Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.6% → 97.6%

Time: 3.9s

Alternatives: 13

Speedup: 0.9×

• 32.4% 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: 92.6% 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: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{elif}\;a \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right) + \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, z, \frac{\mathsf{fma}\left(z, y, x\right)}{a}\right) + t\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.1e+171)
   (fma (fma b z t) a x)
   (if (<= a 0.05)
     (+ (fma a t x) (* (fma b a y) z))
     (* (+ (fma b z (/ (fma z y x) a)) t) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.1e+171) {
		tmp = fma(fma(b, z, t), a, x);
	} else if (a <= 0.05) {
		tmp = fma(a, t, x) + (fma(b, a, y) * z);
	} else {
		tmp = (fma(b, z, (fma(z, y, x) / a)) + t) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.1e+171)
		tmp = fma(fma(b, z, t), a, x);
	elseif (a <= 0.05)
		tmp = Float64(fma(a, t, x) + Float64(fma(b, a, y) * z));
	else
		tmp = Float64(Float64(fma(b, z, Float64(fma(z, y, x) / a)) + t) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.1e+171], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[a, 0.05], N[(N[(a * t + x), $MachinePrecision] + N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * z + N[(N[(z * y + x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.1000000000000001e171

   1. Initial program 92.6%

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

   2. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 75.2

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

   4. Applied rewrites 75.2%

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

   if -2.1000000000000001e171 < a < 0.050000000000000003

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 94.3

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

   4. Applied rewrites 94.3%

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

   if 0.050000000000000003 < a

   1. Initial program 92.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 82.7

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

   4. Applied rewrites 82.7%

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

Alternative 2: 95.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{elif}\;a \leq 1.12 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right) + \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.1e+171)
   (fma (fma b z t) a x)
   (if (<= a 1.12e+61)
     (+ (fma a t x) (* (fma b a y) z))
     (fma (fma b z t) a (* z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.1e+171) {
		tmp = fma(fma(b, z, t), a, x);
	} else if (a <= 1.12e+61) {
		tmp = fma(a, t, x) + (fma(b, a, y) * z);
	} else {
		tmp = fma(fma(b, z, t), a, (z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.1e+171)
		tmp = fma(fma(b, z, t), a, x);
	elseif (a <= 1.12e+61)
		tmp = Float64(fma(a, t, x) + Float64(fma(b, a, y) * z));
	else
		tmp = fma(fma(b, z, t), a, Float64(z * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.1e+171], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[a, 1.12e+61], N[(N[(a * t + x), $MachinePrecision] + N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.1000000000000001e171

   1. Initial program 92.6%

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

   2. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 75.2

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

   4. Applied rewrites 75.2%

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

   if -2.1000000000000001e171 < a < 1.12e61

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 94.3

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

   4. Applied rewrites 94.3%

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

   if 1.12e61 < a

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 70.7

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

   4. Applied rewrites 70.7%

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

Alternative 3: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.8e-20)
   (fma (fma b z t) a x)
   (if (<= a 2e-22) (fma (fma b a y) z x) (fma (fma b z t) a (* z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.8e-20) {
		tmp = fma(fma(b, z, t), a, x);
	} else if (a <= 2e-22) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(fma(b, z, t), a, (z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.8e-20)
		tmp = fma(fma(b, z, t), a, x);
	elseif (a <= 2e-22)
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(fma(b, z, t), a, Float64(z * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.8e-20], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[a, 2e-22], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.8000000000000003e-20

   1. Initial program 92.6%

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

   2. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 75.2

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

   4. Applied rewrites 75.2%

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

   if -2.8000000000000003e-20 < a < 2.0000000000000001e-22

   1. Initial program 92.6%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 73.9

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

   4. Applied rewrites 73.9%

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

   if 2.0000000000000001e-22 < a

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-out N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 70.7

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

   4. Applied rewrites 70.7%

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

Alternative 4: 87.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{if}\;a \leq -2.8 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (fma b z t) a x)))
   (if (<= a -2.8e-20) t_1 (if (<= a 9e+15) (fma (fma b a y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(fma(b, z, t), a, x);
	double tmp;
	if (a <= -2.8e-20) {
		tmp = t_1;
	} else if (a <= 9e+15) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(fma(b, z, t), a, x)
	tmp = 0.0
	if (a <= -2.8e-20)
		tmp = t_1;
	elseif (a <= 9e+15)
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[a, -2.8e-20], t$95$1, If[LessEqual[a, 9e+15], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.8000000000000003e-20 or 9e15 < a

   1. Initial program 92.6%

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

   2. Taylor expanded in y around 0

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

      1. distribute-lft-out N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 75.2

