Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.3% → 96.4%

Time: 4.1s

Alternatives: 12

Speedup: 0.5×

• 31.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b))))
   (if (<= t_1 INFINITY) t_1 (* (fma b z t) a))))

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(b * z + t), $MachinePrecision] * a), $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 97.7%

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

   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. 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 74.8

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

   4. Applied rewrites 74.8%

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

Alternative 2: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)}{t} + a\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.85e+138)
   (* (fma b a y) z)
   (if (<= z 9.6e-67)
     (fma a t (fma z y x))
     (* (+ (/ (fma (fma b a y) z x) t) a) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.85e+138) {
		tmp = fma(b, a, y) * z;
	} else if (z <= 9.6e-67) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = ((fma(fma(b, a, y), z, x) / t) + a) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.85e+138)
		tmp = Float64(fma(b, a, y) * z);
	elseif (z <= 9.6e-67)
		tmp = fma(a, t, fma(z, y, x));
	else
		tmp = Float64(Float64(Float64(fma(fma(b, a, y), z, x) / t) + a) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.85e+138], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 9.6e-67], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision] / t), $MachinePrecision] + a), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8499999999999999e138

   1. Initial program 81.0%

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

   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 84.6

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

   4. Applied rewrites 84.6%

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

   if -1.8499999999999999e138 < z < 9.6e-67

   1. Initial program 97.4%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x + \left(a \cdot t + y \cdot z\right)} \]
   3. 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

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

      5. *-commutative N/A

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

      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. lower-fma.f64 84.4

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

   4. Applied rewrites 84.4%

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

   if 9.6e-67 < z

   1. Initial program 88.1%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. div-add-rev N/A

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

      3. div-add N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 83.5%

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

Alternative 3: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{elif}\;z \leq 4:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.85e+138)
   (* (fma b a y) z)
   (if (<= z 4.0) (fma a t (fma z y 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 (z <= -1.85e+138) {
		tmp = fma(b, a, y) * z;
	} else if (z <= 4.0) {
		tmp = fma(a, t, fma(z, y, 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 (z <= -1.85e+138)
		tmp = Float64(fma(b, a, y) * z);
	elseif (z <= 4.0)
		tmp = fma(a, t, fma(z, y, 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[z, -1.85e+138], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 4.0], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8499999999999999e138

   1. Initial program 81.0%

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

   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 84.6

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

   4. Applied rewrites 84.6%

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

   if -1.8499999999999999e138 < z < 4

   1. Initial program 97.5%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x + \left(a \cdot t + y \cdot z\right)} \]
   3. 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

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

      5. *-commutative N/A

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

      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. lower-fma.f64 83.9

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

   4. Applied rewrites 83.9%

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

   if 4 < z

   1. Initial program 85.8%

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

   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-in 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 76.7

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

   4. Applied rewrites 76.7%

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

Alternative 4: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b a y) z)))
   (if (<= z -1.85e+138) t_1 (if (<= z 5.3e+27) (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(b, a, y) * z;
	double tmp;
	if (z <= -1.85e+138) {
		tmp = t_1;
	} else if (z <= 5.3e+27) {
		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 = Float64(fma(b, a, y) * z)
	tmp = 0.0
	if (z <= -1.85e+138)
		tmp = t_1;
	elseif (z <= 5.3e+27)
		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), $MachinePrecision]}, If[LessEqual[z, -1.85e+138], t$95$1, If[LessEqual[z, 5.3e+27], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8499999999999999e138 or 5.3000000000000004e27 < z

   1. Initial program 83.4%

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

   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 79.9

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

   4. Applied rewrites 79.9%

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

   if -1.8499999999999999e138 < z < 5.3000000000000004e27

   1. Initial program 97.4%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x + \left(a \cdot t + y \cdot z\right)} \]
   3. 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

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

      5. *-commutative N/A

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

      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. lower-fma.f64 83.6

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

   4. Applied rewrites 83.6%

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

Alternative 5: 72.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{if}\;a \leq -4.5 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b z t) a)))
   (if (<= a -4.5e-42) t_1 (if (<= a 6e-112) (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 <= -4.5e-42) {
		tmp = t_1;
	} else if (a <= 6e-112) {
		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 <= -4.5e-42)
		tmp = t_1;
	elseif (a <= 6e-112)
		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, -4.5e-42], t$95$1, If[LessEqual[a, 6e-112], N[(z * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e-42 or 6.0000000000000002e-112 < a

   1. Initial program 88.3%

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

   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 68.3

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

   4. Applied rewrites 68.3%

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

   if -4.5e-42 < a < 6.0000000000000002e-112

   1. Initial program 99.1%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   3. 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 79.1

