Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.9% → 96.5%

Time: 4.0s

Alternatives: 11

Speedup: 0.5×

• 32.0% 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: 92.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 81.3

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

   5. Applied rewrites 81.3%

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

Alternative 2: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+284}:\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;z \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot a\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -5.2e+284)
   (* (* b a) z)
   (if (<= z -7.5e+29)
     (fma z y x)
     (if (<= z 0.0001) (fma a t x) (* (* z a) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -5.2e+284) {
		tmp = (b * a) * z;
	} else if (z <= -7.5e+29) {
		tmp = fma(z, y, x);
	} else if (z <= 0.0001) {
		tmp = fma(a, t, x);
	} else {
		tmp = (z * a) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -5.2e+284)
		tmp = Float64(Float64(b * a) * z);
	elseif (z <= -7.5e+29)
		tmp = fma(z, y, x);
	elseif (z <= 0.0001)
		tmp = fma(a, t, x);
	else
		tmp = Float64(Float64(z * a) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -5.2e+284], N[(N[(b * a), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, -7.5e+29], N[(z * y + x), $MachinePrecision], If[LessEqual[z, 0.0001], N[(a * t + x), $MachinePrecision], N[(N[(z * a), $MachinePrecision] * b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.1999999999999997e284

   1. Initial program 75.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.5

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

   8. Applied rewrites 87.5%

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

   if -5.1999999999999997e284 < z < -7.49999999999999945e29

   1. Initial program 93.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   4. 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

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

   5. Applied rewrites 60.3%

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

   if -7.49999999999999945e29 < z < 1.00000000000000005e-4

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if 1.00000000000000005e-4 < z

   1. Initial program 83.8%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{a \cdot \left(b \cdot z\right)} \]
   4. 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 49.8

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

   5. Applied rewrites 49.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. 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 51.8

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

   7. Applied rewrites 51.8%

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

Alternative 3: 63.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+284}:\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+29} \lor \neg \left(z \leq 31000\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -5.2e+284)
   (* (* b a) z)
   (if (or (<= z -7.5e+29) (not (<= z 31000.0))) (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 (z <= -5.2e+284) {
		tmp = (b * a) * z;
	} else if ((z <= -7.5e+29) || !(z <= 31000.0)) {
		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 (z <= -5.2e+284)
		tmp = Float64(Float64(b * a) * z);
	elseif ((z <= -7.5e+29) || !(z <= 31000.0))
		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[z, -5.2e+284], N[(N[(b * a), $MachinePrecision] * z), $MachinePrecision], If[Or[LessEqual[z, -7.5e+29], N[Not[LessEqual[z, 31000.0]], $MachinePrecision]], N[(z * y + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.1999999999999997e284

   1. Initial program 75.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.5

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

   8. Applied rewrites 87.5%

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

   if -5.1999999999999997e284 < z < -7.49999999999999945e29 or 31000 < z

   1. Initial program 88.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   4. 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 57.0

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

   5. Applied rewrites 57.0%

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

   if -7.49999999999999945e29 < z < 31000

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+284}:\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+29} \lor \neg \left(z \leq 31000\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+137} \lor \neg \left(z \leq 7.3 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.1e+137) (not (<= z 7.3e-6)))
   (fma (fma b a y) z x)
   (fma a t (fma z y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.1e+137) || !(z <= 7.3e-6)) {
		tmp = fma(fma(b, a, y), z, x);
	} else {
		tmp = fma(a, t, fma(z, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.1e+137) || !(z <= 7.3e-6))
		tmp = fma(fma(b, a, y), z, x);
	else
		tmp = fma(a, t, fma(z, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.1e+137], N[Not[LessEqual[z, 7.3e-6]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z + x), $MachinePrecision], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000008e137 or 7.30000000000000041e-6 < z

   1. Initial program 84.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if -1.10000000000000008e137 < z < 7.30000000000000041e-6

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x + \left(a \cdot t + y \cdot z\right)} \]
   4. 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. lift-*.f64 N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 85.5

