Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.4% → 96.5%

Time: 3.6s

Alternatives: 13

Speedup: 0.5×

• 32.2% 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: 92.4% 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: 96.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   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 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-in 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 75.5

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

   4. Applied rewrites 75.5%

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

Alternative 2: 83.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot z, b, x\right)\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.45e+106)
   (fma (* a z) b x)
   (if (<= b 8.6e+144) (fma a t (fma z y x)) (fma (fma b z t) a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.45e+106) {
		tmp = fma((a * z), b, x);
	} else if (b <= 8.6e+144) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = fma(fma(b, z, t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.45e+106)
		tmp = fma(Float64(a * z), b, x);
	elseif (b <= 8.6e+144)
		tmp = fma(a, t, fma(z, y, x));
	else
		tmp = fma(fma(b, z, t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.45e+106], N[(N[(a * z), $MachinePrecision] * b + x), $MachinePrecision], If[LessEqual[b, 8.6e+144], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.4500000000000001e106

   1. Initial program 92.4%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 71.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 71.0

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

      7. +-commutative 71.0

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

      8. *-commutative 71.0

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

      9. *-commutative 71.0

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

      10. +-commutative 71.0

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

   6. Applied rewrites 71.0%

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

   if -1.4500000000000001e106 < b < 8.59999999999999968e144

   1. Initial program 92.8%

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

   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 86.9

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

   4. Applied rewrites 86.9%

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

   if 8.59999999999999968e144 < b

   1. Initial program 90.7%

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

   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-in 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 84.5

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

   4. Applied rewrites 84.5%

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

Alternative 3: 82.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a \cdot z, b, x\right)\\ \mathbf{if}\;b \leq -1.45 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (* a z) b x)))
   (if (<= b -1.45e+106) t_1 (if (<= b 1.85e+145) (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((a * z), b, x);
	double tmp;
	if (b <= -1.45e+106) {
		tmp = t_1;
	} else if (b <= 1.85e+145) {
		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(Float64(a * z), b, x)
	tmp = 0.0
	if (b <= -1.45e+106)
		tmp = t_1;
	elseif (b <= 1.85e+145)
		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[(a * z), $MachinePrecision] * b + x), $MachinePrecision]}, If[LessEqual[b, -1.45e+106], t$95$1, If[LessEqual[b, 1.85e+145], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.4500000000000001e106 or 1.84999999999999997e145 < b

   1. Initial program 91.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 72.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 72.8

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

      7. +-commutative 72.8

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

      8. *-commutative 72.8

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

      9. *-commutative 72.8

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

      10. +-commutative 72.8

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

   6. Applied rewrites 72.8%

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

   if -1.4500000000000001e106 < b < 1.84999999999999997e145

   1. Initial program 92.8%

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

   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 86.8

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

   4. Applied rewrites 86.8%

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

Alternative 4: 74.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{if}\;a \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b z t) a)))
   (if (<= a -3.4e-6) t_1 (if (<= a 6.2e-34) (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 <= -3.4e-6) {
		tmp = t_1;
	} else if (a <= 6.2e-34) {
		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 <= -3.4e-6)
		tmp = t_1;
	elseif (a <= 6.2e-34)
		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, -3.4e-6], t$95$1, If[LessEqual[a, 6.2e-34], N[(z * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.40000000000000006e-6 or 6.1999999999999996e-34 < a

   1. Initial program 86.6%

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

   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 74.0

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

   4. Applied rewrites 74.0%

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

   if -3.40000000000000006e-6 < a < 6.1999999999999996e-34

   1. Initial program 99.0%

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

   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 74.9

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

   4. Applied rewrites 74.9%

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

Alternative 5: 73.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (fma b a y) z)))
   (if (<= z -2.15e-41) t_1 (if (<= z 2.8e-82) (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 <= -2.15e-41) {
		tmp = t_1;
	} else if (z <= 2.8e-82) {
		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 <= -2.15e-41)
		tmp = t_1;
	elseif (z <= 2.8e-82)
		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, -2.15e-41], t$95$1, If[LessEqual[z, 2.8e-82], N[(a * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1499999999999999e-41 or 2.80000000000000024e-82 < z

   1. Initial program 88.2%

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

   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 70.1

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

   4. Applied rewrites 70.1%

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

   if -2.1499999999999999e-41 < z < 2.80000000000000024e-82

   1. Initial program 99.0%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 79.6

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

   4. Applied rewrites 79.6%

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

Alternative 6: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot a\right) \cdot b\\ \mathbf{if}\;a \leq -1.45 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;a \leq 0.038:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* z a) b)))
   (if (<= a -1.45e+227)
     t_1
     (if (<= a -0.3)
       (fma a t x)
       (if (<= a 0.038) (fma z y x) (if (<= a 3.3e+118) (fma a t x) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * a) * b;
	double tmp;
	if (a <= -1.45e+227) {
		tmp = t_1;
	} else if (a <= -0.3) {
		tmp = fma(a, t, x);
	} else if (a <= 0.038) {
		tmp = fma(z, y, x);
	} else if (a <= 3.3e+118) {
		tmp = fma(a, t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * a) * b)
	tmp = 0.0
	if (a <= -1.45e+227)
		tmp = t_1;
	elseif (a <= -0.3)
		tmp = fma(a, t, x);
	elseif (a <= 0.038)
		tmp = fma(z, y, x);
	elseif (a <= 3.3e+118)
		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[(z * a), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[a, -1.45e+227], t$95$1, If[LessEqual[a, -0.3], N[(a * t + x), $MachinePrecision], If[LessEqual[a, 0.038], N[(z * y + x), $MachinePrecision], If[LessEqual[a, 3.3e+118], N[(a * t + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.4499999999999999e227 or 3.3e118 < a