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

   4. Applied rewrites 79.1%

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

Alternative 6: 60.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{if}\;z \leq -3 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-230}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-145}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq 0.0009:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b a y) z)))
   (if (<= z -3e+57)
     t_1
     (if (<= z -2e-230)
       (fma z y x)
       (if (<= z 1.45e-145) (* a t) (if (<= z 0.0009) (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, a, y) * z;
	double tmp;
	if (z <= -3e+57) {
		tmp = t_1;
	} else if (z <= -2e-230) {
		tmp = fma(z, y, x);
	} else if (z <= 1.45e-145) {
		tmp = a * t;
	} else if (z <= 0.0009) {
		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, a, y) * z)
	tmp = 0.0
	if (z <= -3e+57)
		tmp = t_1;
	elseif (z <= -2e-230)
		tmp = fma(z, y, x);
	elseif (z <= 1.45e-145)
		tmp = Float64(a * t);
	elseif (z <= 0.0009)
		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 * a + y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3e+57], t$95$1, If[LessEqual[z, -2e-230], N[(z * y + x), $MachinePrecision], If[LessEqual[z, 1.45e-145], N[(a * t), $MachinePrecision], If[LessEqual[z, 0.0009], N[(z * y + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3e57 or 8.9999999999999998e-4 < z

   1. Initial program 85.0%

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

   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 76.9

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

   4. Applied rewrites 76.9%

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

   if -3e57 < z < -2.00000000000000009e-230 or 1.44999999999999992e-145 < z < 8.9999999999999998e-4

   1. Initial program 98.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   3. 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 50.0

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

   4. Applied rewrites 50.0%

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

   if -2.00000000000000009e-230 < z < 1.44999999999999992e-145

   1. Initial program 99.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 42.2

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

   4. Applied rewrites 42.2%

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

Alternative 7: 58.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+213}:\\ \;\;\;\;\left(b \cdot z\right) \cdot a\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-32}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+268}:\\ \;\;\;\;\left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.9e+213)
   (* (* b z) a)
   (if (<= a -1.35e-32)
     (* a t)
     (if (<= a 7.6e+83)
       (fma z y x)
       (if (<= a 1.75e+268) (* (* a z) b) (* a t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.9e+213) {
		tmp = (b * z) * a;
	} else if (a <= -1.35e-32) {
		tmp = a * t;
	} else if (a <= 7.6e+83) {
		tmp = fma(z, y, x);
	} else if (a <= 1.75e+268) {
		tmp = (a * z) * b;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.9e+213)
		tmp = Float64(Float64(b * z) * a);
	elseif (a <= -1.35e-32)
		tmp = Float64(a * t);
	elseif (a <= 7.6e+83)
		tmp = fma(z, y, x);
	elseif (a <= 1.75e+268)
		tmp = Float64(Float64(a * z) * b);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.9e+213], N[(N[(b * z), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, -1.35e-32], N[(a * t), $MachinePrecision], If[LessEqual[a, 7.6e+83], N[(z * y + x), $MachinePrecision], If[LessEqual[a, 1.75e+268], N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision], N[(a * t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.8999999999999999e213

   1. Initial program 76.2%

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

   2. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 49.4

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

   4. Applied rewrites 49.4%

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

   if -1.8999999999999999e213 < a < -1.3499999999999999e-32 or 1.74999999999999986e268 < a

   1. Initial program 88.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 37.7

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

   4. Applied rewrites 37.7%

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

   if -1.3499999999999999e-32 < a < 7.6000000000000004e83

   1. Initial program 98.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   3. 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 71.3

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

   4. Applied rewrites 71.3%

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

   if 7.6000000000000004e83 < a < 1.74999999999999986e268

   1. Initial program 84.0%

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

   2. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 43.1

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

   4. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 44.5

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

   6. Applied rewrites 44.5%

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

Alternative 8: 58.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a \cdot z\right) \cdot b\\ \mathbf{if}\;a \leq -1.9 \cdot 10^{+213}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-32}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+268}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* a z) b)))
   (if (<= a -1.9e+213)
     t_1
     (if (<= a -1.35e-32)
       (* a t)
       (if (<= a 7.6e+83) (fma z y x) (if (<= a 1.75e+268) t_1 (* a t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * z) * b;
	double tmp;
	if (a <= -1.9e+213) {
		tmp = t_1;
	} else if (a <= -1.35e-32) {
		tmp = a * t;
	} else if (a <= 7.6e+83) {
		tmp = fma(z, y, x);
	} else if (a <= 1.75e+268) {
		tmp = t_1;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * z) * b)
	tmp = 0.0
	if (a <= -1.9e+213)
		tmp = t_1;
	elseif (a <= -1.35e-32)
		tmp = Float64(a * t);
	elseif (a <= 7.6e+83)
		tmp = fma(z, y, x);
	elseif (a <= 1.75e+268)
		tmp = t_1;
	else
		tmp = Float64(a * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * z), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[a, -1.9e+213], t$95$1, If[LessEqual[a, -1.35e-32], N[(a * t), $MachinePrecision], If[LessEqual[a, 7.6e+83], N[(z * y + x), $MachinePrecision], If[LessEqual[a, 1.75e+268], t$95$1, N[(a * t), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8999999999999999e213 or 7.6000000000000004e83 < a < 1.74999999999999986e268