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

   5. Applied rewrites 85.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+137} \lor \neg \left(z \leq 7.3 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, a, y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+132} \lor \neg \left(a \leq 9.5 \cdot 10^{+92}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -1.45e+132) (not (<= a 9.5e+92)))
   (* (fma b z t) a)
   (fma a t (fma z y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.45e+132) || !(a <= 9.5e+92)) {
		tmp = fma(b, z, t) * a;
	} else {
		tmp = fma(a, t, fma(z, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.45e+132) || !(a <= 9.5e+92))
		tmp = Float64(fma(b, z, t) * a);
	else
		tmp = fma(a, t, fma(z, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -1.45e+132], N[Not[LessEqual[a, 9.5e+92]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a), $MachinePrecision], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4499999999999999e132 or 9.4999999999999995e92 < a

   1. Initial program 85.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(t + b \cdot z\right)} \]
   4. 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 87.7

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

   5. Applied rewrites 87.7%

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

   if -1.4499999999999999e132 < a < 9.4999999999999995e92

   1. Initial program 96.1%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x + \left(a \cdot t + y \cdot z\right)} \]
   4. 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. lift-*.f64 N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 83.6

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

   5. 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. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+132} \lor \neg \left(a \leq 9.5 \cdot 10^{+92}\right):\\ \;\;\;\;\mathsf{fma}\left(b, z, t\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+37} \lor \neg \left(z \leq 1.85 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.15e+37) (not (<= z 1.85e-5))) (* (fma b a y) z) (fma a t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.15e+37) || !(z <= 1.85e-5)) {
		tmp = fma(b, a, y) * z;
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.15e+37) || !(z <= 1.85e-5))
		tmp = Float64(fma(b, a, y) * z);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.15e+37], N[Not[LessEqual[z, 1.85e-5]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15000000000000001e37 or 1.84999999999999991e-5 < z

   1. Initial program 87.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      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 77.2

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

   5. Applied rewrites 77.2%

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

   if -1.15000000000000001e37 < z < 1.84999999999999991e-5

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 76.6

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

   5. Applied rewrites 76.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+37} \lor \neg \left(z \leq 1.85 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+29}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-187}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;z \leq 7.9 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -7.6e+29)
   (* z y)
   (if (<= z -1.15e-187) (* a t) (if (<= z 7.9e-57) x (* z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.6e+29) {
		tmp = z * y;
	} else if (z <= -1.15e-187) {
		tmp = a * t;
	} else if (z <= 7.9e-57) {
		tmp = x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-7.6d+29)) then
        tmp = z * y
    else if (z <= (-1.15d-187)) then
        tmp = a * t
    else if (z <= 7.9d-57) then
        tmp = x
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.6e+29) {
		tmp = z * y;
	} else if (z <= -1.15e-187) {
		tmp = a * t;
	} else if (z <= 7.9e-57) {
		tmp = x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -7.6e+29:
		tmp = z * y
	elif z <= -1.15e-187:
		tmp = a * t
	elif z <= 7.9e-57:
		tmp = x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -7.6e+29)
		tmp = Float64(z * y);
	elseif (z <= -1.15e-187)
		tmp = Float64(a * t);
	elseif (z <= 7.9e-57)
		tmp = x;
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -7.6e+29)
		tmp = z * y;
	elseif (z <= -1.15e-187)
		tmp = a * t;
	elseif (z <= 7.9e-57)
		tmp = x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -7.6e+29], N[(z * y), $MachinePrecision], If[LessEqual[z, -1.15e-187], N[(a * t), $MachinePrecision], If[LessEqual[z, 7.9e-57], x, N[(z * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.59999999999999942e29 or 7.8999999999999999e-57 < z

   1. Initial program 88.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 40.0

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

   5. Applied rewrites 40.0%

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

   if -7.59999999999999942e29 < z < -1.14999999999999999e-187

   1. Initial program 96.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 47.0

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

   5. Applied rewrites 47.0%

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

   if -1.14999999999999999e-187 < z < 7.8999999999999999e-57

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 49.5%

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

Alternative 8: 63.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+29} \lor \neg \left(z \leq 31000\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -7.5e+29) (not (<= z 31000.0))) (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 ((z <= -7.5e+29) || !(z <= 31000.0)) {
		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 ((z <= -7.5e+29) || !(z <= 31000.0))
		tmp = fma(z, y, x);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -7.5e+29], N[Not[LessEqual[z, 31000.0]], $MachinePrecision]], N[(z * y + x), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.49999999999999945e29 or 31000 < z