   1. Initial program 80.8%

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

   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 47.5

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\left(b \cdot z\right) \cdot a} \]
   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. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 47.2

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

   6. Applied rewrites 47.2%

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

   if -1.4499999999999999e227 < a < -0.299999999999999989 or 0.0379999999999999991 < a < 3.3e118

   1. Initial program 89.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 53.1

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

   4. Applied rewrites 53.1%

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

   if -0.299999999999999989 < a < 0.0379999999999999991

   1. Initial program 99.0%

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

   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 73.4

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

   4. Applied rewrites 73.4%

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

Alternative 7: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.42 \cdot 10^{+227}:\\ \;\;\;\;\left(b \cdot z\right) \cdot a\\ \mathbf{elif}\;a \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;a \leq 0.038:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.42e+227)
   (* (* b z) a)
   (if (<= a -0.3)
     (fma a t x)
     (if (<= a 0.038)
       (fma z y x)
       (if (<= a 6e+108) (fma a t x) (* (* b a) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.42e+227) {
		tmp = (b * z) * a;
	} else if (a <= -0.3) {
		tmp = fma(a, t, x);
	} else if (a <= 0.038) {
		tmp = fma(z, y, x);
	} else if (a <= 6e+108) {
		tmp = fma(a, t, x);
	} else {
		tmp = (b * a) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.42e+227)
		tmp = Float64(Float64(b * z) * a);
	elseif (a <= -0.3)
		tmp = fma(a, t, x);
	elseif (a <= 0.038)
		tmp = fma(z, y, x);
	elseif (a <= 6e+108)
		tmp = fma(a, t, x);
	else
		tmp = Float64(Float64(b * a) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.42e+227], N[(N[(b * z), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, -0.3], N[(a * t + x), $MachinePrecision], If[LessEqual[a, 0.038], N[(z * y + x), $MachinePrecision], If[LessEqual[a, 6e+108], N[(a * t + x), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.41999999999999992e227

   1. Initial program 78.4%

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

   2. Taylor expanded in b around 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 49.3

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

   4. Applied rewrites 49.3%

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

   if -1.41999999999999992e227 < a < -0.299999999999999989 or 0.0379999999999999991 < a < 5.99999999999999968e108

   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 z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 53.2

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

   4. Applied rewrites 53.2%

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

   if -0.299999999999999989 < a < 0.0379999999999999991

   1. Initial program 99.0%

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

   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 73.4

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

   4. Applied rewrites 73.4%

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

   if 5.99999999999999968e108 < a

   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 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 48.8

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

   4. Applied rewrites 48.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 33.5

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

   7. Applied rewrites 33.5%

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

Alternative 8: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot a\right) \cdot z\\ \mathbf{if}\;a \leq -1.45 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;a \leq 0.038:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* b a) z)))
   (if (<= a -1.45e+227)
     t_1
     (if (<= a -0.3)
       (fma a t x)
       (if (<= a 0.038) (fma z y x) (if (<= a 6e+108) (fma a t x) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * a) * z;
	double tmp;
	if (a <= -1.45e+227) {
		tmp = t_1;
	} else if (a <= -0.3) {
		tmp = fma(a, t, x);
	} else if (a <= 0.038) {
		tmp = fma(z, y, x);
	} else if (a <= 6e+108) {
		tmp = fma(a, t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * a) * z)
	tmp = 0.0
	if (a <= -1.45e+227)
		tmp = t_1;
	elseif (a <= -0.3)
		tmp = fma(a, t, x);
	elseif (a <= 0.038)
		tmp = fma(z, y, x);
	elseif (a <= 6e+108)
		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), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[a, -1.45e+227], t$95$1, If[LessEqual[a, -0.3], N[(a * t + x), $MachinePrecision], If[LessEqual[a, 0.038], N[(z * y + x), $MachinePrecision], If[LessEqual[a, 6e+108], N[(a * t + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.4499999999999999e227 or 5.99999999999999968e108 < a

   1. Initial program 80.8%

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

   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 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 46.0

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

   7. Applied rewrites 46.0%

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

   if -1.4499999999999999e227 < a < -0.299999999999999989 or 0.0379999999999999991 < a < 5.99999999999999968e108

   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 z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 53.2