   1. Initial program 81.3%

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

   2. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 45.3

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

   4. Applied rewrites 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

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

   if -1.8999999999999999e213 < a < -1.3499999999999999e-32 or 1.74999999999999986e268 < a

   1. Initial program 88.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 37.7

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

   4. Applied rewrites 37.7%

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

   if -1.3499999999999999e-32 < a < 7.6000000000000004e83

   1. Initial program 98.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   3. 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 71.3

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

   4. Applied rewrites 71.3%

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

Alternative 9: 57.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+88}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -3.2e+88) (* a t) (if (<= t 2.5e+119) (fma z y x) (* a t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -3.2e+88) {
		tmp = a * t;
	} else if (t <= 2.5e+119) {
		tmp = fma(z, y, x);
	} else {
		tmp = a * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -3.2e+88)
		tmp = Float64(a * t);
	elseif (t <= 2.5e+119)
		tmp = fma(z, y, x);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -3.2e+88], N[(a * t), $MachinePrecision], If[LessEqual[t, 2.5e+119], N[(z * y + x), $MachinePrecision], N[(a * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1999999999999999e88 or 2.5e119 < t

   1. Initial program 90.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 56.1

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

   4. Applied rewrites 56.1%

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

   if -3.1999999999999999e88 < t < 2.5e119

   1. Initial program 93.4%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   3. 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 60.3

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

   4. Applied rewrites 60.3%

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

Alternative 10: 39.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{-53}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-106}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.6e-53) (* a t) (if (<= a 1.06e-106) (* z y) (* a t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.6e-53) {
		tmp = a * t;
	} else if (a <= 1.06e-106) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.6d-53)) then
        tmp = a * t
    else if (a <= 1.06d-106) then
        tmp = z * y
    else
        tmp = a * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.6e-53) {
		tmp = a * t;
	} else if (a <= 1.06e-106) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -2.6e-53:
		tmp = a * t
	elif a <= 1.06e-106:
		tmp = z * y
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.6e-53)
		tmp = Float64(a * t);
	elseif (a <= 1.06e-106)
		tmp = Float64(z * y);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -2.6e-53)
		tmp = a * t;
	elseif (a <= 1.06e-106)
		tmp = z * y;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.6e-53], N[(a * t), $MachinePrecision], If[LessEqual[a, 1.06e-106], N[(z * y), $MachinePrecision], N[(a * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.59999999999999996e-53 or 1.06e-106 < a

   1. Initial program 88.4%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 36.8

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

   4. Applied rewrites 36.8%

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

   if -2.59999999999999996e-53 < a < 1.06e-106

   1. Initial program 99.1%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 40.8

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

   4. Applied rewrites 40.8%

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

Alternative 11: 38.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+22}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -4.1e+22) (* a t) (if (<= t 1.8e-48) x (* a t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -4.1e+22) {
		tmp = a * t;
	} else if (t <= 1.8e-48) {
		tmp = x;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-4.1d+22)) then
        tmp = a * t
    else if (t <= 1.8d-48) then
        tmp = x
    else
        tmp = a * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -4.1e+22) {
		tmp = a * t;
	} else if (t <= 1.8e-48) {
		tmp = x;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -4.1e+22:
		tmp = a * t
	elif t <= 1.8e-48:
		tmp = x
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -4.1e+22)
		tmp = Float64(a * t);
	elseif (t <= 1.8e-48)
		tmp = x;
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -4.1e+22)
		tmp = a * t;
	elseif (t <= 1.8e-48)
		tmp = x;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -4.1e+22], N[(a * t), $MachinePrecision], If[LessEqual[t, 1.8e-48], x, N[(a * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.09999999999999979e22 or 1.8000000000000001e-48 < t

   1. Initial program 91.0%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 45.4

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

   4. Applied rewrites 45.4%

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

   if -4.09999999999999979e22 < t < 1.8000000000000001e-48

   1. Initial program 93.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 33.6%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 26.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 92.3%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 26.5%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(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)))