   1. Initial program 87.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + y \cdot z} \]
   4. 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 55.1

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

   5. Applied rewrites 55.1%

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

   if -7.49999999999999945e29 < z < 31000

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+29} \lor \neg \left(z \leq 31000\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+128} \lor \neg \left(z \leq 8.5 \cdot 10^{+174}\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.35e+128) (not (<= z 8.5e+174))) (* z y) (fma a t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.35e+128) || !(z <= 8.5e+174)) {
		tmp = z * y;
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.35e+128) || !(z <= 8.5e+174))
		tmp = Float64(z * y);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.35e+128], N[Not[LessEqual[z, 8.5e+174]], $MachinePrecision]], N[(z * y), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35000000000000001e128 or 8.5000000000000007e174 < z

   1. Initial program 83.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -1.35000000000000001e128 < z < 8.5000000000000007e174

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.3

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

   5. Applied rewrites 66.3%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+128} \lor \neg \left(z \leq 8.5 \cdot 10^{+174}\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.26 \cdot 10^{+39} \lor \neg \left(t \leq 3.5 \cdot 10^{+120}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.26e+39) (not (<= t 3.5e+120))) (* a t) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.26e+39) || !(t <= 3.5e+120)) {
		tmp = a * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-1.26d+39)) .or. (.not. (t <= 3.5d+120))) then
        tmp = a * t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.26e+39) || !(t <= 3.5e+120)) {
		tmp = a * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -1.26e+39) or not (t <= 3.5e+120):
		tmp = a * t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.26e+39) || !(t <= 3.5e+120))
		tmp = Float64(a * t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -1.26e+39) || ~((t <= 3.5e+120)))
		tmp = a * t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.26e+39], N[Not[LessEqual[t, 3.5e+120]], $MachinePrecision]], N[(a * t), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < -1.26000000000000001e39 or 3.50000000000000007e120 < t

   1. Initial program 86.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 55.5

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

   5. Applied rewrites 55.5%

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

   if -1.26000000000000001e39 < t < 3.50000000000000007e120

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 34.1%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.26 \cdot 10^{+39} \lor \neg \left(t \leq 3.5 \cdot 10^{+120}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.1% accurate, 30.0× 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 93.5%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 28.9%

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

Developer Target 1: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* z (+ (* b a) y)) (+ x (* t a)))))
   (if (< z -11820553527347888000.0)
     t_1
     (if (< z 4.7589743188364287e-122)
       (+ (* (+ (* b z) t) a) (+ (* z y) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * ((b * a) + y)) + (x + (t * a))
    if (z < (-11820553527347888000.0d0)) then
        tmp = t_1
    else if (z < 4.7589743188364287d-122) then
        tmp = (((b * z) + t) * a) + ((z * y) + x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z * ((b * a) + y)) + (x + (t * a));
	double tmp;
	if (z < -11820553527347888000.0) {
		tmp = t_1;
	} else if (z < 4.7589743188364287e-122) {
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z * ((b * a) + y)) + (x + (t * a))
	tmp = 0
	if z < -11820553527347888000.0:
		tmp = t_1
	elif z < 4.7589743188364287e-122:
		tmp = (((b * z) + t) * a) + ((z * y) + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z * Float64(Float64(b * a) + y)) + Float64(x + Float64(t * a)))
	tmp = 0.0
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = Float64(Float64(Float64(Float64(b * z) + t) * a) + Float64(Float64(z * y) + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z * ((b * a) + y)) + (x + (t * a));
	tmp = 0.0;
	if (z < -11820553527347888000.0)
		tmp = t_1;
	elseif (z < 4.7589743188364287e-122)
		tmp = (((b * z) + t) * a) + ((z * y) + x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * N[(N[(b * a), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(x + N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -11820553527347888000.0], t$95$1, If[Less[z, 4.7589743188364287e-122], N[(N[(N[(N[(b * z), $MachinePrecision] + t), $MachinePrecision] * a), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2025057 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -11820553527347888000) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 47589743188364287/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a))))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))