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

   4. Applied rewrites 75.2%

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

   if -2.8000000000000003e-20 < a < 9e15

   1. Initial program 92.6%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 73.9

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

   4. Applied rewrites 73.9%

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

Alternative 5: 86.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (fma b a y) z x)))
   (if (<= z -4.6e+18) t_1 (if (<= z 1.45e-63) (fma a t (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(fma(b, a, y), z, x);
	double tmp;
	if (z <= -4.6e+18) {
		tmp = t_1;
	} else if (z <= 1.45e-63) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(fma(b, a, y), z, x)
	tmp = 0.0
	if (z <= -4.6e+18)
		tmp = t_1;
	elseif (z <= 1.45e-63)
		tmp = fma(a, t, 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 * a + y), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -4.6e+18], t$95$1, If[LessEqual[z, 1.45e-63], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6e18 or 1.44999999999999987e-63 < z

   1. Initial program 92.6%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 73.9

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

   4. Applied rewrites 73.9%

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

   if -4.6e18 < z < 1.44999999999999987e-63

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 94.3

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

   4. Applied rewrites 94.3%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-fma.f64 78.4

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

   7. Applied rewrites 78.4%

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

Alternative 6: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot a, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -8.5e+78)
   (* (fma b z t) a)
   (if (<= b 3.6e+148) (fma a t (fma z y x)) (fma (* b a) z x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -8.5e+78) {
		tmp = fma(b, z, t) * a;
	} else if (b <= 3.6e+148) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = fma((b * a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -8.5e+78)
		tmp = Float64(fma(b, z, t) * a);
	elseif (b <= 3.6e+148)
		tmp = fma(a, t, fma(z, y, x));
	else
		tmp = fma(Float64(b * a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -8.5e+78], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[b, 3.6e+148], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * z + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.50000000000000079e78

   1. Initial program 92.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -8.50000000000000079e78 < b < 3.60000000000000006e148

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 94.3

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

   4. Applied rewrites 94.3%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-fma.f64 78.4

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

   7. Applied rewrites 78.4%

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

   if 3.60000000000000006e148 < b

   1. Initial program 92.6%

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

   2. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.2

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

   7. Applied rewrites 50.2%

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

Alternative 7: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b z t) a)))
   (if (<= a -7.5e-26) t_1 (if (<= a 4e-14) (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 <= -7.5e-26) {
		tmp = t_1;
	} else if (a <= 4e-14) {
		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 <= -7.5e-26)
		tmp = t_1;
	elseif (a <= 4e-14)
		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, -7.5e-26], t$95$1, If[LessEqual[a, 4e-14], N[(z * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.4999999999999994e-26 or 4e-14 < a

   1. Initial program 92.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -7.4999999999999994e-26 < a < 4e-14

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto z \cdot y + x \]

      3. lower-fma.f64 52.0

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

   7. Applied rewrites 52.0%

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

Alternative 8: 74.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{if}\;z \leq -2 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.38 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b a y) z)))
   (if (<= z -2e+42) t_1 (if (<= z 1.38e-61) (fma a t x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, a, y) * z;
	double tmp;
	if (z <= -2e+42) {
		tmp = t_1;
	} else if (z <= 1.38e-61) {
		tmp = fma(a, t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(b, a, y) * z)
	tmp = 0.0
	if (z <= -2e+42)
		tmp = t_1;
	elseif (z <= 1.38e-61)
		tmp = fma(a, t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2e+42], t$95$1, If[LessEqual[z, 1.38e-61], N[(a * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000009e42 or 1.37999999999999992e-61 < z

   1. Initial program 92.6%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.0

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

   4. Applied rewrites 50.0%

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

   if -2.00000000000000009e42 < z < 1.37999999999999992e-61

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 53.4

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

   7. Applied rewrites 53.4%

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

Alternative 9: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot z\right) \cdot a\\ \mathbf{if}\;b \leq -8.5 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* b z) a)))
   (if (<= b -8.5e+78)
     t_1
     (if (<= b -4.2e-223)
       (fma z y x)
       (if (<= b 9.6e-214) (fma a t x) (if (<= b 2e+153) (fma z y x) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * z) * a;
	double tmp;
	if (b <= -8.5e+78) {
		tmp = t_1;
	} else if (b <= -4.2e-223) {
		tmp = fma(z, y, x);
	} else if (b <= 9.6e-214) {
		tmp = fma(a, t, x);
	} else if (b <= 2e+153) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * z) * a)
	tmp = 0.0
	if (b <= -8.5e+78)
		tmp = t_1;
	elseif (b <= -4.2e-223)
		tmp = fma(z, y, x);
	elseif (b <= 9.6e-214)
		tmp = fma(a, t, x);
	elseif (b <= 2e+153)
		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), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[b, -8.5e+78], t$95$1, If[LessEqual[b, -4.2e-223], N[(z * y + x), $MachinePrecision], If[LessEqual[b, 9.6e-214], N[(a * t + x), $MachinePrecision], If[LessEqual[b, 2e+153], N[(z * y + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.50000000000000079e78 or 2e153 < b

