor \neg \left(t \leq 8 \cdot 10^{+40}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 28.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, 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 = z * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 96.1

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 96.1

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

4. Applied rewrites 96.1%

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

5. Taylor expanded in x around -inf

   \[\leadsto \mathsf{fma}\left(t, a, \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{a \cdot \left(b \cdot z\right) + y \cdot z}{x} - 1\right)\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. div-add-rev N/A

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

   6. lower--.f64 N/A

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

7. Applied rewrites 87.6%

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

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. *-commutative N/A

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

   3. lower-fma.f64 50.4

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

10. Applied rewrites 50.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 23.6%

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

Developer Target 1: 97.4% 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 2024363 
(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)))