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

   4. Applied rewrites 53.2%

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

   if -0.299999999999999989 < a < 0.0379999999999999991

   1. Initial program 99.0%

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

   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 73.4

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

   4. Applied rewrites 73.4%

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

Alternative 9: 61.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -4.6e+17)
   (fma a t x)
   (if (<= t 1.15e+149) (fma z y x) (fma a t x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -4.6e+17) {
		tmp = fma(a, t, x);
	} else if (t <= 1.15e+149) {
		tmp = fma(z, y, x);
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -4.6e+17)
		tmp = fma(a, t, x);
	elseif (t <= 1.15e+149)
		tmp = fma(z, y, x);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -4.6e+17], N[(a * t + x), $MachinePrecision], If[LessEqual[t, 1.15e+149], N[(z * y + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.6e17 or 1.1499999999999999e149 < t

   1. Initial program 90.3%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 67.9

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

   4. Applied rewrites 67.9%

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

   if -4.6e17 < t < 1.1499999999999999e149

   1. Initial program 93.7%

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

   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 59.5

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

   4. Applied rewrites 59.5%

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

Alternative 10: 58.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+187}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -7.5e+187) (* z y) (if (<= z 6e+187) (fma a t x) (* z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.5e+187) {
		tmp = z * y;
	} else if (z <= 6e+187) {
		tmp = fma(a, t, x);
	} else {
		tmp = z * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -7.5e+187)
		tmp = Float64(z * y);
	elseif (z <= 6e+187)
		tmp = fma(a, t, x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -7.5e+187], N[(z * y), $MachinePrecision], If[LessEqual[z, 6e+187], N[(a * t + x), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5000000000000002e187 or 5.9999999999999998e187 < z

   1. Initial program 78.5%

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

   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 48.2

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

   4. Applied rewrites 48.2%

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

   if -7.5000000000000002e187 < z < 5.9999999999999998e187

   1. Initial program 95.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 60.8

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

   4. Applied rewrites 60.8%

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

Alternative 11: 38.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+17}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-125}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+149}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -5.2e+17)
   (* a t)
   (if (<= t 2.05e-125) x (if (<= t 1.15e+149) (* z y) (* a t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -5.2e+17) {
		tmp = a * t;
	} else if (t <= 2.05e-125) {
		tmp = x;
	} else if (t <= 1.15e+149) {
		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 (t <= (-5.2d+17)) then
        tmp = a * t
    else if (t <= 2.05d-125) then
        tmp = x
    else if (t <= 1.15d+149) 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 (t <= -5.2e+17) {
		tmp = a * t;
	} else if (t <= 2.05e-125) {
		tmp = x;
	} else if (t <= 1.15e+149) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -5.2e+17:
		tmp = a * t
	elif t <= 2.05e-125:
		tmp = x
	elif t <= 1.15e+149:
		tmp = z * y
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -5.2e+17)
		tmp = Float64(a * t);
	elseif (t <= 2.05e-125)
		tmp = x;
	elseif (t <= 1.15e+149)
		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 (t <= -5.2e+17)
		tmp = a * t;
	elseif (t <= 2.05e-125)
		tmp = x;
	elseif (t <= 1.15e+149)
		tmp = z * y;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -5.2e+17], N[(a * t), $MachinePrecision], If[LessEqual[t, 2.05e-125], x, If[LessEqual[t, 1.15e+149], N[(z * y), $MachinePrecision], N[(a * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.2e17 or 1.1499999999999999e149 < t

   1. Initial program 90.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 51.2

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

   4. Applied rewrites 51.2%

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

   if -5.2e17 < t < 2.0499999999999999e-125

   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 33.1%

      \[\leadsto \color{blue}{x} \]

   if 2.0499999999999999e-125 < t < 1.1499999999999999e149

   1. Initial program 93.6%

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

   2. Taylor expanded in y around 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 28.8

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

   4. Applied rewrites 28.8%

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

Alternative 12: 38.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{+17}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+149}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -5.2e+17) (* a t) (if (<= t 1.15e+149) x (* a t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -5.2e+17) {
		tmp = a * t;
	} else if (t <= 1.15e+149) {
		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 <= (-5.2d+17)) then
        tmp = a * t
    else if (t <= 1.15d+149) 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 <= -5.2e+17) {
		tmp = a * t;
	} else if (t <= 1.15e+149) {
		tmp = x;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -5.2e+17:
		tmp = a * t
	elif t <= 1.15e+149:
		tmp = x
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -5.2e+17)
		tmp = Float64(a * t);
	elseif (t <= 1.15e+149)
		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 <= -5.2e+17)
		tmp = a * t;
	elseif (t <= 1.15e+149)
		tmp = x;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -5.2e+17], N[(a * t), $MachinePrecision], If[LessEqual[t, 1.15e+149], x, N[(a * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.2e17 or 1.1499999999999999e149 < t

   1. Initial program 90.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-*.f64 51.2

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

   4. Applied rewrites 51.2%

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

   if -5.2e17 < t < 1.1499999999999999e149

   1. Initial program 93.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 31.1%

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

Alternative 13: 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 92.4%

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

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 2025117 
(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)))