   1. Initial program 92.6%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.0

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 26.6

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

   7. Applied rewrites 26.6%

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

   if -8.50000000000000079e78 < b < -4.19999999999999965e-223 or 9.60000000000000081e-214 < b < 2e153

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto z \cdot y + x \]

      3. lower-fma.f64 52.0

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

   7. Applied rewrites 52.0%

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

   if -4.19999999999999965e-223 < b < 9.60000000000000081e-214

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 53.4

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

   7. Applied rewrites 53.4%

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

Alternative 10: 56.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.55e+156)
   (fma z y x)
   (if (<= z 3.5e-63) (fma a t x) (fma z y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.55e+156) {
		tmp = fma(z, y, x);
	} else if (z <= 3.5e-63) {
		tmp = fma(a, t, x);
	} else {
		tmp = fma(z, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.55e+156)
		tmp = fma(z, y, x);
	elseif (z <= 3.5e-63)
		tmp = fma(a, t, x);
	else
		tmp = fma(z, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.55e+156], N[(z * y + x), $MachinePrecision], If[LessEqual[z, 3.5e-63], N[(a * t + x), $MachinePrecision], N[(z * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5500000000000001e156 or 3.50000000000000003e-63 < z

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto z \cdot y + x \]

      3. lower-fma.f64 52.0

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

   7. Applied rewrites 52.0%

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

   if -1.5500000000000001e156 < z < 3.50000000000000003e-63

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 53.4

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

   7. Applied rewrites 53.4%

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

Alternative 11: 53.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, t, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 84.2%

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

5. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 53.4

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

7. Applied rewrites 53.4%

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

Alternative 12: 40.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+74}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-22}:\\ \;\;\;\;t \cdot a\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -8.5e+74) (* 1.0 x) (if (<= x 5e-22) (* t a) (* 1.0 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -8.5e+74) {
		tmp = 1.0 * x;
	} else if (x <= 5e-22) {
		tmp = t * a;
	} else {
		tmp = 1.0 * x;
	}
	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 (x <= (-8.5d+74)) then
        tmp = 1.0d0 * x
    else if (x <= 5d-22) then
        tmp = t * a
    else
        tmp = 1.0d0 * x
    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 (x <= -8.5e+74) {
		tmp = 1.0 * x;
	} else if (x <= 5e-22) {
		tmp = t * a;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -8.5e+74:
		tmp = 1.0 * x
	elif x <= 5e-22:
		tmp = t * a
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -8.5e+74)
		tmp = Float64(1.0 * x);
	elseif (x <= 5e-22)
		tmp = Float64(t * a);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -8.5e+74)
		tmp = 1.0 * x;
	elseif (x <= 5e-22)
		tmp = t * a;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -8.5e+74], N[(1.0 * x), $MachinePrecision], If[LessEqual[x, 5e-22], N[(t * a), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.50000000000000028e74 or 4.99999999999999954e-22 < x

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 26.1%

      \[\leadsto 1 \cdot x \]

   if -8.50000000000000028e74 < x < 4.99999999999999954e-22

   1. Initial program 92.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 82.7

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \left(\frac{x}{a} + t\right) \cdot a \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{x}{a} + t\right) \cdot a \]

      3. lower-/.f64 45.9

         \[\leadsto \left(\frac{x}{a} + t\right) \cdot a \]

   7. Applied rewrites 45.9%

      \[\leadsto \left(\frac{x}{a} + t\right) \cdot a \]

   8. Taylor expanded in x around 0

      \[\leadsto t \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 29.4%

       \[\leadsto t \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 29.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ t \cdot a \end{array} \]

FPCore

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

C

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

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 = t * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

2. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. div-add-rev N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 82.7

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

4. Applied rewrites 82.7%

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

5. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\frac{x}{a} + t\right) \cdot a \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\frac{x}{a} + t\right) \cdot a \]

   3. lower-/.f64 45.9

      \[\leadsto \left(\frac{x}{a} + t\right) \cdot a \]

7. Applied rewrites 45.9%

   \[\leadsto \left(\frac{x}{a} + t\right) \cdot a \]

8. Taylor expanded in x around 0

   \[\leadsto t \cdot a \]
9. Step-by-step derivation

10. Applied rewrites 29.4%

    \[\leadsto t \cdot a \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64
  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))